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An Explanation of Computation Theory for Lawyers
Wednesday, November 11 2009 @ 07:48 PM EST

If I had to describe the fairly universal geek reaction to the oral argument at the US Supreme Court on Monday in In Re Bilski, I would have to say it's a worry that some of the participants didn't seem to understand computers or the tech behind software very well.

Groklaw member PolR has written an explanation for lawyers of computation theory to try to fill a gap in their knowledge that he has observed from reading legal briefs.

A brief extract:

Lawyers and judges know about modern electronics in general and computers in particular. They know about code and the compilation process that turns source code into binary executable code. Computation theory is something different. It is an area of mathematics that overlaps with philosophy. Computation theory provides the mathematical foundations that make it possible to build computers and write programs. Without this knowledge several of the fundamental principles of computer science are simply off the radar and never taken into consideration.

The fundamentals of computation theory are not obvious. A group of the greatest mathematicians of the twentieth century needed decades to figure them out. If this information is not communicated to lawyers and judges they have no chance to understand what is going on and mistakes are sure to happen. All the errors I have found result from this omission. The purpose of this text is to help fill the gap. I try to explain all the key concepts in a language most everyone will understand. I will underline the key legal questions they raise or help answer as they are encountered. I will conclude the article with examples of common mistakes and how the knowledge of computation theory helps avoid them.

Yes. We might as well give it a try. After all, we've spent a lot of time and effort explaining the legal process to geeks; now it's time for the geeks to help the lawyers out with the tech. They actually do want to get it right, you know. You'll find links to the main cases cited at the end, along with some helpful resources.

Feel free, then, those of you who are also programmers and engineers, to amplify, correct, explain, or if there is another topic you feel lawyers and judges need to understand, email me about it, and let's see if we can help them out.

Similarly, if you are a lawyer or judge, feel free to ask questions or ask that further information be provided.

What I get from the paper is that there isn't really a difference between what a human does with a paper and pencil and what a computer does, except speed, and that neither method of computation should be patentable subject matter. Also, it's now clear that just as you wouldn't go into litigation without a lawyer, if you are wise, because that is their area of expertise, lawyers also shouldn't assume they understand software and patents and how they relate unless they get help from technology experts who know how software and computers, and the underlying math, really work.

Update: If you'd like to print the article without the introduction, I've placed a copy here. And here's a PDF.

And for those of you who are interested in digging a bit more into computational theory, with an emphasis on the theory as opposed to the consequences with regard to patents, here's a reference for you, A Problem Course in Mathematical Logic, by Stefan Bilaniuk. Here's a description of what you'll find:

Parts I and II, Propositional Logic and First-Order Logic respectively, cover the basics of these topics through the Soundness, Completeness, and Compactness Theorems, plus a little on applications of the Compactness Theorem. [...]

Part III, Computability, covers the basics of computability using Turing machines and recursive functions; it could be used as the basis of a one-term course.

Part IV, Incompleteness, is concerned with proving the Gödel Incompleteness Theorems.


An Explanation of Computation Theory for Lawyers

By PolR
[This article is licensed under
a Creative Commons License.]

I am a computer professional with over 25 years of experience. I have a Master's degree in computer science and no legal expertise beyond what I acquired reading Groklaw. Yet I have developed an interest in understanding US patent law and how it applies to software. I have read a few court cases and legal briefs. Based on my professional knowledge I can say that in some precedent-setting cases the legal understanding of computers and software is not technologically correct.

I see the situation like this. The authors of legal briefs and court rulings have enough of an understanding to feel confident they can write meaningful arguments on the topic. But yet they do not understand computers and software well enough to reach technically correct conclusions. The unfortunate result is legal precedents that do not connect with reality. Computers don't work the way some legal documents and court precedents say they do1.

I think I can identify the root cause of this. It seems that nobody has ever taught the legal community the basic concepts of computation theory. At the very least I don't find any evidence that computation theory is known in the cases and legal briefs I have read so far. Lawyers and judges know about modern electronics in general and computers in particular. They know about code and the compilation process that turns source code into binary executable code. Computation theory is something different. It is an area of mathematics that overlaps with philosophy2. Computation theory provides the mathematical foundations that make it possible to build computers and write programs. Without this knowledge several of the fundamental principles of computer science are simply off the radar and never taken into consideration.

The fundamentals of computation theory are not obvious. A group of the greatest mathematicians of the twentieth century needed decades to figure them out. If this information is not communicated to lawyers and judges they have no chance to understand what is going on and mistakes are sure to happen. All the errors I have found result from this omission.

The purpose of this text is to help fill the gap. I try to explain all the key concepts in a language most everyone will understand. I will underline the key legal questions they raise or help answer as they are encountered. I will conclude the article with examples of common mistakes and how the knowledge of computation theory helps avoid them.

Let's illustrate the importance of computation theory with one of the legal issues computation theory will help resolve. Consider the following list of statements.

  • All software is data.
  • All software is discovered and not invented.
  • All software is abstract.
  • All software is mathematics.

If my understanding of the US patent law is correct, whether or not any of these statements is true could determine whether or not software is patentable subject matter. The resolution of this issue has serious consequences to the computer electronics and software industries.

When you know computation theory, you know without a shred of a doubt that each of these statements states a fact that is grounded in well-established mathematics. If you don't know computation theory, these statements will probably look to you like debatable issues.

Effective methods

Historically, modern computation theory started in the 1920s and 1930s with research on "effective methods". In a totally untypical manner, the definition of this mathematical concept is written in plain English as opposed to some mathematical notation that would be impenetrable to laymen. Here it is3:

A method, or procedure, M, for achieving some desired result is called 'effective' or 'mechanical' just in case

  1. M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols);

  2. M will, if carried out without error, always produce the desired result in a finite number of steps;

  3. M can (in practice or in principle) be carried out by a human being unaided by any machinery save paper and pencil;

  4. M demands no insight or ingenuity on the part of the human being carrying it out.

The terms 'effective' or 'mechanical' are terms of art in mathematics and philosophy. They are not used in their ordinary, everyday meaning. The phrase "effective method" is a term of art. This term has nothing to do with the legal meaning of "effective" and "method". The fact that these two words also have a meaning in patent law is a coincidence.

An effective method is a procedure to perform a computation that is entirely defined by its rules. The human that carries out the computation must follow the rules strictly; otherwise the answer he gets is not the one that is intended. Examples of effective methods were taught to us in grade school when we learned how to perform ordinary arithmetic operations like additions, subtractions, multiplications and divisions. We were taught procedures that are to be carried out with pencil and paper. You have to perform these operations exactly how they were taught, otherwise the result of the calculation is wrong.

This definition was written in the early part of the 20th century, before the invention of the electronic computer. This was before the word "algorithm" came to be used in computer science. Effective methods are precursors to computer algorithms. The definition works from the assumption that you need a human to do a computation4. Clause 3 explicitly mentions that the computation must be one that it is possible for a human to carry out with pencil and paper. Aid from machinery is explicitly forbidden. In effect this definition spells out what kind of human activity would qualify as an algorithm, but without using that terminology. The term "effective method" was used instead.

Of course this concept has since been superseded with the modern concept of computer algorithm. Mathematical discoveries from the 1930s have shown the human requirement may be dropped from the definition without changing the mathematical meaning. Effective methods and computer algorithms are the same thing, except that computers are allowed to execute algorithms without running afoul of the definition. This is known as the Church-Turing thesis. This is one of the most important results of computation theory and theoretical computer science. A big part of this text will be devoted at explaining exactly what this thesis is, how mathematicians discovered it and what is the evidence we have of its truth.

There are a few fine points in this definition that deserve further explanations. Clause number 3 states that an effective method can be carried out "in practice or in principle" by a human being. This language is there because mathematics is about abstract concepts. You don't want an effective method to stop being an effective method when a human runs out of pencils or paper. You don't want the method to stop being an effective method when a human dies of old age before being done with the calculation. This is the sort of thing that may happen if, for example, you try to calculate a ridiculously large number of decimals of π5. By stating the method may be carried out "in principle" the definition makes abstraction of these practical limitations.

Clause number 4 states that no demands are made on insight or ingenuity. This clause implies that all the decisions that are required to perform the calculation are spelled out explicitly by the rules. The human must make no judgment call and use no skill beyond his ability to apply the rules with pencil and paper. The implication is that the human is used as a biological computing machine and serves as a reference for defining what a computation is.

Effective methods and the Church-Turing thesis are relevant to an issue that has legal implications. What is the relationship between an algorithm executed by a computer and a human working with pencil and paper? Abstract methods made of mental steps are not patentable subject matter6. When is software such a method? When is it a physical process? With knowledge of computation theory and the Church-Turing thesis, lawyers and courts will be able to tap on the discoveries of mathematics to answer these questions with greater accuracy.

Formal Systems

Why were mathematicians so interested in effective methods in the 1920s? They wanted to solve a problem that was nagging them at the time. Sometimes someone discovered a theorem and published a proof. Then some time passes, months, years or decades, and then someone discovered another theorem that contradicted the first and published a proof. When this happens, mathematicians are left scratching their heads. Both theorems cannot be true. There must be a mistake somewhere. If the proof of one of the theorems is wrong then the problem is solved. They revoke the theorem with a faulty proof and move on. But what if both proofs are sound and stand scrutiny?

These kinds of discoveries are called paradoxes. They are indicative that there is something rotten in the foundations that define how you prove theorems. Mathematical proofs are the only tool that can uncover mathematical truth. If you never know when the opposite theorems will be proven, you can't trust your proofs. If you can't trust your proofs, you can't trust mathematics. This is not acceptable. The solution is to look at the foundations, find the error and correct it so paradoxes don't occur any more. In the early twentieth century, mathematicians were struggling with pretty nasty paradoxes. They felt the need to investigate and fix the foundations they were working on7.

The relationship between algorithms and mathematical proofs is an important part of computation theory. It is part of the evidence that software is abstract and software is mathematics. The efforts to fix the foundations of mathematics in the early twentieth century are an important part of this story.

One of the most prominent mathematicians of the time, David Hilbert, proposed a program that, if implemented, would solve the problem of paradoxes. He argued that mathematical proofs should use formal methods of manipulating the mathematical text, an approach known as formalism. He proposed to base mathematics on formal systems that are made of three components.

  1. A synthetic language with a defined syntax
  2. An explicit list of logical inference rules
  3. An explicit list of axioms

Let's review all three components one by one to see how they work. Mathematics uses special symbols to write propositions like a+b=c or E=mc2. There are rules on how you use these symbols. You can't write garbage like +=%5/ and expect it to make sense. The symbols together with the rules make a synthetic language. The rules define the syntax of the language. Computer programmers use this kind of synthetic language every day when they program the computer. The idea of such a special language to express what can't be easily expressed in English did not originate with computer programmers. Mathematicians thought of it first8.

How do you test if your formula complies with the syntax? Hilbert required that you use an effective method that verifies that the rules are obeyed. If you can't use such a method to test your syntax then the language is unfit to be used as a component of a formal system9.

Inference rules are how you make deductions10. A deduction is a sequence of propositions written in the language of mathematics that logically follows. At each step in the sequence there must be a rule that tells you why you are allowed to get there given the propositions that were previously deduced. For example suppose you have a proof of A and separately you have a proof of B. Then the logical deduction is you can put A and B together to make "A and B". This example is so obvious that it goes without saying. In a formal system you are not allowed to let obvious things be untold. All the rules must be spelled out before you even start making deductions. You can't use anything that has not been spelled out no matter how obvious it is. This list of rules is the second component of a formal system.

It was known since the works of philosopher Gottlob Frege, Bertrand Russell, and their successors that all logical inference rules can be expressed as syntactic manipulations. For example, take the previous example where we turn separate proofs of A and B into a proof of "A and B" together. The inference consists of taking A, taking B, put an "and" in the middle and we have "A and B". You don't need to know what A and B stand for to do this. You don't need to know how they are proved. You just manipulate the text according to the rule. All the inferences that are logically valid can be expressed by rules that work in this manner11. This opens an interesting possibility. You don't use a human judgment call to determine if a mathematical proof is valid. You check the syntax and verify that all inferences are made according to the rules that have been listed12. This check is made by applying an effective method13. You may find a computerized language for writing and verifying proofs in this manner at the Metamath Home Page.

If there is no human judgment call in verifying proofs and syntax, where is human judgment to be found? It is when you find a practical application to the mathematical language. The mathematical symbols can be read and interpreted according to their meanings. You don't need to understand the meaning to verify the proof is carried out according to the rules of logic, but you need to understand the meaning to make some real-world use of the knowledge you have so gained. For example when you prove a theorem about geometry, you don't need to know what the geometric language means to verify the correctness of the proof, but you will need to understand what your theorem means when you use it to survey the land.

Another place where human judgment is required is in the choice of the axioms. Any intuitive element that is required in mathematics must be expressed as an axiom. Each axiom must be written using the mathematical language. The rules of logic say you can always quote an axiom without having to prove it. This is because the axioms are supposed to be the embodiment of human intuition. They are the starting point of deductions. If some axiom doesn't correspond to some intuitive truth, it has no business in the formal system. If some intuitive truth is not captured by any axiom, you need to add another axiom14.

The list of axioms are the third component of a formal system. Together the syntax, the rules of inferences and the axioms include everything you need to write mathematical proofs. You start with some axioms and elaborate the inferences until you reach the desired conclusion. Once you have proven a theorem, you add it in the list of propositions you can quote without proof. The assumption is that the proof of the theorem is included by reference to your proof. And because all the rules and axioms have been explicitly specified from the start and meticulously followed, there is no dark corner in your logic where something unexpected can hide and eventually spring out to undermine your proof.

To recapitulate, effective methods play a big role in this program. They are used (1) to verify the correctness of the syntax of the mathematical language and (2) to verify that the mathematical proofs comply with the rules of inferences. The implication is that there is a tie between formal methods and abstract mathematical thinking. If you consider the Church-Turing thesis, there is a further implication of a tie between computer algorithms and abstract mathematical thinking. This article will elaborate on the nature of this tie.

Let's return to Hilbert's program. When you make explicit all part of the mathematical reasoning in a formal system, the very concept of mathematical proof is amenable to mathematical analysis. You can write proof about how proofs are written and analyze what properties various formal systems have15. Hilbert intended to use this property of formal systems to bring the paradoxes to a permanent end. Hilbert's program was to find a formal system suitable to be the foundations of all of mathematics that would meet these requirements:

  1. The system must be consistent. This means there must be no proposition that could be proven both true and false. Consistency ensures there is no paradoxes.
  2. The system must be complete. This means every proposition that could possibly be written in the system could be either proven true or proven false. Completeness ensures that every mathematical question that could be formulated with mathematical language could be solved within the confine of the system.
  3. The system must be decidable. This means that there must be an effective method to find at least one actual proof of a proposition when it is true or at least one actual refutation of the proposition when it is false16. Decidability ensures that when confronted with a mathematical problem we always know how to solve it.

If the formal system is amenable to mathematical analysis then, according to Hilbert, it would be possible to find a mathematical proof that the system meet the requirements of consistency, completeness and decidability17. Completing such a program would be a formidable intellectual accomplishment. All of mathematics would be contained in a system based on axioms, inference rules and effective methods. Every mathematical question would be solved by carrying out a decision method where human judgment plays no part. In this approach, the role of human intuition is limited to the selection of the axioms.

From Effective Methods to Computable Functions

Hilbert's program was successful in part and failed in part. Useful formal systems that appear to be free of paradoxes were developed18. This is the successful part. The failed part is there is no formal system that can be used as the foundation of the entirety of mathematics and is also consistent, complete and decidable. Such formal systems are not possible. Theorems that show that these properties are mutually incompatible have been found. But before the mathematicians reached this conclusion, substantial research was done. It is in the course of this research that the fundamental concepts of computation theory were developed. Ultimately these findings have led to the development of the modern programmable electronic computer19.

When working on Hilbert's program, mathematicians were struggling with another nagging issue. Hilbert's program called for every mathematical concept to be defined using a special mathematical language. How do we write the definition of effective method in such a language? The reference to a human being calculating with pencil and paper is not easily amenable to this kind of treatment. This is why the original definition is written in plain English. Some specific effective methods could be defined mathematically. Effective methods in general were something a mathematician would know when he sees one but couldn't be defined with mathematical formulas.

The work on Hilbert's program is what led to the discovery of how to define mathematically the concept of effective methods. Several different but equivalent definitions were found. To reflect the definitions being used, effective methods are called "computable functions" when we refer to the mathematical definitions. When effective methods are considered for use with a computer instead of being carried out by a human, they are called "computer algorithms". When translated in a programming language for use on an actual computer, they are called "computer programs".

These multiple definitions of computable functions led to very different methods of how to actually make the calculations. A human being that would use one of these system would write on paper something very different from someone using the other system. But still they would all do the same calculations. The reason is that algorithms are abstractions different from the specific steps of how the calculation is done20.

For example, consider a sorting algorithm written in a programming language like C. You give the program a list of names in random order and the program prints them sorted in alphabetic order. The program may be compiled for Intel x86, for Sun SPARC, for ARM or any other make and brand of CPUs. The binary instructions generated by the compiler for one CPU will be very different from the instructions for the other CPUs because this is how CPUs are made. If you trace the hardware instructions that are actually executed with a debugger, you will verify that the execution of the code is different from CPU to CPU because of the difference in the instructions. But still it is the same C program that executes. The names will be printed out in sorted order on all CPUs. The abstract sorting algorithm is the same on all CPUs.

This difference between specific instructions and abstract algorithm has its basis in computation theory. By the time you will be done reading this article you will know what that basis is.

There are three of the definitions of computable functions that stand as the most important ones. The Church-Turing thesis states that computable functions as defined by these definitions are the same functions as those one may compute with effective methods.

  1. Kurt Gödel's recursive functions
  2. Alonzo Church's lambda-calculus
  3. Alan Turing's Turing machines

One needs to know all three definitions to understand computation theory. All three definitions played a role in the discovery that Hilbert's program is impossible to accomplish. All three definitions bring considerable insights as to what an algorithm is. All three definitions are equivalent to both effective methods and computer algorithms. Together they explain why the work of a human with pencil and paper and the work of a computer manipulating bits follow the exact same abstract processes.

Now let's look each of these three definitions one by one and flesh out the most legally relevant concepts of computation theory.

Kurt Gödel's Recursive Functions

In mathematics the word "function" is a term of art. A function is a correspondence between a pair of mathematical objects that is such that the second object is determined by the first. For example the area of a circle is a function. When you know the circle, you can find out the area. Addition is a function. You know a pair of values21 and you add them up to get the result. Often functions are associated with a method or formula that state how to calculate the value of the function.

This terminology issue being settled, let's go back to the main topic. Kurt Gödel wasn't looking to define computable functions. This happened almost by accident. He was trying to do something very different and a few years later other mathematicians found out the implications of what he did.

Kurt Gödel was working on the Hilbert program. He was working on a formal system called "Peano arithmetic" that defines the arithmetic of positive integers. This is the numbers 0, 1, 2, 3 .... to infinity. The basic principle of Peano arithmetic is to base the entirety of arithmetic on the ability to count. This is a foundation made of two basic components. The first one is the number zero (0). The other one is the successor function that increases a number by one. The successor of 0 is 1, the successor of 1 is 2 etc. But the language of Peano arithmetic is too rudimentary to write numbers this way. It can only write the successor by appending an apostrophe. Zero is written 0, but 1 is written 0' and two is written 0'' and you continue like this appending one apostrophe every time you increment by one. You can identify a number by counting the apostrophes that are appended to 0.

Why are mathematicians bothering to do this? It is because this is research on the foundations. The goal is to find out what are the most minimal concepts that could serve as the basis of arithmetic. Do we need addition? Do we need subtraction? Do we need Arabic numbers? Or can we do everything with zero and successors alone? If the foundational concepts are kept very minimal, it reduces the risks that some complexity hides paradoxes that will pop up after a decade or two of research. It also makes proofs about the formal system simpler.

Remember the Hilbert program. There is a requirement to analyze the formal system to prove consistency, completeness and decidability. If the formal system is complex, these proofs will be hard to write. The hope is to make these proofs easier by keeping the formal system simple.

Gödel was studying a method of defining arithmetic functions called recursion. When you write a recursive definition22 you must specify two rules.

  1. You define the value of the function for the number zero
  2. Assuming you know the value of the function for an arbitrary number n, you define how you compute the function for n'.

With these two rules you know how to compute the value of the function for all integers. Rule 1 tells you the value of the function for 0. Then you use rule 2 to compute the value of the function for 1 given that you know the value for 0. Then you proceed to compute the value for 2 and then 3 etc. As an example here is how you define addition using recursion.

  1. m + 0 = m
  2. m + n' = (m + n)'

For the sake of example assume you want to compute 5+3. Starting with 5=0''''' and 3=0''' the steps go as follow:

5 + 3 = 0''''' + 0''' Translation into the Peano arithmetic language
0''''' + 0''' = (0''''' + 0'')' by rule no 2
(0''''' + 0'')' = (0''''' + 0')'' by rule no 2
(0''''' + 0')'' = (0''''' + 0)''' by rule no 2
(0''''' + 0)''' = (0''''')''' by rule no1
(0''''')''' = 0'''''''' because the parentheses can be removed at this point
0'''''''' = 8 Translate back into Arabic numbers

This illustrates how recursion works as an effective method. Every time you define something using recursion, you actually define an effective method to perform the calculation.

This example also shows why no one will want to actually make additions in this manner. The process is way too cumbersome. The goal is not to provide an alternative to the usual methods of doing arithmetic. The goal is to provide a foundation that is sound and free of paradoxes.

The purpose of the recursive definition of addition is to provide a reference. Other methods to add numbers are allowed but only if we can prove a theorem that proves these methods will yield the same answer as the recursive definition. Without the theorem we have no guarantee that the alternative calculation will yield the correct answer. The point of the theorem is to provide the guarantee that the procedures we use to calculate additions rest on the foundations.

Formal systems work like an edifice. You have a small number of concepts and associated axioms and rules of inference that provide a foundation. But the foundation is not used directly because it is too minimal to be usable in a practical situation. You must build theorems and theorems over theorems until you have a body of knowledge that is usable. Part of the task required by the Hilbert program was to build the edifice to show the proposed foundations are suitable for the intended purpose. This is what Gödel's concept of recursion is contributing to.

Gödel Numbers

When working on recursion, Gödel got a revolutionary insight. He noticed something useful about prime numbers.

I suppose you remember prime numbers from your high school arithmetic. But if you don't let's recapitulate. A prime number is a number greater than 1 that can be divided evenly only by 1 or itself. There is an infinite sequence of prime numbers: 2, 3, 5, 7, 11, 13 ... etc. Every number can be factored into the product of prime numbers. Here are a few examples.

  • 12 = 2231

  • 1125 = 203253

When writing a prime number factorization it is customary to omit the multiplication symbols. They are implicit. 2231 actually means 22 x 31.

Look at the exponents. Sometimes you need to multiply the same prime number multiple times to get the result. 2231 actually means 2 x 2 x 3. An exponent of 1 means you multiply one times. An exponent of 0 means you don't need to multiply the prime number. 203253 actually means 3 x 3 x 5 x 5 x 5.

What if you assign numbers to an alphabet? For instance suppose you assign a=1, b=2 until z=26? Then you can use prime numbers to encode text into numbers. You can turn "Pamela" into something like 2p3a5m7e11l13a. Replace the letters with their corresponding numbers and the text becomes a number. You can also do the reverse. When you want to read the text you factor the number into primes and then turn the exponents back into letters. This works because there is only one way to factor a number into prime numbers; therefore a number can only translate into one text. There can be no ambiguity.

Numbers that are used to encode text in this manner are called Gödel numbers.

Ordinary text editing operations people can do with pencil and paper can also be defined using recursion. For example you can use arithmetic to put 2g3r5o7k and 2l3a5w together to make 2g3r5o7k11l13a17w. This corresponds to the use of pencil and paper to put "grok" and "law" together to make "groklaw". Arithmetic operations that manipulate the exponents of prime numbers are the mirror images of the symbolic manipulations on text.

This is time to go back to formal systems. Put this idea together with what we have just said on prime numbers. We get that formal systems themselves have a mirror image in arithmetic. This is the case for Peano arithmetic. There the logic goes:

  • All the symbols can be translated into numbers.
  • The effective methods that test when a mathematical proposition made of symbol is syntactically correct can be translated into a recursive function that makes the same test on exponents of prime numbers.
  • The effective method that tests when a proof is made according to the inference rules can also be translated into a recursive function that makes the same test on exponents on prime numbers.
  • All the axioms can be translated into exponents on prime numbers by translating their symbols.

Then the entire formal system has been translated into arithmetic23. This is called the "arithmetization of syntax". It is one of the greatest mathematical discoveries of the twentieth century. It has many far reaching consequences24, some of them are germane to patent law.

Symbols and Reducibility

Gödel numbers are evidence that the symbols of the mathematical language need not be visual patterns written on paper. They can also be something abstract like the exponents of prime numbers. Or if we take a more modern look, they can be 0s and 1s stored as electromagnetic signals in electronic devices. In all cases the symbols have the same meaning and when we use them to write mathematical statements they allow to write the same logical inferences. This observation raise more issues. What do we mean, "have the same meaning"? What do we mean "allow to write the same logical inferences"? This asks for a human intelligence to look at the symbols and make such determination. This is in apparent conflict with the principle of a formal system where the validity of mathematical proofs is verified without using human judgment calls.

Gödel's work brings a solution to this apparent conflict. Syntactic translations between the different representations of the formal system can be defined mathematically. Do you remember when Hilbert pointed out that formal systems may be subject to mathematical analysis? Here we are. Syntactic translations are such mathematical analysis. Once you have shown mathematically that there is a translation you can conclude that whatever the meaning is in the original it will be preserved in the translation. This is called reducibility25. We say syntax is reduced to arithmetic when a translation from the syntax to arithmetic is available.

Reducibility is a concept that should be important to patent law. Abstract ideas are not patentable subject matter. In order to prevent abstract ideas from being indirectly patented, there are exceptions in the law that will not allow patenting an "invention" made of printed text based on the content of the text. The implication of Gödel's discovery is that the symbols that represent an abstract idea need not be printed text. The symbols can be something abstract like exponents of prime numbers. They can be something physical like electromagnetic signals or photonic signals. Symbols can take a variety of forms26. Does the law exempt abstract ideas from patenting only when they are written in text or does it exempt all physical representations? In the latter case what is the test that can tell a physical representation of an abstract idea from a patentable physical invention? Mathematicians know of such a test and it is reducibility.


Gödel numbers are also used to define computable enumerability. This is a term of art for a mathematical concept of computation theory. Imagine a list of all numbers that have a specific properties. For example these lists are computable enumerable:

  • The even numbers: 0, 2, 4, 6 ...
  • The odd numbers: 1, 3, 5 ,7 ...
  • The prime numbers: 2, 3, 5, 7, 11 ...

Computable enumerability occurs when you can test with a recursive function whether the number has the property or not. You build your list by running all the numbers 0, 1, 2 in sequence and run them against the test. The first number that passes the test is the first on the list, The next number that passes the test is the next one etc.

Thanks to Gödel numbers, it is possible to define an arithmetical test for the syntax of the definition of a recursive functions. Then we can enumerate all the possible definitions. More specifically each recursive function corresponds to the Gödel number of its definition. What we enumerate is the Gödel numbers and this is the same thing as enumerating the definitions themselves because the Gödel numbers can be translated back into the corresponding text.

Now consider the Church-Turing thesis that computer programs are the same thing as recursive functions. This is not obvious now, but by the time we are done with this article all the mathematical evidence will have been presented to you. Would this be of consequence to the patentability of software? Developing a computer program that fulfills a specific purpose is mathematically the same thing as finding a number that has some specific properties. Can you patent the discovery of a number?

Another way to look at it is that there is a list of all the possible programs. We can make one by listing all possible programs in alphabetic order27. Any program that one may write is sitting in this list waiting to be written. In such a case, could a program be considered an invention in the sense of patent law? Or is it a discovery because the program is a preexisting mathematical abstraction that is merely discovered?

This idea that programs are enumerable numbers is supported by the hardware architecture of computers. Programs are bits in memory. Strings of 0s and 1s are numbers written in binary notations. Every program reads as a number when the bits are read in this manner. Programming a computer amounts to discovering a number that suits the programmer's purpose.

Alonzo Church's Lambda-Calculus

Lambda-calculus is the second definition of effective method that uses mathematical language, recursive functions being the first. I follow the plan of explaining each of these definitions one by one and now is the turn of lambda-calculus.

It is the brainchild of Alonzo Church. Like Gödel he was doing research in the foundations of mathematics but he used a very different angle. Instead of working on the arithmetic of positive integers he was trying to use the abstract concept of mathematical function as his foundation.

Functions occur in all areas of mathematics: arithmetic, geometry, boolean algebra, set theory etc. Church was concerned with the properties of functions that belonged to all types of functions independently of the branch of mathematic. When Peano arithmetic uses zero and the successor function as the most elementary concept, lambda-calculus gives this role to the operations of abstraction and application28.

What is abstraction? Do you remember in high school how shocking it was when the teacher showed you for the first time an algebraic formula that mixed up letters and numbers? How could you add a letter and a number like in x+7? For most students this concept is hard to grasp. People can understand 3+7 or 9+7 but x+7? What kind of number would be the result of such addition? The point that was hard to learn is abstraction. The point of writing x+7 is not to calculate an answer. The point is to define a pattern, the act of adding 7 to something. The pattern must be applicable to any number whatever it is. This is the purpose of using a letter. The formula doesn't want to refer to a specific number. The letter marks the place where a number belongs but this number is unknown and must not be specified. This finding of a pattern is abstraction.

Application is the reverse of abstraction. It is what you do when you know that x is the number 5 and that at last you can turn x+7 into 5+7.

To put it simply, abstractions are when you make functions by stating how they are calculated. Application is the actual use of the rule of the function.

The task Church set to himself is to develop a formal system based on abstraction and application. These two operations require the exercise of human intelligence. A formal system requires rules that can be followed without the use of human ingenuity. How do you reconcile the two? Church found a solution that involved breaking down abstraction and application into more elementary steps.

He handled abstraction with a syntactic trick. He required that the variable being abstracted is explicitly identified with the Greek letter λ. The expression x+7 would have to be written λx.x+7. Then you no longer need to use intelligence to know that it is an abstraction and that x is the marker for what is being abstracted. This information is written in the formula.

Application is expressed by juxtaposition. If you decide that x is 5 you would write (λx.x+7)5. Parentheses are used to group symbols when how they read may otherwise be ambiguous. This example means the abstraction x+7 is applied to the value 5. At this point the formal system would want you to look at the variable after the λ, in this example it is x, and you would replace it with the expression it is applied to, 5 in this example. Then (λx.x+7)5 is transformed into 5+7. This operation is called beta-reduction29. This is a fancy name Church has cooked up to mean what a modern word processor would call a search and replace operation.

This technique allows multiple levels of abstraction. For example consider the concept of addition. It can be expressed as an abstraction of both arguments λyλx.x+y. An expression like this would capture the very concept of addition independently of the values being added. Also each value can be applied separately. For example (λyλx.x+y)7 becomes after beta reduction λx.x+7. Functions like λyλx.x+y that produce another abstraction are called high order functions. A complete computation would require to apply beta reduction repeatedly until no abstractions are left. For example (λyλx.x+y)(7)(5) would turn into 5+7 after two beta reductions.

The combination of the λ notation and the beta reduction operation allowed Church to define his lambda-calculus. That is a formal system that captures the properties of abstraction and application. The repeated use of beta reduction until no abstraction is left is an effective method that is used to perform all computations in this system.

There is a twist. My examples used arithmetic operations because every one understands elementary arithmetic. This helps make the concepts clear. But the goal of Church was to study functions independently of any branch of mathematics. He did not include arithmetic operations in lambda-calculus30. Surprisingly this doesn't matter. Abstraction and application alone are sufficient to let you count. And then you can use recursion to reconstruct arithmetic from this basis.

Here is how you may represent numbers in lambda-calculus.

  1. λsλx.x
  2. λsλx.s(x)
  3. λsλx.s(s(x))
  4. λsλx.s(s(s(x)))

Remember Peano arithmetic when we identified the numbers by counting the apostrophes? Now we count the occurrences of s. The writing system is different but the idea is the same.

If you find yourself scratching your head at stuff like λsλx.s(x) wondering what it actually means this is normal. You need to be skilled at lambda calculus to make sense out of this kind of formula. I am not giving you a course in lambda calculus. There is no way you can learn how to read this language from the explanations I am giving you.

The point I am making is there is a translation from the concepts of 0 and successor numbers to other concepts that belong to lambda-calculus. The result of the translation is less important than the fact that there is a translation at all. The reason is that if there is a translation then mathematicians may invoke the principle of reducibility. The implication is that any computation31 that can be expressed in the language of Peano arithmetic can also be expressed with the language of lambda-calculus.

The existence of a translation is why mathematicians consider that recursive functions and lambda-calculus are equivalent. All computations that can be done by means of recursion can be translated into an equivalent computation made with lambda calculus. And conversely the language of lambda-calculus can be transformed into arithmetic by means of Gödel numbers. There is no computation that one of the languages can do that the other can't do.

There is a fundamental difference between Gödel numbers and this translation. Gödel numbers are translating the symbols of the formal language one by one. Therefore you have a mirror image of the text of the language hidden in the exponents of prime numbers. The translation into lambda-calculus is different. It translates not the symbols but the numbers themselves into functions. You have a mirror image of the numbers hidden into an enumeration of functions.

This technique of translation is called modeling. The mirror image in lambda-calculus of Peano arithmetic is called a model.

Imagine a geometric pattern like a circle. It remains a circle whether it is a wheel, a plate or a jar cover. You can see the pattern is the same by comparing the shapes. But how do you define "the same shape" mathematically? You don't eyeball the object and call them the same. You verify that all of these objects have a boundary where every point is at the same distance from the center. This is the mathematical definition of a circle and everything that meets the definition is a circle.

Now imaging the whole formal system is an abstract pattern made of relationships between formulas. The axioms and inference rules of the formal system are the equivalent of the definition of the circle. Everything that obeys these same rules is matching the definition of the pattern. The translation is proof that the model obeys the rules of Peano arithmetic. Therefore the abstract pattern that is characteristic of the arithmetic of positive integers is found within lambda-calculus32.

Models work a bit like a mock up. In the olden times, engineers designing a car would build a mock up before building a real car. The examination of the mock up helped them finding faults in their design before incurring the expense of making the real thing. The model of arithmetic is a mathematical mock up except that unlike the car mock up, the model has the full expressive power of the original.

The Church-Turing Thesis

Alonzo Church was so impressed by the power of Gödel numbers that he argued every effective method would be reproducible by some recursive function based on a Gödel number encoding of the effective method. This idea implies that recursive functions, lambda-calculus and effective methods are equivalent concepts.

This is known as the Church Thesis. It didn't convince everyone when it was first formulated. A more convincing formulation had to wait until the discovery of Turing machines. Nowadays a revised version of the Church Thesis that adds Turing machines to this list of equivalent concepts is accepted by mathematicians under the name of the Church-Turing thesis.

A Universal Algorithm

In Peano arithmetic each recursive function makes a different effective method. The rules to carry out the computation are provided by the definition. In lambda-calculus the situation is different. There is only one effective method that is used for all computations. This effective method is to keep doing beta reduction until there are no applications left that could be reduced.

Since lambda-calculus has the capability to perform every possible computation (in the extensional sense) then one algorithm is all one needs to perform every possible computation33. The difference between one computation and another is in the data that is supplied to the algorithm. This is one of the reasons we argue that software is data.

Lambda-calculus can be and has been implemented as a programming language. The language LISP is a prominent example.

This raises an interesting possibility. Suppose I am sued for infringement on a software patent. I may try defending myself arguing I did not implement the method that is covered by the patent. I implemented the lambda-calculus algorithm which is not covered by the patent. And even if it did, lambda-calculus is old. There is prior art. What happens to software patents then?

Anyone that argues in favor of method patents on software must be aware that the current state of technology permits this possibility.


Consider two of these idealized humans that are required to carry out effective methods. One of them is carrying out, say, a multiplication using Peano arithmetic. The other does the same multiplication using lambda-calculus. If we would compare their writings on paper, they will be different. The formal languages are not the same. the symbols are not the same and one follows the rules of recursion directly while the other follows them indirectly through repeated beta reduction. How can it be said the two computations are the same?

This is a very important question because it speaks to what constitutes "is the same" from a mathematician's perspective. For example when one argues software is mathematics, the argument is that the computation done by the computer is the same thing as a mathematical computation, therefore this is mathematics that is being done by the machine. Such argument can be understood only when one knows the mathematical meaning of "being the same" computation.

This question can be answered two different ways. There is the "intensional" definition and the "extensional" definition34.

The intensional definition considers the details of how the computation is done. If you change the details of the computation, from the intensional perspective you get a different computation even though the two computations always produce the same answer. By this perspective recursive functions and lambda-calculus are different.

The extensional definition ignores the details of the calculation and considers only the equivalence by means of reducibility. If the two calculations can be matched with a translation from one language to another, then they are the same from the extensional perspective. From the extensional perspective recursive functions and lambda-calculus are equivalent because every calculation made in one system is matched by a calculation made in the other system..

The law has a similar35 concept when it makes a difference between ideas and expressions of ideas. For example consider I write a program. Some outside party has written a similar program. This party sees my program and thinks I infringe on his proprietary rights. There are two scenarios depending on whether he makes his claim according to copyright law or patent law.

For claims of copyright infringement the text of the source code matters. If I wrote my program independently and I can prove the texts are different, I don't infringe on his rights. This is an intensional point of view.

For claims of patent infringement then differences in the text of the source code won't matter. It won't even matter if the code is written in a different programming language. If my program uses the same method that is covered by the patent according to whatever legal test of "same method" is applicable, I will infringe. This is an extensional perspective.

Which of the two perspectives is correct? It depends on the degree of abstraction that is wanted. The intensional perspective is the less abstract. It is the perspective you need when you study the properties of the written expressions of the symbols. It may be used to answer questions like how many steps are required to produce an answer or how much space in memory is required to store the symbols.

The extensional perspective is more abstract. It is the one you need when you are concerned with the expressive power of the language. It answers questions like: whether you can say the same things in both languages or how to tell when two expressions in different languages have the same meaning.

The intensional vs extensional dichotomy explains the situation of the C sorting program that is the same when run on different hardware architectures. When you compile the code, you produce different instructions depending on which hardware architecture is targeted. These difference are significant only when you look at the code from the intensional perspective. When you take the extensional perspective, all executables produce the same output. Therefore they are the same. The consequence is that software is abstract. The details of the instructions don't matter36.

Denotational Semantics

C is a programming language, not a mathematical language. How do we define its meaning mathematically? This is done using a technique called denotational semantics37.

For example consider the statement i=i+1 that could be written in many programming languages such as C. It looks like an equation but it is not an equation. Here the letter i is called a variable. Imagine a box with a label. The box contains information. The variable is the label for such a box. The statement i=i+1 means open the box labeled i, get the number that is in the box, add 1 to it and put it back in the same box. If there is no number in the box your program has a bug and it will fail. The question is what is the mathematical meaning of this operation.

Suppose you run your program in your favorite debugger38. You can stop the execution of the program just before the i=i+1 statement. Then you can use the debugger to inspect the memory of the computer. What do you see? There are many variables in use. Each of them is a labeled box containing some information. You may have a variable called "FirstName", another called "LastName", another one called "Salary" and one named "HiringDate". The memory is assigning each of these names a value. This is a mathematical function. It doesn't happen to be one you calculate because the values are explicitly stored and not computed with a formula or anything. It is a mathematical function nonetheless.

Now use the debugger to execute the i=i+1 statement and stop the execution again. What do you see in the memory? None of the variables have seen their values changed except i which has been incremented by one. The memory is a new mathematical function that is identical to the previous one except for the change in the value of i. This is the meaning of the statement. When you consider the content of the memory as a function, the statement is a high order function that changes this function into another function.

This is how denotational semantics works. Lambda-calculus is good at handling high order functions. You build a mathematical model of the memory content of the computer using lambda-calculus. Then each statement in the language can be translated into a high order function that manipulates this model. The result is a definition of the mathematical meaning of the program.

This programming statement is discussed in a brief to the Supreme Court of the United States by Professor Hollaar and IEEE-USA. When arguing that software is not mathematics, the brief states39:

But in most instances, the correspondence between computer programs and mathematics is merely cosmetic. For example, the equation E = MC2 expresses a relationship between energy and matter first noted by Einstein, while the computer program statement E = M * C ** 2 represents the calculation of M time C raised to the second power and then assigning the result to a storage location named E. It is unfortunate for purposes here that the early developers of programming language made their calculation-and-assignment statements look like mathematical equations so that they would seem familiar to scientists and engineers. But the common programming statement I = I + 1, which increments the value stored in location I, is essentially nonsense as a mathematical equation. Similarly, a computer program is a series of calculation-and-assignment statements that are processed sequentially, not a set of simultaneous mathematical equations that are solved for their variables.

This statement from the brief is incorrect. The correspondence between a statement in a high level language such as C and mathematics is not defined by syntactic similarity. It is defined by denotational semantic. There is another correspondence that involves machine instructions. This correspondence will be discussed when we reach the topic of Turing Machines.

Programming Directly from Mathematical Languages

If programming languages have a mathematical meaning what stops programmers from writing their programs using a mathematical language in the first place? The answer is it can be done. Languages such as LISP in the functional family of languages implement lambda-calculus and can be used to write programs using a mathematically defined notation. Programmers prefer languages in the imperative family40 of languages because the resulting programs is closer to the hardware architecture of the CPU and gives them more control on how the algorithm will execute. This situation may change. The trend is to increase the number of CPUs that are put on a chip. In presence of multiple CPUs imperative language becomes more difficult to use. The more CPUs, the harder it gets write a program that runs well on the computer. Functional languages are more amenable to work well in this kind of environment.

There is another approach. One may take advantage of the relationship between algorithms and logical proofs. Consider this parable.

This is 1963. It is the heart of the Cold War. The head of CIA rushes into the office of his main scientific advisor.

CIA Head: We have received a report from James Bond. The Soviets have developed a new formula that makes our strategic defense obsolete. It will cost us billions to upgrade. But the report says it could be disinformation, that the new theory may be something phony to make us spend billions uselessly. We cannot afford to guess. We need to know. Can you use a computer to find out if the formula is true of false?

Chief Scientist: Sure I do. I can write a program that prints "True" whatever the input is and another one that prints "False" whatever the input is. One of the two programs is bound to provide the correct answer. Therefore I can use a computer to print the solution you seek.

There are two major schools of mathematicians. The classical school would say the answer of the chief scientist is logically correct although inappropriate. One of the two programs does indeed print the correct answer and the chief scientist can indeed write it. Therefore the chief scientist has literally answered the question that has been asked.

The intuitionistic school will protest that the chief scientist's answer is illogical. You can't argue you can write the requested program until you can tell which program is the one you seek.

The parable has been designed to make it sounds like the intuitionistic point of view is the correct one. In this context the whole point of the question is to know which of the two programs provides the answer. I wanted to show that intuitionistic logic has merits. Such logic requires building a tie between a proof that a problem is solvable and the construction of an effective method that actually solves the problem. Their point is that if you can't actually solve the problem, how can you argue a solution exists?41

If you work out this tie using the language of lambda-calculus you get something called the Curry-Howard correspondence. It is a mathematical translation that takes a proof and turn it into an algorithm in the language of lambda-calculus or vice-versa. The implication is you can write a mathematical proof and use an automatic process to extract the built-in algorithm and compile it into an executable program. Or conversely you write a program and the compiler verifies that it is bug-free by extracting a proof of its correctness from the code or reporting an error if the proof doesn't work out. This is a topic for research. The mathematical principles are known but their application is not ready for use outside of the research laboratory yet.42 Despite this limitation some specific programs that have been verified in this manner have been developed.43

A question for patent law is what do you do when this research comes to fruition? At this point there will be no actual difference between the work of computer programmer and the work of a mathematician. How would you patent software without directly patenting mathematics?

The merging of programming and mathematical proofs has industrial applications. [See Campbell for an example.] If you can prove a program is mathematically correct you verify that its logic will perform according to the intent of the programmer. This eliminates a large quantity of bugs.44

Several formal verification methods exist outside of the Curry-Howard correspondence. One of the most popular is called Hoare logic45. It is a formal system where every statement in a programming language corresponds to a rule of inference. Programmers annotate their code with mathematical propositions and verify that the rules of inferences are obeyed. If they do, the program is proven mathematically correct.

Alan Turing's Turing Machines

Turing machines are the third of the major mathematical definitions of computable functions. They were proposed specifically to resolve an open question with the first two definitions. Alonzo Church put forth the thesis that Gödel numbers were powerful enough to reproduce every effective method that could be. But at the time the evidence was intuition. It looked like Gödel numbers have that power but how do you prove it? Alan Turing's work brought convincing evidence.

The definition of effective method requires a human being doing the work with pencil and paper. Turing proceeded from an analysis of what such a person is able to do. He eliminated all redundant possibilities until he was left with what is essential.46 Let's look at this elimination process.

Do we need to write things in columns or otherwise arrange symbols in two dimensions? This is not necessary. One can do the same work by writing everything on the same line and visualizing how it fits two-dimensionally. It is inconvenient, but since the task is to find out the essential minimum, writing things in two dimensions can be done away with. This suggests that regular sheet paper is not necessary. The paper may be in the form of a long tape where symbols are written sequentially. The human winds the tape back and forth as he works with the information.

How long the tape must be? Our idealized human who carries out the effective method never runs out of time and paper. The tape has to be infinitely long to reflect this capacity.

A human often needs to refer to information already written. How fast does he need to shuffle paper to find out what he wants? Moving the tape one symbol at the time is sufficient. If you need to go farther away you can repeat moving the tape one symbol until you get where you want to be.

When reading the information, how many symbols do you need to look at at the time? The answer is one. If you need to look at more symbols you can examine them one by one.

When faced with making a choice, on what basis would our person make his decision? Choices are made on the basis of the symbols being read and on his state of mind. He must track where he is in the execution of the effective method. This tracking is done by changing his state of mind. Then, when faced with a choice, the human will consider his state of mind and the symbol being read and make the next move as is required by the method. This implies we don't need to delve into human psychology or the biology of the brain. All we need to know is (1) that there are states of mind and (2) that the human changes his states as he tracks where he is in the execution of the effective methods.

How many states of mind are required? Only a finite number is required. They can be inventoried by analyzing the rules of the method and finding the junctures where choices are made. This means that if we write down the inventory of states of mind we may represent them with a finite number of symbols.

These considerations are summarized in the list of capabilities below. They are the minimal requirements to perform any effective method.

  • There is an infinitely long tape where symbols can be recorded sequentially.
  • The tape has a "current" position where a symbol can be read. This is the current symbol.
  • A symbol may be written at the current position overwriting any symbol already there.
  • The tape may be moved one symbol position either to the left or to the right. It may also stand still.
  • The human carrying out the method has a number of distinct possible states of mind.
  • The states of mind are finite in number and may be represented with a finite number of symbols.
  • The human changes his state of mind based on the previous state and the current symbol on the tape.

The implication is that it is possible to describe every effective method in these terms. The rules for carrying out the method are written as a list of quintuples. This is a list of line items. Each of them is made of five elements of information.

  1. The current state of mind that is required for the rule to apply.
  2. The current symbol that is required for the rule to apply.
  3. The symbol that is written on the type when the rule applies. If this is the same symbol as the one mentioned in item 2, then this is a do-nothing operation.
  4. Whether the tape moves right, left or stands still after writing the symbol if the rule applies.
  5. The state of mind one must adopt after the application of the rule.

This process is transforming the broad scope of effective methods into something that has a standardized form. You analyze the original method rules, translate them into a suitable list of quintuples, and you are all set.

Now imagine that our idealized human carrying out the method chooses to use an idealized typewriter instead of pencil and paper. The typewriter writes on an infinite tape that can be moved left or right one symbol at a time at the press of a key. The user may see the current symbol or overwrite it with a new symbol by the press of a key. There is a blackboard on the wall where the list of quintuples is written. The user uses his current state of mind and the current symbol to select the applicable quintuple, and then he performs the actions dictated by the quintuple. He repeats this process until it is over. This human is doing the same calculation as if he were carrying out the original effective method.

Now imagine you could endow the typewriter with the ability to read the symbols and track its internal state by itself. The quintuples could be mechanized, allowing the typewriter to carry out the calculation all by itself, without the help of an actual human. What you get when you do this is a Turing machine.

The Evidence for the Church-Turing Thesis

Now we have closed the loop. Effective methods are reduced to Turing machines. But by definition a Turing machine is an effective method because the human typing on the keyboard of the idealized typewriter might just as well use pencil and paper. Therefore the equivalence between effective methods and Turing machines is a two-way street.

Turing machines can be translated into recursive arithmetic with Gödel numbers. Recursive arithmetic can be translated into lambda-calculus with the method we already saw. Turing found a set of rules for a Turing machine that can do the beta reduction algorithm of lambda-calculus. The cycle is complete. All three definitions of computable functions may be reduced to each other. The argument in favor of the Church-Turing thesis is now complete.

Quite a lot of evidence of the Church-Turing Thesis has been found beyond what has been presented here. An article on the Church-Turing Thesis on the Turing Archive summarizes it as follow:

Much evidence has been amassed for the 'working hypothesis' proposed by Church and Turing in 1936. Perhaps the fullest survey is to be found in chapters 12 and 13 of Kleene (1952). In summary: (1) Every effectively calculable function that has been investigated in this respect has turned out to be computable by Turing machine. (2) All known methods or operations for obtaining new effectively calculable functions from given effectively calculable functions are paralleled by methods for constructing new Turing machines from given Turing machines. (3) All attempts to give an exact analysis of the intuitive notion of an effectively calculable function have turned out to be equivalent in the sense that each analysis offered has been proved to pick out the same class of functions, namely those that are computable by Turing machine. Because of the diversity of the various analyses, (3) is generally considered to be particularly strong evidence. Apart from the analyses already mentioned in terms of lambda-definability and recursiveness, there are analyses in terms of register machines (Shepherdson and Sturgis 1963), Post's canonical and normal systems (Post 1943, 1946), combinatory definability (Schonfinkel 1924, Curry 1929, 1930, 1932), Markov algorithms (Markov 1960), and Godel's notion of reckonability (Godel 1936, Kleene 1952).

This same article also makes a point of what the Church-Turing thesis does not say. The thesis says Turing machines are equivalent to computations made by human using effective methods. It doesn't say they are equivalent to any possible computation made by a machine. The reason is that the idealized typewriter argument doesn't discuss equivalence with other machines, and therefore we don't have a basis to reach such a conclusion.

The article provides a list of published articles where mathematicians have constructed abstract computing machines that are not equivalent to Turing machines. These calculations cannot be done by a human and are not effective methods. Nobody has constructed a physical device that implements any of these machines. It is unknown whether you can build a physical computing device that can perform a computation a Turing machine can't do.

The Working Principle of the Modern Computer

What if we design an effective method that directs the idealized human to read the quintuples of a Turing machine and do as they tell him to do? What if we make a Turing machine out of this method? You get a universal Turing machine. Here "universal" means that it can do the work of all other Turing machines.

There are two lists of quintuples here. The universal Turing machine uses a list of quintuples to define its activity like any other Turing machine. If you supply this machine with a tape where the quintuples of another machine are written, the universal machine will emulate the behavior of the other machine.

This is a major discovery in the history of computing as the Stanford Encyclopedia of Computing points out:

Turing's construction of a universal machine gives the most fundamental insight into computation: one machine can run any program whatsoever. No matter what computational tasks we may need to perform in the future, a single machine can perform them all. This is the insight that makes it feasible to build and sell computers. One computer can run any program. We don't need to buy a new computer every time we have a new problem to solve. Of course, in the age of personal computers, this fact is such a basic assumption that it may be difficult to step back and appreciate it.

This is the point where computation theory stopped being the exclusive province of mathematicians and became relevant to engineers. The question was how do you build an actual universal Turing machine?47 Today we know the answer. We use electronics. The tape corresponds to the computer memory. The table of quintuples corresponds to the CPU instruction set. The CPU itself has the ability to change states. The Turing machine you want to emulate is programming data that you load in memory for execution. When we assemble the pieces, we have a modern digital computer.

When making an actual computer, engineers had to cross the line that separates the abstract world of the mathematician from reality. The Turing machine is an abstraction suitable to provide a foundation. Do you remember what we said about foundations when discussion Peano arithmetic? They are not meant to be used directly. You need to build an edifice on top of them to obtain a usable result. The engineers had to build the edifice.

Moving a tape one symbol at the time is inefficient. You need to be able to get the information directly. You need to access memory randomly.

There are many types of storage devices with different characteristics: RAM, ROM, Flash, magnetic disks. optical disks etc. You need to mix and match them as required. Where the Turing machine needed one storage device, the tape, in real life computer need many.

The Turing machine can do only one operation: it writes one symbol on the tape based on the appropriate quintuple. This is too rudimentary for real-life computing. An actual computer needs an elaborate instruction set.

You get the idea. The Turing machines provided the starting point but engineers had to elaborate on the concept. The result is what is now known as the Von Neuman architecture.48

Why do we still think computers are Turing machines after these engineering changes? The answer depends on if you look at an intensional definition or an extensional definition. Mathematicians didn't stop working and reducibility didn't go away. They studied the mathematical effect of changes in the architecture of the Turing machine.49 What if the machine has more than one tape? What if the tapes are replaced with random access memory? The answer is these machines can be reduced to Turing machines. You don't bring into existence new computations by doing these changes. From an extensional perspective the modified machines perform the same computations as a plain Turing machine.

There is one difference that has an effect. The idealized human never runs out of time and paper. This is why the Turing machine has an infinite tape. But in the real world, there are physical limits. Memory is not infinite, and time is limited. This restricts the computations that are possible. A real-life computer is an approximation of a Turing machine. Whether or not the calculations it carries out corresponds to one performed by a Turing machine depends on whether or not this calculation will fit within the physical constraints. If it doesn't fit, the calculation will not be carried out to completion even though the Turing machine has the theoretical ability to do it. But when a computation fits within the constraints, it is always the same as one that is done by a Turing machine.

The Nature of Software

Let's recapitulate. A universal Turing machine is one that, when given the quintuples of another Turing machine, will do as instructed. This principle is implemented in an actual computer as follows:

  • The tape of a Turing machine corresponds to the memory of a computer.
  • The read/write capability of a Turing machine corresponds to the memory bus of the computer.
  • The changing states of the Turing machine corresponds to the changing states in the CPU electronics.
  • The quintuples of a universal Turing machine correspond to the CPU electronics.
  • The quintuples of the other Turing machine correspond to the computer program.

The last bullet is an important one for patent law. It says software is the information you need to give to the universal Turing machine to make it perform the desired computation. This simple fact has consequences.50

  • Software is data.

    This is because information is synonymous with data. When you look at a physical computer, software is stored in a data file like any other form of data. It is loaded in memory and stored in exactly the same chips of RAM as data. The CPU manipulates the software with the same instructions as it uses to manipulate other forms of data. The fact that software is data is the mathematical principle that makes it possible to build and sell modern digital computers.

  • Software is abstract.

    This is a consequence of how Turing machines are constructed. A computer program is equivalent to the work of an idealized human being carrying out an effective method. By definition this work is an abstraction written down on paper. Another way to look at it is that a computer program is equivalent to some Turing machine. If we apply the discoveries of Alan Turing, we never need to physically build this Turing machine. It is sufficient to find a description of it and pass it to a universal Turing machine. The only device we need to actually build is the universal Turing machine. The computer is the universal Turing machine. The software is a description of an abstract machine that is never physically implemented.

  • Software is mathematics.

    This is because all the instructions the CPU is able to execute are mathematical operations. Physical computers are implementations of universal Turing machines. Universal Turing machines are part of mathematics. Therefore, whatever they do is mathematics. Another way to look at it is software is the description of some abstract Turing machine. Since all Turing machines are part of mathematics software is mathematics

  • Software is discovered as opposed to invented.

    This is a consequence of the fact that software is abstract and software is mathematics. You can't invent abstract ideas and mathematics. You can only discover them.

What is Not Computable?

Computable functions are defined through an exercise in reducibility. Whatever is reducible to a Turing machine is computable. The flip side of the question is, what cannot be reduced to a Turing machine?

Alan Turing discussed this question in his doctoral thesis.51 He studied what he called an oracle machine, which is a Turing machine augmented with the capability to ask a question to an oracle when the computation reaches a certain point. Then the oracle answers the question and the computation resumes, taking this information into consideration.

The oracle could be anything.52 For example, imagine our idealized human typing on his idealized typewriter doing calculations about sunspots. Every now and then, he picks up the phone and asks an astronomer the percentage of the surface of the sun that is currently covered with sunspots. The astronomer looks into the telescope and provides the answer. Then our idealized human resumes typing on the idealized typewriter carrying out the rest of the computation. The astronomer is the oracle. Observing the sun is not a computation even if the result of the measurement is a number. Therefore the whole process including the astronomical observation is not something that could be done by a Turing machine alone.

A Turing oracle machine brings the concept of computations relative to the oracle. This means part of the process is a computation and part is not. It is possible to draw a line between the two by sorting out what is the oracle and what is the Turing machine.

Consider that instead of a human being typing at the typewriter, we have a computer that interfaces directly to the telescope. It takes photographs of the sun and computes the surface of the sunspots from the photos. This computer is doing mathematical calculations relative to the photographs, but taking the picture is not a mathematical operation. This kind of integration with non-computational devices is where the boundaries of computation theory lie.

This very issue has been contemplated by the Supreme Court of the United States in two celebrated cases, Parker v. Flook and Diamond v. Diehr.

In Parker v. Flook the court ruled on how mathematical algorithms should be treated by patent law:

Respondent's process is unpatentable under § 101 not because it contains a mathematical algorithm as one component, but because once that algorithm is assumed to be within the prior art, the application, considered as a whole, contains no patentable invention. Even though a phenomenon of nature or mathematical formula may be well known, an inventive application of the principle may be patented. Conversely, the discovery of such a phenomenon cannot support a patent unless there is some other inventive concept in its application.

In Diamond v. Diehr the court looked at the flip side of the situation:

In contrast, the respondents here do not seek to patent a mathematical formula. Instead, they seek patent protection for a process of curing synthetic rubber. Their process admittedly employs a well-known mathematical equation, but they do not seek to preempt the use of that equation. Rather, they seek only to foreclose from others the use of that equation in conjunction with all of the other steps in their claimed process. These include installing rubber in a press, closing the mold, constantly determining the temperature of the mold, constantly recalculating the appropriate cure time through the use of the formula and a digital computer, and automatically opening the press at the proper time. Obviously, one does not need a "computer" to cure natural or synthetic rubber, but if the computer use incorporated in the process patent significantly lessens the possibility of "overcuring" or "undercuring," the process as a whole does not thereby become unpatentable subject matter.
These two cases address this question: when is an invention an application of a mathematical algorithm as opposed to being the algorithm itself? With knowledge of computation theory, we may say the answer must have to do with the interactions with the real world. Everything that is the symbolic manipulations done by a CPU or equivalent devices is a mathematical computation. But when the device interacts with the outside world, whether or not you have a patentable invention, according to those cases, depends on the nature of this interaction. This is what we learn from Turing's oracle machine.

Why Lawyers Need to Know Computation Theory

What is a mathematical formula when implemented in digital electronics? What is an application of the formula as opposed to the formula itself? These questions have confounded lawyers for decades for good reason. These are hard questions. But they are the questions they have to answer when software patents are discussed.

For this type of question, guidance from mathematicians is appropriate. This is why patent lawyers should learn computation theory. This is especially important in software because the relevant mathematical concepts are hard. It took decades for the best mathematicians of the first half of the twentieth century to figure them out. Their findings are among the greatest intellectual achievements of their time. It is appropriate to stand on the shoulders of giants because this is not the sort of thing anyone can figure out by himself merely by pondering dictionary definitions and thinking things through. But many courts have been trying to do just that in software patents cases. For example consider this sentence from In re Alappat:

We have held that such programming creates a new machine, because a general purpose computer in effect becomes a special purpose computer once it is programmed to perform particular functions pursuant to instructions from program software.

In a single sentence the court tosses out the fundamental principle that makes it possible to build and sell digital computers. You don't need to create a new machine every time you perform a different computation; a single machine has the capability to perform all computations. This is what universal Turing machines are doing.

In the 1940s, the ENIAC used to be programmed by physically reconnecting circuits. This design was awkward. Engineers took pains to eliminate this need and designed the next generations of computers by using the ideas of universal Turing machines as guidance. The explicit goal was to make sure you don't need to make new circuitry, much less make a new machine, when you program the computer. This history is documented on the Stanford Encyclopedia of Philosophy.53 See also this article on Groklaw.

How could the court make such a holding? It is because it didn't know about computation theory. And I suspect it didn't know because no one ever put an explanation of computation theory into the court record.

Let's look at another example. This one is from Application of Walter D. BERNHART and William A. Fetter. It discusses a patent on a software algorithm to plot lines based on Cartesian coordinates. In this case, several issues where computation theory is relevant show very clearly. The first one is about determining what is a method made of mental steps:

Looking first at the apparatus claims, we see no recitation therein of mental steps, nor of any element requiring or even permitting the incorporation of human faculties in the apparatus. These claims recite, and can be infringed only by, a digital computer in a certain physical condition, i. e., electro-mechanically set or programmed to carry out the recited routine. The claims also define the invention as having plotting means for drawing lines or for illustrating an object. When such functional language is used in a claim, 35 U.S.C. § 112 states that "such claim shall be construed to cover the corresponding structure, material, or acts described in the specification and equivalents thereof." The specification here mentions only mechanical drafting machines. The claims therefore cover, under section 112, only such mechanical drafting machines and their equivalents. We know of no authority for holding that a human being, such as a draftsman, could ever be the equivalent of a machine disclosed in a patent application, and we are not prepared to so hold in this case. Accordingly, we think it clear that applicants have not defined as their invention anything in which the human mind could be used as a component.


In the case now before us, the disclosure shows only machinery for carrying out the portrayal process. In fact it is the chief object of the invention to eliminate the drudgery involved in a draftsman's making the desired portrayals. Accordingly, a statutory process is here disclosed. Looking then to method claim 13, we find that it in no way covers any mental steps but requires both a "digital computer" and a "planar plotting apparatus" to carry it out. To find that the claimed process could be done mentally would require us to hold that a human mind is a digital computer or its equivalent, and that a draftsman is a planar plotting apparatus or its equivalent. On the facts of this case we are unwilling so to hold. We conclude that the method defined by claim 13 is statutory, and its patentability must be judged in light of the prior art.

Imagine how different this passage would have been if the court knew about Turing machines and their equivalence with effective methods. Would the idealized human being typing on his idealized typewriter count as proof that the claims are made of mental steps? Or would a court hold that the very same steps are mental steps only when they are carried out by an actual human? In this case, the court clearly looked at the problem from an intensional perspective and took note of the difference between actual humans and electronic devices. But what if the extensional perspective is the correct one? This is the way mathematicians would look at it. Without knowledge of computation theory, the court has no way even to take the mathematical view into consideration.

This Bernhart case is really fascinating. It is a concentrate of some of the hardest issues where computation theory is relevant. Here is another extract that speaks to what constitutes a symbol:

Nor are the "printed matter" cases, cited by the board, supra, controlling as to these apparatus claims either on the facts or in principle. On their facts, those cases dealt with claims defining as the invention certain novel arrangements of printed lines or characters, useful and intelligible only to the human mind. Here the invention as defined by the claims requires that the information be processed not by the mind but by a machine, the computer, and that the drawing be done not by a draftsman but by a plotting machine. Those "printed matter" cases therefore have no factual relevance here.

What if the judge had factored in the equivalence of a Turing machine and effective methods? He would have understood that the information stored in memory is symbolic in nature. He would have understood that there is no difference between a mental step of a human being and the computations of the computer. He would have understood that the symbols written in computer memory have the same meaning as those written on paper. Besides, some humans can read information meant to be processed by machines. There are tools like debuggers that have been made expressly for this purpose.

Let's take this from another angle. What kind of files contain instructions that cause the computer to perform calculations? Here are a few examples. This list is by no means complete:

  • Executable binaries: these files contain instructions that are executed directly by the CPU. These files are the result of the compilation of source code. When legal cases discuss the programming of a computer, they think almost exclusively of these files. I never see an analysis of how the case reasoning would extend to other kinds of instructions.
  • Interpretable code: these files contain programs in human-readable form that may also be executed directly by the computer without an intermediate compilation step. This requires a special program called an interpreter that reads the code and executes the instruction. Several programming languages are interpreted. This court didn't consider the possibility that the patented algorithm could be implemented in such a language. Would such code infringe? But in this case it is an invention made of a particular arrangement of characters that is intelligible to the human mind. Would this change the assessment of the court regarding the factual relevance of the "printed matter" cases?
  • PDF files: these files contain instructions to be executed by a virtual machine that are specialized to the visual rendering of written documents. How do the printed matter court cases apply to patents written to algorithms made of the execution of such instructions?

The distinction between symbols readable by humans and machine-readable information is artificial. Computation theory teaches us that symbols remain symbols whether they are ink on paper or recorded in electronic form. In both cases they can be used to implement the same abstract processes.

Let's take this same idea from still another angle. Suppose a court determines that software in its executable binary form is patentable because it is turned into something physical. This is an argument we see frequently. For purpose of this discussion it doesn't matter if the court thinks the physical something is a machine, a process or whatnot. The question is what happens to the patentability of code in forms other than executable binaries?

  • The court doesn't want to patent data like a novel or music. Therefore the physical something that makes binaries patentable will not be sufficient to make data patentable.
  • The court doesn't want to patent textual data in a PDF file no matter how novel and nonobvious the text may be. Such files are made of executable instructions. Therefore it must take more than the presence of novel and non obvious instructions that are executed to make something patentable.
  • How about patenting source code in textual form? This will raise free speech issues. How can you discuss code if you can't write the text without infringing? The text of source code should not be patentable.
  • How about code written in an interpreted language? This code is never compiled into executable binaries. the only executable binary is the interpreter. This is the same code that runs for all programs. It does not implement the patented algorithm and by itself it doesn't infringe. The interpreter implements the language. The text of the source code (which is not patentable) is given to the interpreter. The interpreter reads the text and does what it is told. All data is manipulated by the interpreter on behalf of the program. On what basis should this code be patentable? The physical something that is the basis of patentability never comes into existence. The interpreter doesn't infringe. The data is not patentable. The source code is not patentable. Executing instructions doesn't make patentable something that isn't.

This analysis is meant to communicate that software is about manipulating symbols. It is not about physical circuitry or processes. Whatever perceived circuitry is used to claim software is patentable, programmers have the means to make an end-run around this alleged circuitry. It is possible to arrange the manipulation of the symbols in such a way that the circuitry that is the alleged basis of patenting software never comes into existence.

We have already encountered a theme like this when we discussed the beta reduction algorithm of lambda-calculus. One algorithm can do the work of every other algorithm. We can make sure the method patents are never infringed because the claimed methods are never physically implemented. Interpreted languages are built on this basis.

From the same Benrhart case, there is a section that is reminiscent of the "software makes a new machine" from Alappat:

There is one further rationale used by both the board and the examiner, namely, that the provision of new signals to be stored by the computer does not make it a new machine, i. e. it is structurally the same, no matter how new, useful and unobvious the result. This rationale really goes more to novelty than to statutory subject matter but it appears to be at the heart of the present controversy. To this question we say that if a machine is programmed in a certain new and unobvious way, it is physically different from the machine without that program; its memory elements are differently arranged. The fact that these physical changes are invisible to the eye should not tempt us to conclude that the machine has not been changed. If a new machine has not been invented, certainly a "new and useful improvement" of the unprogrammed machine has been, and Congress has said in 35 U.S.C. § 101 that such improvements are statutory subject matter for a patent. It may well be that the vast majority of newly programmed machines are obvious to those skilled in the art and hence unpatentable under 35 U.S.C. § 103. We are concluding here that such machines are statutory under 35 U.S.C. § 101, and that claims defining them must be judged for patentability in light of the prior art.

These physical changes are the normal operation of the computer. It would probably not have occurred to this judge to rule that turning the steering wheel to the left is an improvement to a car that is patentable subject matter and could be barred from patenting only because it is old and obvious. But this is a physical change that is applied to the car, and this change does give the car the ability to turn left. It is a useful improvement of the car based on the logic of this paragraph.

Machines have moving parts. Moving the moving parts as they are intended to be moved is not an invention. What are the moving parts of a computer? This information is in the definition of the Turing machine. The moving parts include the ability to read and write symbols in memory. More, the operating principle of the computer requires that programs are stored in memory and are modifiable by the computer as the execution of the program progresses. The changes to the memory are not an improvement of the computer. They are the operation of the computer as it is intended to be operated. The knowledge of the Turing machine enables one to make this determination.

What if we hold the other view? Suppose we agree with this court that writing something in memory is a patentable improvement of the computer? Then we get absurdities. Modern computers change the state of their memory billions of times per second. Do we have billions of improvements per second, each of them being patentable subject matter? How does one perform due diligence to avoid patent infringement then?

Looking only at memory changes that concern programming instructions doesn't help much. Such instruction is data held in memory and may be changed as fast as the speed of memory permits.

Let's take a look at some of the instances where the actions of a user may cause a new program to be loaded in memory.

  • You click on a menu item.
  • You visit a web site and Javascript is embedded in the page.
  • You visit a web site and a Flash or Java applet is downloaded.
  • Your operating system downloads a security patch.
  • You use the add/remove application menu option of your Ubuntu system to get application from the web.
  • You open a PDF file. The PDF document is a series of instructions written for execution by a virtual machine.
  • You open a text document that contains macros.
  • You open a spreadsheet that contains formulas. Solving formulas in a spreadsheet counts as an algorithm, doesn't it?
  • You use a program that generates and executes (or interprets) code on the fly. This includes many circumstances when your program accesses a relational database using SQL, because this language does not specify the algorithm in the code. The algorithm is determined dynamically by the language interpreter. Since SQL is the dominant standard for databases this situation covers most programs that use databases.

From a user perspective these activities are the normal operation of the computer. It is not reasonable to ask the users to conduct a patent search for due diligence against infringement every time they perform one of these actions. This cannot be the correct interpretation of patent law.

Software is Mathematics

The preceding section was a series of case studies explaining why lawyers should know computation theory. Now it is time to return to the questions that started this section. What is a mathematical formula when implemented in digital electronics? What is an application of the formula as opposed to the formula itself? We already know the answer. Software is mathematics. Everything that is executed by a CPU is mathematics. The boundaries of mathematics lie in the interaction of the computer with the outside world.

This idea is a hard one to understand unless you know computation theory. When you don't know computation theory, all sorts of misconceptions occur. For example patent attorney Gene Quinn has written an argument against software being mathematics. Computation theory is conspicuous by its absence in this article. He is an eloquent man. Someone who doesn't know computation theory will likely be convinced. If you bring computation theory into the picture, however, his argument falls apart.

I think this quote summarizes the core of his position:

I think the disagreement between me and my critics is largely due to the way patent law views mathematics, which seems admittedly quite different from the way that many computer programmers view mathematics.

It is impossible to argue that software code does not employ mathematical influences, because it does. I think Dijkstra's explanation that a "programmer applies mathematical techniques" is completely true and accurate. Having said this, the fact that mathematical techniques are employed does not as a matter of fact mean that software is mathematical. Under the US patent laws you cannot receive a patent that covers a mathematical equation or a law of nature. You can certainly use mathematical equations and laws of nature as the building blocks to create something that is new and nonobvious that is patentable. So even if software used mathematical equations there would be no prohibition against the patenting of software under a true and correct reading of the US patent laws.

If I paraphrase the argument, Quinn thinks there are some mathematical equations or methods on one side and the programmer uses them to write software on the other side. Then he says "Look, the maths are over there. This is not the same as the software over here." Of course when couched in these terms, software is different from mathematics. For example if someone write a program simulating chemical reactions, the laws of chemistry are mathematically defined and they are different from the software. Or similarly if one uses mathematical influences to reason about the code, the influences are not the code.

The error is that the mathematics of computation theory is not the mathematics Quinn is pointing to. You can't understand what is meant by "software is mathematics" unless you understand computation theory.

What does it take for the law to find as a matter of fact that software is mathematical? Do the opinions of mathematicians and computer scientists trained in mathematics counts? Or only lawyers?

Since Quinn mentioned Dijkstra, let's explain what are the mathematical methods he advocated.54 He wanted the programmers to use Hoare logic to formally prove the programs before they are compiled and run for the first time. As we have said, Hoare logic is a formal system wrapped around an imperative programming language. Here is what Hoare himself says in the introduction of the seminal paper where he first proposed his logic:

Computer programming is an exact science in that all the properties of a program and all the consequences of executing it in any given environment can, in principle, be found out from the text of the program itself by means of purely deductive reasoning. Deductive reasoning involves the application of valid rules of inference to sets of valid axioms. It is therefore desirable and interesting to elucidate the axioms and rules of inference which underlie our reasoning about computer programs. The exact choice of axioms will to some extent depend on the choice of programming language. For illustrative purposes, this paper is confined to a very simple language, which is effectively a subset of all current procedure-oriented languages.

When one writes programs using this method, the program is a by-product of writing a mathematical proof. Software is not "employing mathematical influences" to use the words of Quinn. The statements in the program are providing the applicable inference rules that are used to write the proof. When you are done with the proof, all that is left to do to get actual running code is insignificant post-solution activity. You take the code that has been proven and compile it.

When software is written is this manner, it is as mathematical as anything could be. But if this is not sufficient to convince a court, how about software developed using the Curry-Howard correspondence? This improves on the above methodology in that there is no difference between the text of a mathematical proof and the text of the program. You can compile the proof directly to get executable code.

I will admit not all software is developed in such manner. In fact most programmers never use such methods. Does the way software is developed count for patent infringement determination? Do we need to make a difference between software developed mathematically which is not patentable and software which is developed using a non-mathematical method which is patentable? Perhaps once it were decided that formal verification techniques bring the code into a patent-free haven, these methods would grow more popular. The industry could avoid billions of dollars in patent infringement liabilities. A side-effect would be great improvements in the reliability of software due the the drastic reduction in the number of bugs. But in such scenario what would be the point of software patents?

We don't need to consider such hypotheticals. The activity of the computer itself is mathematics no matter how the source code is written. There the argument goes:

  • Is the activity of a human carrying out an effective method mathematics? Mathematicians think so.
  • Is the activity of the same human typing on an idealized typewriter performing the same effective method mathematics? This is the same work using a different tool.
  • If we automate the idealized typewriter to work by itself without a human, is it doing mathematics? Again this is the same work being done.
  • If we implement the idealized typewriter in actual real-world electronics, is the computer doing mathematics? You get the picture here. If the human at the start of the chain is doing mathematics, so does the computer.

We may also ask, how about denotational semantics? It is a translation of the code into the language of lambda-calculus. It doesn't matter whether the programmer uses a mathematical methods or not when developing software. The source code has a mathematical meaning regardless of how the developer wrote it.

Now contrast the above with this quote from Quinn:

Returning to the matter of software generally, the logic employed by the computer programmer is certainly of a mathematic nature, but that is quite different than saying that the code created by the programmer is a mathematical algorithm. Now I am speaking in generalities here, but what I am trying to address is whether all computer software should be considered unpatentable subject matter. I see absolutely no justification for all software to be considered unpatentable subject matter because it is simply not correct to say that software code is the equivalent of a mathematical equation or a mathematical algorithm. Employing the same logical structure is certainly wise, and complies with best practice standards for programming, but at the core computer software directs. The code is a series of instructions written using mathematical logic as its foundation. In the patent arena this does not and cannot mean that the patenting of software is the equivalent of patenting mathematics. It merely means that the instructions are written in a language and format that are heavily influenced by mathematics.

This whole paragraph betrays a complete lack of knowledge of computation theory. From an extensional perspective, software is data that represents the quintuples of a Turing machine. This is a mathematical entity. Or if you prefer to take an intensional view and consider the specifics of the hardware architecture, the instruction set of the CPU is a machine of some sort based on a computational model that is different from but still reducible to a Turing machine. This machine is universal in the sense that it can perform any computation provided it is given a symbolic description of this computation. Software is this description. Both the universal machine and the description of the computation are mathematical entities. The patentable physical devices are the actual CPU as it is etched in silicon and the computer, including memory, buses, motherboard etc. When the computer computes, it is doing mathematical work exactly like a human being working with pencil and paper.

We can find the root of Quinn's misconceptions in this quote:

The influence mathematics has on programmers is largely or perhaps even solely related to work-flow. Mathematics teaches us how to manage a problem by going through the steps to solve the problem in a predictable and traceable manner. By employing the same type of thoughtful, step-by-step approach to writing code the programmer can manage write segments of code and tie everything together into an overall structure that will deliver the desired functionality.

Contrast this view with the use of Hoare logic or Curry-Howard correspondence. Contrast this view with denotational semantics. Contrast this view with the fact that the computer itself is executing an effective method. The role of mathematics in software is not limited to work flow and debugging. Without knowledge of computation theory there is no way to understand how deep the rabbit hole goes.

There is one more interesting argument in this quote:

I just cannot reconcile in my mind why employing lessons learned from mathematics would, could or ought to render software unpatentable. If you take this logic to its extreme you are left with absolutely nothing being patentable, which I suspect is the goal of some, but certainly not all, of those who would rather software not be patentable subject matter. Again, it is undeniable that mathematics is the backbone to virtually everything. Virtually everything can be explained using mathematical models. If you combine mathematics and physics there is virtually no mechanical, electrical or chemical phenomena that cannot be explained in a descriptive way. So does that mean that mechanical, electrical and chemical inventions ought not to be patentable?

Here we are looking at a two-way street. It is correct that we can describe just about everything mathematically. It is correct that such descriptions don't make anything unpatentable. But we can also use something physical to describe mathematics. In such case patenting the physical representation is patenting mathematics. We don't patent mathematical textbooks. Quinn's error is he is looking at one side of the street without considering the other. This is understandable considering the fact he is not looking at the same mathematics as we are discussing.

When we want to invoke this argument to dismiss the use of mathematics and claim the physical reality is patentable, we have an obligation. We need to make sure we correctly understand what part of mathematics is involved, what is the corresponding physical entity, and how they relate. If mathematics describes the physical entity, then the physical entity may be patentable. But what if things go the other way round? In the case of software the analysis goes like this:

  • When a human being writes something mathematical with pencil and paper, does the text describes mathematics? Or does the mathematics describe the text? The text describes mathematics of course.
  • When the human uses a typewriter to write the same mathematics, the written text still describes the maths. The maths don't describe either the text or the typewriter.
  • If we automate the typewriter to execute an effective method, the relationship between the text and the typewriter doesn't change. The text describes the maths. The maths don't describe the automated typewriter nor the text.
  • If we build an electronic device that does the same calculations as the automated typewriter but with symbols encoded electronically instead, does their meaning change? Of course not, the symbols still mean the mathematics. The maths don't describe the computer or anything physical in the computer. The maths don't describe the software either.

As an concluding remark, did you notice how I have used reducibility to track where the meaning of symbols lies? This is exactly the sort of thing mathematicians have used reducibility for. This sort of thing is exactly why mathematicians prefer an extensional view over an intensional one to track abstract concepts across different representations.



Application of Walter D. BERNHART and William A. Fetter. 417 F.2d 1395

Bilski v. Kappos, Amicaus Brief of Professor Lee A. Hollaar and IEEE-USA.

Diamond v. Diehr, 450 U.S. 175, 182 (1981)

In re Alappat, U.S. Court of Appeals Federal Circuit, July 29, 1994

Parker v. Flook 437 U.S. 584 (1978)

On the Web

[Agda] Agda: an interactive proof editor

[Campbell] MacGregor Campbell, Universal kernel code to keep computers safe, from New Scientist,

[Cayenne] Cayenne hotter than Haskell

[CLISP] GNU CLISP home page.

[Compcert] Compcert compilers you can formally trust

[Coq] The Coq proof assistant

[Dictionary] Dictionary of Algorithms and Data Structures

[Dijkstra] On the cruelty of really teaching computer science (manuscript) (html)

[Epigram] Home page of the epigram project

[Groklaw] Correcting Microsoft's Bilski Amicus Brief -- How Do Computers Really Work?

[Metamath] Metamath Home Page

[Quinn] Groklaw Response: Computer Software is Not Math

[Turing Archives] The Turing Archives for the History of Computing. This site is maintained by Jack Copeland, professor at the University of Canterbury, New Zealand

[Stanford] The Stanford Encyclopedia of Philosophy

In Print

[Davis 2000] Davis, Martin, Engines of Logic, Mathematicians and the Origin of the Computer, W.W. Norton and Company, 2000. This book was originally published under the title The Universal Computer: The Road from Leibnitz to Turing.

[Gödel 1986] Gödel, Kurt, 1986, Collected Works, vol. 1, Oxford: Oxford University Press.

[Hoare 1969] Hoare, Charles Anthony Richard, An Axiomatic Basis for Computer Programming, Communications of the ACM, October 1969 pp. 580

[Kleene 1950] Kleene, Stephen Cole, Introduction to Metamathematics, D. Van Nostrand Company, 1950

[Soare 2009] Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory, Annals of Pure and Applied Logic , to appear in volume of Proceedings of Computability in Europe, Siena, 2007 meeting. Available on-line [Link] In a private email Dr Soare informs me this has been published from Publisher Elsevier. However I don't have the actual publication title. I have used the on-line version.

[Stoy 1977] Stoy, Joseph E. Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory. The MIT Press, 1977

[Turing 1936] Turing, Alan, On Computable Number with and Application to the Entscheidungsproblem, Proceeding of the London Mathematical Society, ser. 2, vol 42 (1936), pp. 230-67. Correction: vol 43 (1937) pp. 544-546. This paper could be ordered from the publisher here [Link] This paper is available on-line here [link]

[Turing 1939] Turing, Alan, Systems of logic based on ordinals, Proceeding of the London Mathematical Society, vol 45, Part 3 (1939) pp. 161-228. This paper is available on-line here [Link]


1 I believe the rulings in Benson, Flook and Diehr by the Supreme Court of the United States. all reached conclusions that I consider to be technically correct.

2This is why the Stanford Encyclopedia of Philosophy is one of the authorities I will quote. See [Stanford]

3 From the article on the Church-Turing Thesis at the Turing Archives for the History of Computing. See also [Turing Archive] from the references.

4 Prior to the invention of the electronic digital computer, the word "computer" referred to humans doing calculations. See the Brief History of Computing at the Turing archives [Turing].

5 The decimal expansion of the number π has an infinite number of decimals. No matter how fast we calculate them, the calculation will never end. Therefore the definition of what constitutes a mathematical calculation must not be contingent on the practical limitations of a human doing the calculation.

6 This text was written before the Supreme Court ruling on Bilski v. Kappos. I don't know if the ruling will affect the accuracy of this sentence.

7 Some of this history is summarized in chapter III of [Kleene 1950]. Another account that goes way over my head in the mathematical details may be found in the article on Paradoxes and Contemporary Logic of the Stanford Encyclopedia of Philosophy. See also the article on Russell's paradox.

8 Martin Davis traces this idea back to Leibnitz (1646-1716). See [Davis 2000]

9 In this section on formal systems I dig deeper into the connection between algorithms and abstract thinking. Computation theory has very close ties with the very definition of what is a mathematical proof, and I am exploring some of that here. I am laying some of the ground work I need to explain why software is discovered and not invented, why software is abstract and why software is mathematics. T

10 Inference is a fancy word that here means deduction.

11 A complete list of these rules can be found in chapter IV section 19 of [Kleene 1950]

12 This means mathematicians can't argue their theorem with a mesmerizing tap dance and hope nobody will follow up and check it, as sometimes lawyers seem to do in litigation. Other mathematicians will never look at how plausible the argument may sound. They will check each inference one by one and make sure every single one of them follows the rules. As soon as they find a single inference that doesn't clearly follow the rules, the whole proof is ruthlessly rejected.

13 The consequence of this fact is there is no dispute among mathematicians on whether a theorem is proven or not. They check the proof and either it passes the test or it doesn't. This fact may be put to productive use by lawyers. In any issue where mathematical truth is relevant, theorems that are brought into the court record will not be disputed by anyone competent to do so.

14 This is why you have axioms for arithmetic, other axioms for geometry, yet other axioms for set theory etc. The intuitive truths at the foundation of any mathematical discipline are different. The axioms are chosen to reflect these truths.

15 This was called by Hilbert metamathematics which means mathematics about mathematics. See [Kleene 1950]

16 This requirement further expands on the role effective methods play in Hilbert's program. However this part of the program cannot be done. There are theorems that show that effective methods to solve all mathematical questions in a consistent formal system suitable to be the foundation of all of mathematics are not mathematically possible. However it is possible to develop algorithms of a more limited scope that will automatically discover the proof of some theorems. According to the article on Automated Reasoning of the Stanford Encyclopedia of Philosophy, these algorithms have reached the point where they can be used by researchers to attack open questions in mathematics and logic and to solve problems in engineering. This is further evidence of the tie between computer programs and abstract thinking.

17 There is a risk of circular reasoning in this logic. If the mathematical analysis of the formal system is itself subject to paradoxes, then the proofs envisioned by Hilbert would be unreliable and nothing would be really solved. This is why Hilbert further required that the proof of consistency and completeness must be done through finitary means. The reason is all the paradoxes that plagued mathematics during Hilbert's time were located in parts of mathematics that were dealing with infinity in one way or another. Hilbert argued the proofs of consistency, completeness and decidability should only use parts of mathematics that are considered safe from paradoxes and therefore could be relied on.

18 See chapters IV to VIII of [Kleene 1950] for the development and analysis of a formal system sufficient to form the basis of the arithmetic of integers. Most of mathematics can be based on a formal system called the Zermelo-Fraenkel set theory.

19 A very readable account of this history can be found in [Davis 2000]

20 This is where dictionary definitions of algorithm are lacking. For example the Dictionary of Algorithms and Data Structures [Dictionary] defines algorithm as "a computable set of steps to achieve a desired result." Literally the definition is correct. But it doesn't capture nuances like degrees of abstraction such as the one illustrated in the example of the C sorting program. Lawyers working from dictionary definitions without knowledge of computation theory have no chance of really understanding algorithms.

21 In mathematics a pair of mathematical objects is also a mathematical object.

22 This explanation is based on the simplest form of recursion called primitive recursion. Effective methods are equivalent to recursion in its full generality. See [Kleene 1950] for an complete explanation. See also the article on Recursive Functions in the Stanford Encyclopedia of Philosophy.

23 Gödel has provided a full translation of a formal system called Principia Mathematica in his seminal paper On formally undecidable propositions of Principia Mathematica and related systems. The original was in German and published in 1931. An English translation is available in [Gödel 1986] pp 144-195.

24 Two of these consequences are the famous Gödel undecidability theorems. They proved that Hilbert's program is impossible to fulfill because when a formal system is powerful enough to serve as the foundation of Peano arithmetic, then consistency and completeness are mutually incompatible. A system that is consistent cannot be complete and a system that is complete cannot be consistent. These theorems are outside the scope of this text.

25 Reducibility is a term of art in computation theory.

26 It can be said that the bits in computer electronics are numbers. Textual symbols are represented using numeric encodings like ASCII or Unicode. The original Gödel numbering system is not friendly to modern electronics and more convenient encodings were developed. This doesn't change the substance of the argument being made. The bits are symbols that represent something abstract.

27 We can make such a list by listing all series of one symbol A, B .. Z, then all series of two-symbols AA, AB ... ZZ and then series of three symbols, four symbols etc., and we filter out those series of symbols that are not valid programs according to the rules of syntax of the programming language. Whatever program someone writes, it must be on this list.

28 Abstraction and application are terms of art specific to lambda-calculus.

29 There is also an alpha-reduction that let you rename a variable. For example (λx.x+7) may be rewritten (λy.y+7) without changing the meaning of the abstraction.

30 Nowadays this is called "pure" lambda-calculus. There are variants of lambda-calculus that augment lambda-calculus with operators from other branches of mathematics.

31 This translation is provided in detail in chapter 5 of [Stoy 1977] among other places. Note that I said the computations can be translated. This means the recursive functions are translated, and this is the point that is of interest to computation theory. The method I am referring to doesn't have the capability to reproduce predicate calculus which is an integral part of Peano arithmetic. Therefore this part of Peano arithmetic is not included in the model.

32 For those of you who want a taste of how modeling works, I give some more explanations. A model required to reproduce all the base concepts and then demonstrate the translations obeys the same logical rules as the original.

We have seen how zero is translated. We also need a translation for the successor function; this is the apostrophe in Peano arithmetic. The translation is λzλsλx.s(zsx). Why this cryptic formula? It is because when you apply it to one of the formulas that are used to translate numbers and perform beta reduction you will obtain the translation of the next number. Let's see an example of how it works.

The formula below applies the successor to zero. After beta reduction you should get 1.


Now we need to perform beta reduction on z to resolve the application.


Within the parentheses, we have a pair of successive applications (λsλx.x)sx.. In situation like this the rules of lambda-calculus say you must perform the two beta reductions one at a time. The first substitution requires you to replace s with s in λx.x. But there is no s in there. This is just a do-nothing operation that removes what is unnecessary.


The next beta reduction requires to replace x for x. Again this is a do-nothing operation the removes more that is unnecessary.


This result is indeed -- the intended translation for the number 1.

As you can see, these kinds of models are not intended to provide an efficient method to do arithmetic. They are intended to study the expressive power of formal languages and gain understanding of how abstract concepts relate to each other.

33 There exist an infinite number of algorithms that may serve this purpose. As noted above, the native algorithm of lambda-calculus is too inefficient to be put into practical use. It can be improved into something that can be used in practice. Languages such as LISP make use of the improvements.

34 I have found this distinction in Soare [2009] p 13. This is a slightly different concept from extensionality as used in set theory.

35 I write "similar" concept and not "identical" concept. The law will consider issues like derivative works that have no equivalent in mathematics.

36 This fact contradicts the point of view of those that want to patent software on the basis that transistors and other signals are changing state in memory. This is an intensional detail that is absent from both the extensional properties of the software and the extensional test for patent infringement. If the test for infringement is extensional, then the invention subject matter must be defined extensionally.

37 For a more detailed explanation, see for example [Stoy 1977]

38 A debugger is a programming tool that lets a programmer verify the execution of a program and inspect the computer memory as it executes. Its purpose is to help find bugs and understand the error so it can be corrected.

39 See Bilski v. Kappos, amicus brief of Professor Lee A. Hollaar and IEEE-USA.

40 Imperative languages are made of statements that translate directly into instructions that can be executed by a CPU. These languages are well suited for programs meant to be executed into single CPU computers. They are popular because single CPU computers have dominated the market for so many years. Much of the understanding of programming that lawyers have developed is based on compiled imperative languages. This understanding often doesn't apply when other types of languages are considered. Languages can also be interpreted, or they can be something that is not imperative at all. It is dangerous to make case law based on compiled imperative languages without considering how the underlying logic apply to other types of languages.

41 The classical view is you are allowed to argue by contradiction. You assume a solution doesn't exist and prove a contradiction. Therefore some solution must exist somehow, even if you don't actually know what it is. Intuitionistic logic disallows this form of reasoning.

42 Examples of research in this direction are Coq, Agda, Cayenne and Epigram.

43 The Compcert compiler is an example. From the Compcert page: "The Compcert verified compiler is a compiler for a large subset of the C programming language that generates code for the PowerPC processor. The distinguishing feature of Compcert is that it has been formally verified using the Coq proof assistant: the generated assembly code is formally guaranteed to behave as prescribed by the semantics of the source C code.".

44 Namely it eliminates all bugs that are due to the actual code not matching the intent of the programmer as expressed in the proof. It doesn't eliminate the bugs due to the programmer intending to do the wrong thing, it does not eliminate the bugs due to the programmer proving something that doesn't mean what he thinks it means, and it does not eliminate problems due to the hardware not working as expected. But still this is a very large category of bugs that is gotten rid of.

45 This method has been first proposed by C.A.R. Hoare in [Hoare 1969]

46 An account of this story can be found in [Davis 2000] chapter 7. Turing's original analysis is found in [Turing 1936] section 9.

47 A summary of this history can be found in the Stanford Encyclopedia of Computing. This is also the topic of chapters 7 and 8 of [Davis 2000]. The Turing Archive brief history of computing reports the following:

Turing's computing machine of 1935 is now known simply as the universal Turing machine. Cambridge mathematician Max Newman has remarked that right from the start Turing was interested in the possibility of actually building a computing machine of the sort that he had described (Newman in interview with Christopher Evans ('The Pioneers of Computing: an Oral History of Computing, London Science Museum, 1976)).

During the Second World War, Turing was a leading cryptanalyst at the Government Code and Cypher School, Bletchley Park. Here he became familiar with Thomas Flowers' work involving large-scale high-speed electronic switching (described below). However, Turing could not turn to the project of building an electronic stored-program computing machine until the cessation of hostilities in Europe in 1945.

See also from the same article Turing's work with Colossus and ACE.

48 The Stanford Encyclopedia of Philosophy describes the contribution of John Von Neuman like this:

In 1944, John von Neumann joined the ENIAC group. He had become ‘intrigued' (Goldstine's word, [1972], p. 275) with Turing's universal machine while Turing was at Princeton University during 1936–1938. At the Moore School, von Neumann emphasised the importance of the stored-program concept for electronic computing, including the possibility of allowing the machine to modify its own program in useful ways while running (for example, in order to control loops and branching). Turing's paper of 1936 (‘On Computable Numbers, with an Application to the Entscheidungsproblem') was required reading for members of von Neumann's post-war computer project at the Institute for Advanced Study, Princeton University (letter from Julian Bigelow to Copeland, 2002; see also Copeland [2004], p. 23). Eckert appears to have realised independently, and prior to von Neumann's joining the ENIAC group, that the way to take full advantage of the speed at which data is processed by electronic circuits is to place suitably encoded instructions for controlling the processing in the same high-speed storage devices that hold the data itself (documented in Copeland [2004], pp. 26–7). In 1945, while ENIAC was still under construction, von Neumann produced a draft report, mentioned previously, setting out the ENIAC group's ideas for an electronic stored-program general-purpose digital computer, the EDVAC (von Neuman [1945]). The EDVAC was completed six years later, but not by its originators, who left the Moore School to build computers elsewhere. Lectures held at the Moore School in 1946 on the proposed EDVAC were widely attended and contributed greatly to the dissemination of the new ideas.

Von Neumann was a prestigious figure and he made the concept of a high-speed stored-program digital computer widely known through his writings and public addresses. As a result of his high profile in the field, it became customary, although historically inappropriate, to refer to electronic stored-program digital computers as ‘von Neumann machines'.

49 Some of this analysis is described in the Stanford Encyclopedia of Computing.

50 These consequences are in addition to the other arguments supporting the same conclusions that are scattered throughout this article.

51 See [Turing 1939]

52 Turing assumed the oracle would be some kind of mathematical function. Given a physical computer it is easy to extend the concept to forms of input that are outside the scope of pure mathematics. Computers can be connected to physical measurement devices.

53 See also [Davis 2000] chapter 8.

54For instance see On the cruelty of really teaching computer science. (html) See [Dijkstra] for a link to a scan of the original manuscript.

Before we part, I would like to invite you to consider the following way of doing justice to computing's radical novelty in an introductory programming course.

On the one hand, we teach what looks like the predicate calculus, but we do it very differently from the philosophers. In order to train the novice programmer in the manipulation of uninterpreted formulae, we teach it more as boolean algebra, familiarizing the student with all algebraic properties of the logical connectives. To further sever the links to intuition, we rename the values {true, false} of the boolean domain as {black, white}.

On the other hand, we teach a simple, clean, imperative programming language, with a skip and a multiple assignment as basic statements, with a block structure for local variables, the semicolon as operator for statement composition, a nice alternative construct, a nice repetition and, if so desired, a procedure call. To this we add a minimum of data types, say booleans, integers, characters and strings. The essential thing is that, for whatever we introduce, the corresponding semantics is defined by the proof rules that go with it.

Right from the beginning, and all through the course, we stress that the programmer's task is not just to write down a program, but that his main task is to give a formal proof that the program he proposes meets the equally formal functional specification. While designing proofs and programs hand in hand, the student gets ample opportunity to perfect his manipulative agility with the predicate calculus. Finally, in order to drive home the message that this introductory programming course is primarily a course in formal mathematics, we see to it that the programming language in question has not been implemented on campus so that students are protected from the temptation to test their programs. And this concludes the sketch of my proposal for an introductory programming course for freshmen.


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Corrections Thread
Authored by: darksepulcher on Wednesday, November 11 2009 @ 08:25 PM EST
Please file any corrections under this thread so PJ can find them with ease.
Thank you.

Had I but time--As this fell Sergeant, Death
Is strict in his arrest--O, I could tell you--
But let it be.
(Hamlet, Act V Scene 2)

[ Reply to This | # ]

Off-Topic Thread
Authored by: darksepulcher on Wednesday, November 11 2009 @ 08:28 PM EST
This thread is for discussions that have nothing to do with the above article.
Posting on-topic material in here will earn you the wrath of some nameless
entity who will proceed to haunt you in some nameless way.

Had I but time--As this fell Sergeant, Death
Is strict in his arrest--O, I could tell you--
But let it be.
(Hamlet, Act V Scene 2)

[ Reply to This | # ]

News Picks Thread
Authored by: darksepulcher on Wednesday, November 11 2009 @ 08:30 PM EST
Discussion of news picks goes here. Please mention the title of the news pick
you are discussing so everyone can follow along. Thank you.

Had I but time--As this fell Sergeant, Death
Is strict in his arrest--O, I could tell you--
But let it be.
(Hamlet, Act V Scene 2)

[ Reply to This | # ]

Long Division
Authored by: Anonymous on Wednesday, November 11 2009 @ 08:40 PM EST
I think I shall have to seek a patent for the method of long division. I can see
no intrinsic difference between software, algorithms, and the method of long

[ Reply to This | # ]

An Explanation of Computation Theory for Lawyers
Authored by: SLi on Wednesday, November 11 2009 @ 08:42 PM EST

Also, it's now clear that just as you wouldn't go into litigation without a lawyer, if you are wise, because that is their area of expertise, lawyers also shouldn't assume they understand software and patents and how they relate unless they get help from technology experts who know how software and computers, and the underlying math, really work.

Yet the most baffling "misunderstandings" usually come from the companies which are most certain to have lots of talented people to explain this stuff to lawyers. Why? Well, simply, because it's the job of, say, IBM or Microsoft lawyers to get the result that IBM (read: the company, its execs, definitely not its programmers and geeks) wants.

Of course, it will still help those IBM lawyers if they really understand the subject matter, but I'm not convinced it helps them a lot. It may help them prepare to answer the most obvious geek responses to their position and to plan how to best steer the discussion (every court has a limited time) so that the issues remain muddy enough to the judges that their position stays plausible enough in the judges' eyes.

It would be curious to know, though, how thoroughly the lawyers on the pro-swpat side really understand the issue. My guess is that some do understand it quite well, but it would be only the very best of them.

[ Reply to This | # ]

Authored by: Anonymous on Wednesday, November 11 2009 @ 08:43 PM EST
I haven't even gotten past the first few paragraphs, and I can already tell this
is one I'll reserve a lazy weekend afternoon reading and digesting. I'll be a
better person for it, and maybe even a better programmer. Thank you, PoIR.

[ Reply to This | # ]

Discovery vs. Invention
Authored by: Anonymous on Wednesday, November 11 2009 @ 08:54 PM EST
The philosophy graduate part of me can't possibly let the statement "all
software is discovered and not invented" stand unchallenged. Please provide
reasonable contrasting definitions of "discovery" and
"invention" which allow both things to be possible and result in
"software" being classified as a discovery rather than an invention.

I suspect there's a relevant legal definition of "discover" vs
"invent" which trumps all other semantic quibbling in the current
context, but from a purely philosophical perspective I see the process of
invention as being a process of discovering techniques which work to achieve a
particular end. In that sense, all invention is discovery, but not all discovery
is invention. The art of designing a system (whether it be software or hardware,
electronic or mechanical) is invention.

[ Reply to This | # ]

Software is also a "Chameleon"; it changes "color" based on how it is used
Authored by: Anonymous on Wednesday, November 11 2009 @ 09:14 PM EST
This is a wonderful post. I only hope that the lawyers who see it will read and
try to understand ALL of it.

Sometimes software is data - i.e. information to be manipulated by the
Sometimes software is an instruction list - i.e. information that tells the
computer how to manipulate the information that is considered to be data.

Software changes "color" depending upon HOW it is used. This fact is
completely overlooked by the people who believe the it should be patentable.
Just like mathematics: software is sometimes used to compute things and
sometimes used to describe how things should be computed.

[ Reply to This | # ]

Lingering doubts
Authored by: rcweir on Wednesday, November 11 2009 @ 09:57 PM EST
The one thing that nags me in the back of my head is this: Isn't the physical
universe itself essentially mathematics? Isn't math intrinsic to the physical
as well as the mental world?

One of the trends of the information age is the virtualization of things that
formerly were only done physically. For example, instead of underground nuclear
tests, today we simulate nuclear bombs on computers. We can design and simulate
the effectiveness of aircraft designs on computers. We can do chemical synthesis
experiments virtually. Even your average video game system today has a powerful
physics simulation on it.

So if we argue that software is not patentable because it is reducible to
mathematics (which itself is not patentable), then what happens when we find
that many or most physical inventions are also (in theory) reducible to
mathematics as well, to the extent they can be simulated by computers running

Help me out here. What am I missing?

[ Reply to This | # ]

An Explanation of Computation Theory for Lawyers
Authored by: Anonymous on Wednesday, November 11 2009 @ 10:07 PM EST
I haven't read the whole thing. It seems really good as
far as I got. You really put a lot of work into it and it
shows. However, the only problem I have with this is not
one of content.

I guess that I have become jaded by what I have observed
of the various lawyer arguments presented here on Groklaw
and how they will do anything to help their clients,
including telling untruths and just plain making things up.

To get to the point, unless a lawyer would find it
necessary to be educated in computation theory for the sake
of their client, this will probably fall on blind eyes.
It seems that most will say whatever suits their clients
best interest and so far, appearing to be an ignorant
expert inorder to misrepresent the real facts seems to be
the norm.

All that said, I hope it gets read and understood, and can
be a positive force in getting lawyers to get see things
as they really are.

Best of luck and thank you.

[ Reply to This | # ]

An Explanation of Computation Theory for Lawyers
Authored by: SLi on Wednesday, November 11 2009 @ 10:42 PM EST
Now that I finally finished reading the text with thought, I must say it's very
good, even slightly enlightening also to me even if I already am a theoretical
computer science and computation theory geek. This is stuff I'm fairly familiar
with, but it was quite enlightening to see it explained so... simply. And I
appreciated the historical details, like the term "effective method",
which I hadn't heard about.

I only hope it's easy enough for a non-mathematical lawyer geek to understand. I
especially loved how the text "connects the dots" by analysing legal
cases and Quinn's arguments. I think I probably could have made each or most of
the arguments for "SW is maths" myself, but where PolR's (I guess)
"over 25 years of experience" show is the ability to explain these to
the layman so comprehensibly and relatively briefly and to connect them in such
an interesting and simple way.

[ Reply to This | # ]

An example to (attempt to) enlighten Quinn
Authored by: gfim on Wednesday, November 11 2009 @ 10:43 PM EST
Congratulations PolR on a wonderful and enlightening essay. Unfortunately, I don't see it having any significant effect - people are being paid to not understand these things!

On the subject of Gene Quinn and your statement:
Quinn's error is he is looking at one side of the street without considering the other. This is understandable considering the fact he is not looking at the same mathematics as we are discussing.
I think is best illustrated by an example. Quinn's idea of the mathematics in software is something like:
m = 100
c = 299792458
e = m * c * c
print "Energy=", e
He is talking about the "e = m * c * c" line as the maths involved. But, considering the computational theory part of it, there is just as much maths in:
print "hello"
Whereas, he would deny that there is any maths there at all. Get him to see that the second contains maths too, and he'll be convinced.

Of course, the cynic in me tells me that he won't ever see it because he doesn't want to be convinced!


[ Reply to This | # ]

Authored by: Anonymous on Wednesday, November 11 2009 @ 10:52 PM EST
As a former maths competition "whiz kid" who subsequently moved on to
an unrelated career, but still loves to spend as much spare time as possible
immersed in mathematics and coding, I'm extremely impressed. I sincerely hope
this will be read by the Supreme Court (or are they supposed to be isolated from
outside info on active cases, like juries?)

I think the arguments of Quinn are pretty well annihilated by this essay. But I
don't think the patent lawyers really care whether software is mathematics. As
long as they can make money by claiming that software programs are inventions,
they will keep doing so, no matter what mathematicians and computer scientists
say about the topic.

In an adversarial system of justice, lawyers are fundamentally not trying to
arrive at the truth - they are trying to win. Our hopes rest with judges.

[ Reply to This | # ]

An Explanation of Computation Theory for Lawyers
Authored by: Anonymous on Wednesday, November 11 2009 @ 10:57 PM EST
One of the best descriptions of the BASIC operations a computer uses was
provided by Theodor Nelson in "Computer Lib"... which, really, should
be required reading for a LOT of people.

Software patents, I suspect, would qualify as "Cybercrud".

[ Reply to This | # ]

Invention Disclosure
Authored by: Anonymous on Wednesday, November 11 2009 @ 11:16 PM EST
This (lovely!) article also presents one of my pet concerns with software
patents. Patents are supposed to disclose the invention to others skilled in the
art. I believe that most current software patents "disclose" the
invention only to those skilled in legal language. This article supports the
idea that if software is patentable, the only proper disclosure is a
turing-equivalent code for the software being patented. In other words, the only
proper disclosure for a software patent is source code for the patented item in
some computer language. Any patent system that includes the idea of software
patents must be open source!

[ Reply to This | # ]

Lambda Calculus -> Xerox Patented Lambda Calculus
Authored by: leopardi on Wednesday, November 11 2009 @ 11:29 PM EST

I think you need to mention that the Lambda Calculus itself has been patented, as U.S. 7,146,604, Program operators for composing abstractions , assigned to Xerox corporation.

Since, by your argument, every computer executes the essential reduction algorithm of the Lambda Calculus as it runs, every computer violates this patent. Also, by your argument, the execution of any algorithm, by any means, including mental steps by a human, violates this patent.

From the patent:

What is claimed is:
1. A method for composing programming abstractions, comprising:
defining a composition operation for composing a first abstraction and a second abstraction with a selected composition operator; each abstraction having a form, a prefix, and at least one expression; unfolding the expressions from the first and the second abstractions by: (a) removing the prefix of each abstraction, and (b) substituting formal parameter names in each expression with a common parameter name to define open variables; transforming the unfolded expressions to a reduced expression using a composition pattern that tunes semantics of the selected composition operator; selecting a prefix that depends on the selected composition operator and the form of the first abstraction and the second abstraction; the selected prefix having a formal parameter with a type; and nesting the reduced expression in a third abstraction that composes the first abstraction and the second abstraction by: (i) appending the selected prefix to the reduced expression, (ii) binding the open variables of the reduced expression to the formal parameter of the selected prefix, and (iii) computing the type of the formal parameter of the selected prefix.

[ Reply to This | # ]

M will, if carried out without error, always produce the desired result in a finite number of st
Authored by: Anonymous on Thursday, November 12 2009 @ 12:42 AM EST
ahhh, but with error, is what ms has a patent on, as well as the infinite
loop(sorry apple)

[ Reply to This | # ]

Very Good
Authored by: IMANAL_TOO on Thursday, November 12 2009 @ 01:13 AM EST
Very Good.

Hopefully this will be forwarded to the relevant decision makers.

Thanks PoIR!



[ Reply to This | # ]

A Prequel to Computation Theory
Authored by: Mike_no_Softie on Thursday, November 12 2009 @ 03:03 AM EST

Very good article; I now understand Computation Theory better. I just want to add some thoughts which were inspired by this article and which may make the argument clearer for lawyers.

You ask the question: Why were mathematicians so interested in effective methods in the 1920s?
You turn to Hilbert but I want to start somewhat earlier.

Sir Isaac Newton introduced the mathematical technique of calculus and used it in physics to formulate his famous laws. Calculus as introduced by Newton lead to some unfortunate mathematics. Just as an inexperienced programmer can botch it and develop a program that leads to absurd results so an inexperienced mathematician could prove absurd theorems as you state in the article.

The reason was that Sir Isaac did some voodoo with dividing by 0. From school everyone should know this is absolutely forbidden. The rules which were needed to do calculus were developed in the 19th century (excluding the division by 0) and further led to the very careful analysis in early 20th century of the foundation of mathematics.

The calculus as developed by Sir Isaac contained a lot of hand-waving. A computer (or a student) could not use it because there was no algorithm. Only by knowing what the result was (by observing Nature) could you use (the division by zero) calculus. The problem with this approach is that this was acceptable and do-able for physicists but unacceptable for mathematicians. So that's why as you describe early 20th century mathematicians developed effective methods etc. and made a new subdivision in mathematics which became known as theoretical computation science and as a by-product led to the art of programming as we now know it.

Computation Science did not only lead to programming but is also used in physics. For a recent successor of Sir Isaac Newton as Lucasian Professor of Mathematics i.e. Stephen Hawking used Computation Science (in the same way his predecessor Newton used calculus) to compute how the world will develop. See for instance Bekenstein -Hawking entropy. The physical 'entropy' is closely linked with the Computation Science 'information'. One lies in physics and the other in mathematics. Just as velocity lies in physics and the derivative lies in mathematics.

I am wondering if a lawyer would dare say that the research Professor Hawking has done is patentable. The argument that software is patentable can be repeated for the work of Professor Hawking making the physics he researched or at least the mathematics (Computation Science) he used patentable!

Nothing witty to say ;(

[ Reply to This | # ]

Congratulations, PoIR
Authored by: halfhuman on Thursday, November 12 2009 @ 04:09 AM EST

This is truly a magnificent achievement: short (yes!! *very* short), beautifully argued, beautifully written.

If ever "techno-legal" becomes as important as "medico-legal", this text should be published as a classic, on thick creamy paper, intaglio printed, bound in vellum!

The challenge of implementing that sort of quality in an ebook makes one think about where ebooks and their readers should be heading.

PS: of course, philosophically, I still have many questions, which chiefly concern the nature of mathematics and are completely irrelevant to software patents.

[ Reply to This | # ]

software is a pattern of machine instructions
Authored by: Anonymous on Thursday, November 12 2009 @ 04:39 AM EST
At different stages, software is different things.

1) At source code stage, the source code is a text file
which it is an abstract representation of an algorithm - the
algorithm on which software patents are now being issued
should not be patentable. Other things in software that have
been granted patents are specifications for protocols or
data formats. Specifications are not inventions and should
not be patentable because they are not inventions. A patent
would not be allowed on a specification for a PC which
consists of a certain combination of known components would
not be entertained even if nobody else has used that
combination before, so why should be allowed in the case of

2) After compilation, the byte code that sits on a CDROM,
USB drive, floppy, hard drive, or RAM is simply data, and as
such is not patentable.

3) When executing, a computer program executes a series of
machine code instructions. These instructions are a small
closed set of standard machine code instructions which are
defined by the manufacturer of the CPU hardware and these
instructions are no more not novel or inventive than bricks
used for building houses are. Hence the only claim for
novelty at this stage is that the sequence of standard
instruction bytecodes is somehow novel. However this use is
very dubious. Computers are intended to be programmed using
different sequences of bytecodes in order achieve different
effects in the same way that bricks can be used in different
patterns and combinations in order to build different
structures - their use in different combinations is
therefore obvious because that is their entire purpose. We
do not allow patenting of different patterns of bricks - for
example "Flemish bond", or "embodiments comprising of single
brick for a variable height of between 900mm to 1.5m with
half brick of 0mm to 600mm high on top with a common flush
face on both sides", so why would analogous patterns of
bytecode be allowed to be patented in software?

4) The key things that are being abused in order to get
software patents issued are process patents and patents on
specifications. Patenting is allowed on the former for
chemical patents, and the latter for alloy patents. However
in both cases, patents are allowed on very narrow and very
specific combinations only - you cannot patent generic
methods. Simply changing one parameter in the process or
specification or using a different physical process with the
same intermediate products will bypass the patent. In
addition, the patent is invalid if the combination does not
offer a specific and definite stated advantage over other
possible parameters or combinations, or if only a finite set
of valid combinations exist. This case has to be argued
properly if software patents are to be defeated.

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An Explanation of Computation Theory for Lawyers
Authored by: Anonymous on Thursday, November 12 2009 @ 04:49 AM EST
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Standing on the shoulders of giants
Authored by: mpellatt on Thursday, November 12 2009 @ 05:42 AM EST
Reading this article, and (a few of) the copious references, especially the
early history of computing and the incredible work of Turing, brings about a
true sense of humility and reminds me how today's tech is truly built by those
who stand on the shoulders of giants.

Not-so-hidden agenda: Software Patents kill innovation.

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Can you patent maths?
Authored by: Ian Al on Thursday, November 12 2009 @ 06:11 AM EST
I think the Supremes want you to be able to. In one of the links that PJ provided, the ex of one of the Supremes referred to the Bilski certiorari petition citing the Supreme Court's precedent declining to limit the broad statutory grant of patent eligibility for "any" new and useful process beyond excluding patents for "laws of nature, physical phenomena, and abstract ideas." The ex suggests that the Supremes are likely to reverse the clear, bright line test of machine or transformation. Therefore the argument you put needs to put software into the exclusion zone defined by the Supremes previous guidance rather than the District test from Bilski.

The maths stuff expands on what happened when modern computation theory began with research on effective methods, which are procedures to perform a computation that is entirely defined by their rules. It seems to me that this is making the point that software to achieve a computation is maths.

Any computation in pure maths must be unpatentable subject matter even if you use a slide rule, computer or pencil and paper to do it because it is dealing with abstract ideas. However, in Diamond v. Diehr the Supremes make the distinction between pure maths and applied maths. Now the conclusions,

  • All software is data.
  • All software is discovered and not invented.
  • All software is abstract.
  • All software is mathematics.

    may still be true, but may not stand in the way of patentability because they may only be true for software that achieves a computation in isolation. In other words, when the software produces an abstract answer to an abstract question.

    You referred to Diamond v. Diehr where the respondents did not seek to patent a mathematical formula.

    Instead, they seek to patent protection for a process of curing synthetic rubber. Their process admittedly employs a well-known mathematical equation, but they do not seek to preempt the use of that equation. Rather, they seek only to foreclose from others the use of that equation in conjunction with all of the other steps in their claimed process.
    So the mathematics is not patentable, but the application of the mathematics to create a useful invention has that potential.

    You also referred to Parker v. Flook where the process is unpatentable not because it contains a mathematical algorithm as one component, but because once that algorithm is assumed to be within the prior art, the application, considered as a whole, contains no patentable invention.

    Lets suppose we find a piece of software that is not a well-known mathematical equation but which, in conjunction with the other steps in the process, is useful, novel and patentable. From Parker v. Flook, the process is now (potentially) patentable because it contains a mathematical algorithm that is not within the prior art and the application, considered as a whole, now contains a patentable invention. Because the maths... sorry, software is no longer an abstract concept, but is applied as a useful art it fails to fall within "laws of nature, physical phenomena, and abstract ideas."

    We know that the rest of the computer, its peripherals and its connected networks are, in effect, prior art and the only thing left is the applied maths that we wish to foreclose from others using with the same other process steps (aka, computer, peripherals and networks).

    So, with the danger of my sounding like a worn record (Pat. Pending) we can do the inverse of Parker v. Flook and disclose where in the software there is a something that meets the prior art and innovation criteria. The syntax of the software is prior art. Even large sections of the syntactic material will be indistinguishable from any other large section. Anything patentable is higher up in the applied maths. What is more, the patentable stuff has to be identified even when expressed in an alternative syntax, otherwise it is a matter for copyright. The innovative process formed in the software is what has to be identified for the patent disclosure. As I might have said before, the challenge of disclosing the chunks of source code that exclude prior art, but include the invention is the real reason why it should not be possible to get a patent grant for software.

    The ENIAC computer applied maths to making atomic bombs. Could this be patentable subject matter? I think this is where the Supremes don't want a clear, bright line because they want to leave patentable subject matter as broad as possible with any grey areas resolved in the courts. (I think this is appallingly dangerous in the area of software, but the amicus briefs make my point for me). I think that the ENIAC calculations when applied to making atomic bombs were no different to using slide rules (well, perhaps faster!) and there was no direct involvement as a component of an invention.

    Matching Diamond v. Diehr to lossy compression of sound data, we might put words into the mp3 inventor's mouth and propose 'Our process admittedly employs a well-known mathematical equation from two centuries ago, but we do not seek to preempt the use of that equation. Rather, we seek only to foreclose from others the use of that equation in conjunction with the compression of audio data files'. So, which side of the Supremes broad and fuzzy gray line does that fit? I suggest that it is on the patentable subject matter side of the line, but only just. If the invention started and stopped at the data files then it would be outside. However, the invention should be considered as starting with the generation of the original sounds and ending with the decoding and playing back of the files as audio. The preclusion of delays in a process or the process being between more than one communicating or interconnected machines would preclude audio recording and radio transmission of sound from being patentable subject matter. The alarming thing about the mp3 patent is that there are no newly invented components and therefore nothing in the software that needs to be disclosed in detail. Any machine that uses the mp3 maths to encode audio into a computer file of any format or file system would be covered.

    Lets play with Diehr a little more by considering other inventions with no new components. A word processor is potentially patentable subject matter. It takes external data from the keyboard, gives the operator feedback to aid editing (but, I won't reveal how I know that!) and, with the possible time delay of saving the file and reloading it, prints a hard copy. The file standard is a matter of international interoperability and should not be patentable, but, that apart, the rest might be standard programming and standard mechanical and electrical components as far as the word processing invention is concerned. The Supremes attitude is likely to mean that, were it not for prior art, the word processor could receive a patent for what it does. It is an inventive application of standard components and maths (the software). Now, I would not be able to use a web-based word processor to contribute to my favorite blog without a paid-for licence. Since this implies that a new use for a general purpose computer could be patented, then the Supremes have got the standard wrong.

    As a general rule for patents, I continue to believe that any invention that is nationally or internationally used as a standard for interoperability of machines or systems should not be patent subject matter. This should be a new rule at the patent offices in addition to inventiveness and prior art. Any patent request that proposes a 'system' is making a call for patenting interoperability and should be rejected. That would mean that Microsoft's API's and file systems would not be patentable. However, their method of implementation (the detailed contents of the code) might contain a protectable invention.

    One final experiment with Diehr, lets consider a patent on sudo whereby a requester pops up whenever one asks for a computer operation for which permission is not given. There is no input data. The requests for operation from the user are not an integral part of the invention. The trigger for the invention to run is purely software/maths. There is no output to a machine or peripheral because the text or requester display is not used apart from the invention. In other words it is pure maths running in the computer and not applied maths with a tangible result. The results cannot be directly observed. They can only be deduced from the operations of the software and are therefore abstract in nature. The invention, itself, no matter how it is constructed, is not patent subject matter. If the implementation of sudo (i.e. the maths/software) contains a significant invention then this invention might be patentable, but only when that maths is applied to patentable subject matter. However, as I point out earlier in this response, it is not possible to form a patent request that discloses the invention in maths/software that forms a component in a patentable subject matter invention and tops the hurdles documented for patent award (words carefully chosen). Also, if it is not part of a patentable invention then, on its own, it is pure maths and is not patentable subject matter.

    Ian Al

    Linux: Viri can't hear you in free space.

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  • Copyrighting numbers
    Authored by: kh on Thursday, November 12 2009 @ 06:44 AM EST
    When I was young I thought it would be impossible to imagine anyone copyrighting
    a number. Now it happens all the time.

    A CD, an album if you like, is a number. A song, a number. There is no doubt
    these days that it is possible to copyright these numbers.

    A book, digitised is a number too. In many ways it always was. The ways of
    putting together a finite series of symbols (letters) are technically in
    mathematics 'discoveries' but no-one can really argue that writing book is
    anything like discovering a number.

    For those reasons, I'm not sure I find the arguments all that convincing.

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    An Explanation of Computation Theory for Lawyers: Adding/Updating and Republishing?
    Authored by: network_noadle on Thursday, November 12 2009 @ 07:06 AM EST

    I know it will mean some effort, but I think it would be worth it to review this article and the comments, and then either update it, or republish it as an updated article, incorporating the additions. I'd like to offer to help with that, but I don't know what help I could offer.

    I have been in programming and IT for the last 20 years, and I found this discussion of my chosen field fascinating.

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    The introduction of interpreted language
    Authored by: Anonymous on Thursday, November 12 2009 @ 07:14 AM EST
    This was a brilliant paper. The implications of interpreted languages has not
    been sufficiently appreciated, I believe. It is a simple way to trump any
    argument based on compiled code. If a program or business method is patentable
    due in part to its manifestation as compiled code; then what about the same
    program or method manifested in an interpreted language? Infringing or not?

    Thanks so much


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    Too late! :(
    Authored by: Anonymous on Thursday, November 12 2009 @ 07:54 AM EST
    I only wish this could have been put into one of the amicus briefs the Supreme
    Court read! But it's too late for that now, unless someone has the means to
    bring it to their judicial notice (we don't have any Supreme Court law clerks
    here... do we?)

    I am a mathematician, and this man is 100% correct in terms of mathematics and
    computer science. I only wish the justices could read this...

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    An Explanation of Patent Law and common sense for Computation Theorist
    Authored by: Anonymous on Thursday, November 12 2009 @ 08:14 AM EST
    I can adapt a pick-up truck to do a particular job by adding a component or
    system of components.

    For example, I can add a trailer hitch to adapt the truck to pull a load. If
    the trailer hitch has a new and useful feature, like a particular safety
    mechanism or quick release, I can patent the modification to the truck to do
    this load pulling job.

    I can add a winch and a set to tie downs to help haul my motorcycle into the bed
    and secure it there for transport. If the winch or tie downs have new and
    useful features, I can get a patent for the modification to the truck to do this
    winching/hauling job.

    These modifications to the truck make a new machine. I can get more for the
    truck should I choose to sell it because it has more uses than when I first got

    If you copy my inventions, one or both at the same time, you are infringing my
    patents. You would likely be charged with a per instance or per installation
    count of infringement. Not a new count for every micro second the devices

    I can adapt my computer to do a job by adding software. I can add a word
    processor to help me type documents and/or I can add a spread sheet program to
    help me keep track of all the money I make by suing other software developers.
    I can get a patent on the modification to my computer to do these jobs.


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    Getting it right?
    Authored by: GuyllFyre on Thursday, November 12 2009 @ 08:40 AM EST
    "They actually do want to get it right, you know."

    I have severe doubts about this. Perhaps some of the judges would like to
    "get it right" but I highly doubt the lawyers want this to be

    If there is only hard fact to work with, the trolls and other lawyers who make a
    living off the ambiguities would be out of jobs. They are most likely NOT
    willing to get it right so that they can keep arguing in circles and playing up
    their little dog and pony shows to keep robbing others of their hard earned

    Lawyers thrive on ambiguity because it gives them a chance to make a living at
    the expense of the rest of us.

    I fear while this is a great explanation, patent lawyer troll persons like Gene
    Quinn will look to argue these facts and keep them away from everyone. They
    will find some nit to pick and make a big fuss about it and nothing will ever
    get better.

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    On hello world
    Authored by: Anonymous on Thursday, November 12 2009 @ 09:42 AM EST
    Sure as
    it does not look like the calculation example. But think of the translation and
    symbolism operations.

    It could also be
    Screen[0][0] = 'H'
    Screen[0][1] = 'E'
    Screen[0][2] = 'L'
    Screen[0][3] = 'L'
    Where screen[row][col] represents a reference to the memory location
    corresponding to the display.

    Or translating the display address and the character data
    0x80000 = 72
    0x80001 = 69
    0x80002 = 76
    0x80003 = 76

    What it would actually look like is (//// comment)

    //// store the string in memory
    0x10000 72, 69, 76, 76, 76 ....
    //// reference the start of the string
    LD X, #0x10000
    //// The $PRINT function will copy the string at the reference location to the

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    Some More Stuff To Think About:
    Authored by: Imaginos1892 on Thursday, November 12 2009 @ 12:07 PM EST
    Thought I'd add a few ruminations about computers, from the intentional aspect

    A digital computer is only capable of performing a small number of simple binary-logic operations.

    These operations - the machine instructions - are a set of state transition rules permanently
    encoded into the computer's hardware. Everything the computer does or can do is accomplished by
    arranging, sequencing and repeating those simple operations. Every computer program consists of
    an ordered list of operations selected from that set of rules. And in case anybody had any doubt,
    binary logic is math.

    Every operation possible to any digital computer can be reduced to Exclusive-OR.

    Any digital computer can be modeled as a large, but finite, array of binary bits. Each bit has a
    value of 0 or 1, and a state consists of the values of all the bits at one instant in time. In any
    transition from one state to another, each bit can either change, or not change. Therefore, every
    possible operation can be exactly represented by an Exclusive-OR function with a sufficiently large
    operand in which bits that change are set to 1, and bits that do not change are set to 0.

    Every computer program consists of a sequence of operations which can be performed by a human.

    Every program is eventually reduced to a sequence of the simple binary-logic operations that the
    computer can perform. A human can mentally perform each of those operations, else no human could
    have designed the CPU that does perform them. In addition, the programmer had to perform some
    equivalent of those steps mentally in order to write the program.

    A digital computer is fully deterministic.

    Once a computer has been set to some initial state, all subsequent states can be predicted by
    mindlessly applying the state transition rules encoded into its hardware, with the exception of
    asynchronous events such as user input. Even then, given full knowledge of the timing and content
    of those events, the results can be unambiguously predicted by application of the same rules. So,
    programming the computer consists solely of establishing the desired initial state and allowing
    it to progress through all of the inevitable successor states. You could say that the whole
    computer program and everything it will ever do is expressed in that initial state.

    Every possible state has already been anticipated by the design of the hardware. The bits are there.
    All a programmer or user can ever do is set them to 1's and 0's.
    Nobody expects the Spanish Inquisition!!

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    An Explanation of Computation Theory for Lawyers
    Authored by: Daddl on Thursday, November 12 2009 @ 12:30 PM EST
    I already posted this in the draft thread, but as this is where this is actually
    discussed, I c&p it here:

    What I always find so hard to understand, is why people need so complicated
    explanations, mathematical theories, etc. to argue why software can't be

    Basically a computer is a machine that a lot of switches with defined states
    ('on' and 'off' for binary logic circuits) - like a traditional machine with
    levers or valves and such. Only it has many more 'valves'. What software is, is
    a defined state for some of the valves (open or closed, on or off, 0 or 1). A
    computer is constructed to act upon those 'valve settings', like a steam machine
    acts upon the opening or closing of valves. Can I patent a computer (or it's
    hardware parts)? Yes, of course - just like any other machine.

    I can patent the 'valves' in a computer (and the technology to produce them,
    they way they are arranged and connected), the hardware used to output 'valve
    states' to humans (e.g. lcd or crt screen technology, light bulbs, speakers),
    input technologies like keyboards or mice, the technology used to store 'valve
    states' (ram, flash memory, harddiscs, holographic memory...), etc.

    Can I patent the actual way I use that patented technology? No. If I used my
    patented cork screw to remove dirt from under my fingernails after working in
    the garden - could I patent this? No. If I used my notebook as a doorstopper,
    could I patent this? No. Would I infringe anothers patent doing so? No. So why
    should I be able to patent a specific use of a machine, just because I in
    exactly the way it was designed (and already patented) for? Any computer is
    build to act in a predictable manner on the state of these 'valves', it's the
    basic function of a computer - so I'm simply using it inside its original

    Whatever code I write and load onto a computer (i.e. whatever states I set the
    'valves' to), the computer acts just the way it is constructed - it does not
    become a different machine. My watch (mechanical by the way) doesn't become a
    'new patentable machine' just because I set it to a different time. Nor can I
    patent the specific time I set it to.

    Software (as in whatever code we write in Assembly, C/C++, C#, Java, or other
    programming languages) is just a convenient way to describe the specific state
    of some of the 'levers' in a computer. Can I patent a description? No. Can I
    copyright it? Yes.

    I don't see any relevant difference between a computer and any other machine or
    technological device ever devised. Be it the revolutionary 'sharpened rock on a
    stick' mammoth hunting device, a steam machine, a laser gun or whatever. Any
    distinction made is purely artificial.

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    "Invented vs. Discovered" is an open question
    Authored by: Anonymous on Thursday, November 12 2009 @ 01:02 PM EST
    Sorry, I couldn't get past the first few paragraphs.

    "You can't invent abstract ideas and mathematics. You can
    only discover them."

    This is completely a matter of opinion. This is an open
    question in the philosophy of mathematics. This particular
    viewpoint is an tenet of the school referred to as
    "Mathematical Realism".

    My opinion, not that it matters, is that maths are invented
    all the time. If you follow the computing discipline of
    formal verification, new logics are created, extended, and
    scrapped based on how useful they are for proving the
    correctness of programs. Separation Logic, for an example
    is turning out to be a useful extension of C. A. R. Hoare's

    You may be tempted to suggest that they are just...
    "taking mathematic constructs out of the ether", and you
    might further argue that there are an infinite amount of
    such constructs.

    But, my friend, what kind of infinity? Countable? Aleph
    One? That slippery infinity in between infinities that
    nobody has invented/discovered yet?

    See my point?

    [ Reply to This | # ]

    A nice exposition of the math(s), but is it convincing against software patents?
    Authored by: Anonymous on Thursday, November 12 2009 @ 01:32 PM EST
    This is a very nice exposition of the mathematics. I have a Ph.D. in mathematics and have many times had to explain to people how mathematics is about discovery not invention; it is nice to see so many people understanding that.

    However, I am not convinced this is the best argument against software patents, for the following reasons:

    Is "software is maths" really true as stated?

    I'm not sure that it is. It depends on the definition of software. If one adopts a definition that probably many non-mathematicians and non-computer-scientists do, that a piece of software is a description, written in a formal programming language, of a series of steps for a computer to perform, then:

    • That piece of software on its face is not mathematics.
    • Rather, that piece of software is equivalent to a mathematical function.

    The distinction may seem pedantic but I think it is important. I think the legal types like Quinn are thinking about software in a very concrete, "what is this on its face" kind of way, while the mathematical types are looking beneath the surface to an underlying mathematical concept.

    The trouble with using this as an argument against software patents is that if you are willing to look at underlying abstract mathematical concepts you can do the same even for traditional nuts and bolts inventions built with materials like wood, steel, nails, and screws. The steps needed to put such an invention together can be mathematically formalized, with invalid steps mathematically distinguished from valid ones, in a way very similar to the way in which essential methods or algorithms have been mathematically formalized. To demonstrate this, let me describe the beginning of such a formalization (non-mathematicians may want to skip over the next few paragraphs):

    Let M be a finite set called the set of material types, to whose elements we give names such as "wood" and "steel". Let M' be a subset of M whose elements we will call hard metals.

    Define an object to be an ordered pair (R,m) where R (called the object's region) is a connnected subset of R3 (3-dimensional space) and m is in M. (An object is a region of space filled with a particular type of material).

    Define a nail with sharpness s to be an object (R,m) where m is in M' (a hard metal) and R has a designated element p called the point, a designated subset H called the head with centre point c in H, and every other point q in R has the property that the vector from p to q makes an angle smaller than s with the vector from p to c.

    Define a screw to be (similar but more elaborate definition here).

    Define an assemblage to be an ordered pair (A, G) where A is a collection of objects whose regions have disjoint interiors and G is a subset of R3 such that every point in G belongs to the boundary of the regions at least two different objects in A. (A collection of objects with gluing or other attachment on some of the joining surfaces).

    Now one can mathematically categorize the set of assemblages that can be built out a starting assemblage of raw materials. This can be done with a recursive definition similar to that for computable functions. First one defines basic building blocks such as moving, gluing, nailing, and screwing. Then one says that assemblage (A',G') can be built out of (A,G) if either

    1. (A',G') is obtained from (A,G) by moving or gluing or nailing or screwing, or
    2. there is an assemblage (A'',G'') that can be built out of (A,G) and (A',G') is obtained from (A'',G'') by moving or gluing or nailing or screwing.

    We can say that (A',G') is obtained from (A,G) by moving if there exists an affine transformation T of R3 and a subset B of A with the following properties. Here we let C be the union of the regions of the objects in B and D be the union of the regions of the objects in A that are not in B.

    1. T(C) is interior-disjoint from D (we can't move one object into the space occupied by another object we're not moving)
    2. G, C, and D have no point in common (if a region we're moving has a boundary surface that overlaps one we're not moving, the overlap can't be an attachment point.)
    3. A' is the union of (T(R),m) over objects (R,m) in B together with the union of (R,m) over objects in A-B.
    4. G' is the union of T(G intersect C) with G intersect D.

    We can say that (A',G') is obtained from (A,G) by gluing if A' = A and G is a subset of G' and (A',G') meets the definition of a valid assemblage.

    We can say that (A',G') is obtained from (A,G) by nailing if there is a nail N=(R,m) (with head H and shaft S=R-H) in A, with R disjoint from G, and a linear transformation T of R3 such that T(H) is interior-disjoint from the regions of all other objects in A (I am disallowing countersinking here for simplicity; a countersunk nail obviously requires a more elaborate definition), such that:

    1. A' is (T(R),m) together with the union of (R' - interior(T(R),m') over all (R',m') in A-{N} (the nail displaces material from the nailed objects in the location where the nail now is), and
    2. G' is the union of G with the union over all (R',m') in A-{N} of R' intersect T(R) (the places where the nail shaft touches the material it was driven into are now attachment points that can't be pulled apart)

    We can say that (A',G') is obtained from (A,G) by screwing if (similar elaborate definition here).

    (Non-mathematicians can return to the discussion here.) The point is that questions about what configurations of objects can and cannot be built mechanically could, at least in principle, be formulated as a purely mathematical question, in much the same way as questions about what can and cannot be computed. A tangible invention out of wood and nails is simply a practical implementation of a series of building steps which are in turn mathematically equivalent to a choice of a particular function between abstractly defined sets. This is similar to the way in which a computer running a program is simply a practical implementation of a piece of software that is in turn equivalent to a purely mathematical abstract concept.

    So, yes, software "is" mathematics with the right definitions, but so too are the means for building physical and mechanical inventions. I don't think the "software is mathematics, mathematics is not patentable, therefore software should not be patentable either" argument will convince the pro-software-patent types that a computer running a particular software program is not patentable. At best it will convince them that the software program in the abstract, written down on paper but not running anywhere, is not patentable, or that the mathematical abstractions to which the software is equivalent are not by themselves patentable -- but that's not the same thing. They will think of the software in the abstract as being like the artificial formalism I gave above of mathematically modelling mechanical steps for building out of wood and metal: not patentable by itself, but as soon as you actually run it on a computer, you could be infringing a patent, much in the same way that as soon as you actually build a device out of wood and metal using that formalism, it becomes an infringement if the design was patented.

    I think that there is a much better chance of convincing the pro-software-patents legal folks that software is distinct from other types of "inventions" if we focus on the fact (which was also explained in the article) that computers are, essentially, universal Turing machines and that there is no fundamental difference between the notion of "a computer running a particular program" and "a computer running the same program it was designed and built with but being given particular input data". That, in my opinion, is where the difference between mechanical inventions and computer software lies. There is currently no "universal invention-building robot" which, when given a precise description of how to make something, will go ahead and make it. Building a mechanical device requires either human action or a specially-constructed, purpose-built robot, and it also requires permanent commitment of materials and physical resources. However, to run software on a computer, nothing is required except to store the program on the computer in the same way any other kind of data is stored. I think it is this fact that makes the difference between "a software program in the abstract" and "a computer running a software program" fundamentally and qualitatively less of a difference than the difference between "a mechanical invention specification in the abstract" and "a physical device built from that specification". It is not the fact that the software program in the abstract is equivalent to a mathematical concept that makes the difference, because the mechanical invention specification could also be represented by a mathematical concept. Rather, the difference is that to move from the equivalent-to-mathematics mechanical invention specification to an actual device requires specific physical action, whereas to move from the equivalent-to-mathematics abstract software design to an actual computer running that software requires nothing more than operating a computer in the way it was originally designed but feeding it specific input data.

    In other words, should merely exposing an existing device to information qualify as creating a patentable new device? I think one can make a very convincing argument that the answer to that question is a resounding "no". And for that reason (not the fact that the information provided to it is equivalent to mathematics), software should not be patentable. The mathematics is a bit of a red herring here; the key is simply that, because information by itself is not patentable (regardless of whether it is mathematical information or not), merely exposing a device to information and letting it process that information as it was designed to do (even if the designer didn't know what future information it was going to be exposed to) is not enough to create a new device that is subject to patent protection.

    [ Reply to This | # ]

    Modern computers are Turing *oracle* machines, not turing machines
    Authored by: vortex on Thursday, November 12 2009 @ 02:38 PM EST

    I should begin by pointing out that I, too, have spent a quarter century working in the software industry – and do not believe that software should be patentable. However, I find the "because it's mathematics" argument weak. Fundamentally, the correct reason why it shouldn't be patentable is that patents are an abysmally poor fit for this industry.

    However, as to the present discussion, I must point out that what computers do is not all mathematics. In so far as what my computer does is purely computation – and it does a great deal of that – it is, indeed, operating as a Turing machine. However, it is also ignoring my mouse and my keyboard.

    To properly model a modern computer and take account of user input it is necessary to model it as a Turing oracle machine – as I type this, the computer isn't working out what characters to display on the screen, it's consulting some hardware that reports what I'm typing. The program then follows purely mathematical rules in deciding what to do with what I'm typing, namely to display that text and add it to the data in a field of a form that I'll later submit. While vast amounts of what a computer does are deterministically controlled by the programs the computer is running, diverse things aren't: aside from what I type and how I move the mouse, the computer also encounters such non-determinacy when it attempts to interact with remote computers.

    To submit this form, this computer shall do lots of purely Turing activity: but when the program asks Linux to connect a socket to a remote computer, the program has absolutely no way of knowing when it'll get its response. As it happens, I am the author of the code that handles that indeterminacy, so I know its details horribly well. The browser sends its request and gets on with something else. Sporadically, thereafter, it checks to see whether it's got an answer. Typically, it'll get an answer eventually, and get back on with being nice and Turing about it: but every time it stopped to ask for that answer, it was consulting an oracle, in the terms of the article. Once the connection is made, this computer shall send some message and wait for an answer: again, the external network functions as an oracle.

    Likewise, in fact, the way the computer gets keyboard and mouse input involves hardware that works in parallel with the CPU to notice such input: when it does, it generates an interrupt that causes the program to deal with that input. Interrupts are expressly not part of the Turing behaviour of computers: they are not determined by the algorithms the computer is running; every interrupt is a manifestation of some sort of oracle, in Turing's terms. We have designed computers so that these oracular interventions disrupt the model as little as possible (we register nice little Turing fragments of code to handle them, and we take care to write those in such a way that the intrinsic indeterminacy of how and when they get called isn't a problem): but they still happen, so the modern computer is, in fact, a Turing oracle machine – at least whenever it is being interactive.

    Consequently, an argument based solely on "software is mathematics" is both false (it presumes software is pure-Turing, not Turing+oracle) and inadequate: those who want software patents shall walk round it by exploiting the oracular character of interrupts. Indeed, while some patents have been laughably blatant in laying claim to pure Turing activity, many have taken the trouble to couch their claims in terms of the user's input and network activity – both of which are emphatically on the oracular side of Turing's analysis. As PoIR says, that's the side where patents are legitimate: but, where the non-Turing activity arises purely from interrupts driven by entirely conventional hardware, we need the law to grok that the involvement of this not-just-mathematics should not serve as carte blanche for patent land-grabs. Only when a computer is entangled with some hardware that does something novel, albeit assisted by the computer and its programs, should the novel thing the hardware is doing be patentable: and, even then, the software that facilitates it in doing this should not be subject to patenting, any more than mathematics is, even despite any degree to which the software's involvement with oracles takes it outside the pure-Turing domain.

    We should not be using "it's all mathematics" as if it were our strongest argument: first, because it isn't all mathematics, and the oracular parts really do matter; second, because there is a far stronger argument, at least where it comes to persuading legislators and judges. That argument is that the idiom of patent protection is unworkable in the case of software, for diverse reasons that RMS and others have expressed far more eloquently than I can. It is unworkable in court because it produces too much unpredictability (where judges think law should produce predictability) and it is unworkable in the economy because it cripples those who are actually doing something that adds value (and value is the foundation of all the wealth that governments tax; the less of it there is, the less they have to spend). Even when software isn't only mathematics, it's still far better for freedom, the economy and justice if the activity of producing it and using it is as unfettered as possible.

    Finally – the SCO group saga's end game.
    Break out the pop-corn, sit back and watch the fire-works.

    [ Reply to This | # ]

    Z Schema
    Authored by: rhdunn on Thursday, November 12 2009 @ 04:28 PM EST
    The section "Programming Directly from Mathematical Languages"
    reminded me about learning Z Schema [].

    Each function in Z Schema is a transformation from the inputs to the output(s)
    using mathematics to express the transformation.

    There are a set of pre-conditions that are expected to be true for the inputs of
    the function (the axioms, if you will). You also specify post-conditions (things
    expected to be true when the program is finished) and invariants (things you
    expect not to have changed in the system).

    With this, you prove that the post-conditions are met and the invariants hold
    after the program is completed using the pre-conditions and set theory. If these
    hold, you can then transform it into a computer program.

    Applications of this are in areas where correctness of the program is critical
    (e.g. in a nuclear power plant).

    [ Reply to This | # ]

    The Big Question
    Authored by: Anonymous on Thursday, November 12 2009 @ 04:33 PM EST

    First, kudos for all that work! We could all do with the revision :-) I'm offering the following as constructive criticism...

    Here we are looking at a two-way street. It is correct that we can describe just about everything mathematically...

    The notion that, if software is unpatentable because it "is maths", then a steam engine must also be unpatentable because it can be completely described by maths, is a very dangerous one, FUD or not. The idea of accidentally pulling the plug on the whole patent system will worry judges. The "two way street" argument applies to you, too: you don't just have to show that software is maths, you have to show why everything else isn't maths.

    (also bear in mind that anybody who's eyes glaze over at the maths might skip to this bit as it really is The Big Question).

    You rightly rebutt Quinn - but he has blown it by showing that he doesn't understand what "algorithms" and "equations" are so he's a bit of a soft target. "We don't patent maths textbooks" is also bit of a straw man.

    I think the argument that would convince me (the language could be tightened up up a bit) is:

    Computers work, fundamentally, by transforming sets of symbols (data) according to a set of instructions (algorithms/software - which are also symbols) in a way that is defined, with full mathematical rigor by the abstract mathematics of computing theory.

    Other, patentable, machines work by (insert the standard wording about forces of nature and transformation of matter). They do not work by "transforming symbols" and the mathematical describing them can eventually be traced back to empirical observations about the real world. Physics has yet to perfect the mathematical "theory of everything" which completely describes the known universe to the extent that mathematics describes the operation of computing devices.

    Note that the box of tricks we call a computer will contain "input/output" devices (keyboards, video displays) which turn forces and energy to/from sets of symbols. These parts may well be patentable, as may any portion of an invention which describes a new such machine. However, the only possible interaction between the software and the computer is to cause the manipuation of symbols - which is mathematics.

    (Note - the other challenge is to avoid the dreaded "software per se" argument... but I think that is an inherent danger in the whole "computing theory" argument).

    [ Reply to This | # ]

    An Explanation of Computation Theory for Lawyers
    Authored by: Anonymous on Thursday, November 12 2009 @ 05:17 PM EST
    Perhaps I don't understand it properly.

    "All software is discovered and not invented."

    I discover a vaccine for H1N1 by testing materials and analyzing the results.

    Now perhaps the meaning of a = b+c was discovered, but when I use a = b+c, in a
    program, I'm not discovering what it does, I'm using a previously discovered
    thing to construct something new. Further when say a = b+c, d = e+f, g = a +d I
    am constructing the same thing I am constructing when I say d = e +f, a = c + b,
    g = d +a but the expression of the thing I am constructing is different.

    So the question is, how do I protect the effort and expense I go through to
    create my new thing? I clearly can not use copyright, because copyright
    protects only expression, and it is not possible to protect every possible
    expression of creating object g.

    But to be honest, it is also impractical every time I create a thing g or h or
    i, to do a patent search to see if someone else might have patented that thing.
    But this objection is outside the bounds of "Software is

    And while I understand that software is based on mathematical principles, the
    pattern of bits stored on a medium to control a machine are a physical
    implementation of a specific concept.

    If I read a book that says do step a then step b then step c, then I carry out
    those steps, are they being carried out in exactly the same way as Sally two
    cubicles over carries out the steps? If someone says step a is to bath an
    object in a chemical bath for five minutes, and operator a knows that is just
    enough time to go outside and smoke a cigarette, but operator b stands by the
    machine and carefully times the period with a stop watch, you would think
    operator b would get the better results. But operator a has been doing it that
    way for years, and knows exactly that it is the length of time to smoke his
    particular brand of cigarettes, and his product always comes out right and
    operator b gets a 50 percent yield. This is the art necessary for the invention
    to work. But if I am using a pattern of bits on a medium, with a machine with a
    known clock speed and data transfer speed, there is no variance possible as
    there is with a person reading a book. This, in my personal opinion is where
    you have a specific implementation of the concept, and the machine loaded with
    running software is no longer an abstract thing, but a concrete thing.

    Now quite frankly, all this is arguing to allow a general purpose computer, or
    group of computers with instructions to direct specific behavior to be a thing
    that can be patented. And I am against the way software patents are granted
    today. So I am painting myself into a corner. But I am going to say that a
    program that is designed to run on an Intel processor is not the same thing as a
    program that does the same thing on a Power PC processor, or a network or a cell
    phone or a virtual machine. I am going to argue each is a separate thing that
    would need to be protected by it's own patent. Now if I have something that can
    read and write fat32 files on a machine driven by an 8080 processor, the thing
    that does the same thing using a Pentium Processor is a different thing and
    would need to be protected by a different patent. And if it wrote those files
    to an IDE drive,it would need to have a different patent to write the files to a
    SCSI drive, and a different patent to write it to a USB drive. You can't patent
    the world with a concept. It has to be a specific thing. Before anyone says
    this is impractical, go back to the example of writing code where you have to do
    a patent search every time you write a function. Besides the patent lawyers
    ought to have a ball with this idea. You want to patent sudo? Fine document
    with specificity the invention. You document the instruction set used on the
    machine where the invention applies,the environment and so on. And when you
    finally get around to porting it to the ARM processor, your patent lawyers
    pocket another fat paycheck.

    [ Reply to This | # ]

    Conway's Game of Life Turing Complete
    Authored by: rhdunn on Thursday, November 12 2009 @ 05:20 PM EST
    John Conway developed the Game of Life
    [], a cellular automata
    based on a simple set of rules.

    The wikipedia article covers it well, and gives examples of possible patterns
    and types.

    What is interesting is that this is turing complete and can even be used to
    create computers that are capable of running software
    [], albeit very slowly!

    [ Reply to This | # ]

    I dislike software patents, but I found this argument completely unpersuasive
    Authored by: Anonymous on Thursday, November 12 2009 @ 05:34 PM EST
    Consider an analogy. What mechanical things can be built from 26 grams of iron?
    (This is a cube just under 2cm on a side.)

    Each of those things can be described by a string of 6.023 x 10 ** 23 bits.

    How? Well, in the cube, there are that many atoms. So, for each possible
    location, either there is an atom there, or there isn't. So, if the bit is '1',
    put an atom there; if the bit is '0', leave it out.

    (Want something bigger? Add more bits. Want something with more different
    elements? Add more bits.)

    What you've argued is that a program can be described by a string of bits, or by
    a Turing machine, or by ...

    Saying that a constructed thing can be described by a string of bits, like those
    bits of iron, or like a program, says nothing at all about whether it can be
    constructed, or does anything meaningful if it is constructed, or does precisely
    what you want it to if it is constructed.

    To construct pieces of iron, or programs, or ... requires engineering. Because
    we won't live long enough to look at all the possible pieces or programs.

    I think you need to go back to the drawing board on this argument. While I
    dislike software patents, I find this argument as to their unpatentability
    completely unpersuasive. It confounds construction with description, and so

    Pat McGee

    [ Reply to This | # ]

    An Explanation of Computation Theory for Lawyers
    Authored by: Anonymous on Thursday, November 12 2009 @ 06:17 PM EST
    How about this much simpler form:

    1. All software capable of being written by humans in finite time, is capable of
    being expressed as a finite sequence of binary digits.

    1a. The number of states on a machine with n bit registers, when expressed as a
    numeral, is no more than 2^n. In other words an n-bit state register can attain
    exactly 2^n different possible states. Therefore there can also be no more than
    2^n different software instructions that can fit into that register. Similarly,
    for any file of n binary bits in length, there can be only 2^n different
    possible files. So for example a 64 bit machine could theoretically address 16
    exbibytes of memory. There can be only 16 exbibytes worth of possible programs
    that would run on said 64 bit machine.

    2. A finite sequence of binary digits is a finite number, also capable of being
    expressed in base 10 numerals.

    3. A number is not patentable. (However it may be copyrightable.)

    4. All programs are numbers, and therefore not patentable.

    [ Reply to This | # ]

    LISP, S-expressions and lambda calculus
    Authored by: Sander Marechal on Thursday, November 12 2009 @ 07:01 PM EST
    the remarks about LISP in this article makes me wonder. Isn't it trivial to
    actually prove to lawyers and judges that software is math?

    The big stumbling block seems to be that most software is a series of
    instructions, while math is formulas and expressions.

    So, take any piece of software and write the equivalent in LISP. You now have a
    set of S-expressions that you can solve/reduce step-by-step. It's simple lambda
    calculus and squarely the domain of mathematics.

    This is also interesting in reverse: Any software implemented in LISP cannot
    possibly infringe a patent because it's clearly just a mathematical expression.

    Perhaps this can easily demonstrate (in an easier way) that there is no
    difference between software-as-a-set-of-instructions and a mathematical
    expression. After all, LISP is turing-complete so *all* software can be written
    in LISP :-)

    Sander Marechal
    Geek, Programmer and many more, but not a lawyer

    [ Reply to This | # ]

    Simple Terms
    Authored by: Anonymous on Thursday, November 12 2009 @ 08:44 PM EST
    There is one thing about software that I must continuosly come back to in any
    discussion about software vrs patents.

    No matter what - software does not function in a void. Software requires a
    computer (hardware) before the process described in the software can be
    understood. It is simply not possible to seperate software and come up with any
    type of meaning.....

    For example: Lets take the most basic of assembly instructions

    LDA #0

    So I am loading the accumulator with the value of 0.

    Tell me what this means in the terms of a method you can follow?

    This instruction could make a white dot appear on your favorite video display,
    could disable the mouse .... could do anything you could imagine since we don't
    have a computer from which to run the instruction and get our result.

    This instruction can also produce different results depending on the hardware it
    is run on.

    So without a specific set of hardware, software holds no value - makes no

    All these Lawyer descriptions of how computers are laid out as a series of
    switches misses the point - unless you know how those switches are laid out,
    only then does the software sequence start to have value. Bring in a different
    machine with a different layout and the software you just produced will not

    Since software does not operate in a stand alone capacity. How can it be thought
    of as a patentable invention?

    [ Reply to This | # ]

    • Simple Terms - Authored by: Anonymous on Thursday, November 12 2009 @ 09:21 PM EST
    This article is to get rid of "plus a computer" patents.
    Authored by: Anonymous on Thursday, November 12 2009 @ 09:29 PM EST
    The fundamental lessons of it for patent lawyers are:
    (1) A computer is *intended to* do information processing which a human could
    do, but very fast. *Any* such information processing. *All* such information
    processing. Adding different software is what it's *designed* to do.
    (2) Therefore, adding "plus a computer" to information processing
    which could theoretically have been done by a human does not fundamentally
    change it, *and is obvious* -- it should be evaluated exactly as it would be if
    it were done by a human (however difficult this might be). Just like "plus
    writing it in a book" doesn't make a story patentable.

    There is a second aspect which has to be done to prevent the patent system from
    issuing disastrously bogus patents: business method patents have to go away.
    (They are a combination of abstract, purely mental, though not necessarily
    mathematical, algorithms and laws of nature.) If judges understand computation
    theory, they'll understand that fundamentally all software patents -- the ones
    we're trying to get rid of -- are business method patents "plus a

    We have to knock out *both* the business method patents *and* the "plus a

    Fundamentally both are information processing. Information processing consists
    of *nothing but* abstract ideas, and is therefore totally unpatentable.

    This still allows patents on a novel industrial procedure which has new aspects
    *outside* the information processing.

    [ Reply to This | # ]

    Discovered vs Invented
    Authored by: ewe2 on Thursday, November 12 2009 @ 11:01 PM EST
    Firstly, you need to do a doctorate :) I learnt more maths with this intelligent

    Although it seems clear to me, I see some respondents resisting the logic of
    your argument, specifically "discovered" vs "invented".
    Allow me to present a little parable of my own:

    Take 2a * b = c. If a = 2, b = 2 then

    (2x2) x 2 = c
    4 x 2 = c
    8 = c

    Try solving the equation using addition instead of multiplication:

    (a+a) + 2 + 2 = c
    4 + 4 = c
    8 = c

    Has anything changed? No, we have just noticed a new way of expressing the same
    relationship between a, b and c. This is what I take to mean
    "discovery" because it certainly isn't "invention". We can
    infer that multiplication is a quicker form of addition but we could continue to
    add if we chose (in fact many logic chips still do).

    Or perhaps you'd prefer solving for b:

    b = c / 2a
    2 = c / 4
    c = 4 * 2
    c = 8

    The relationship stays the same regardless if you use subtraction, addition, or
    even the terms on different sides. This is discovery not invention. Just because
    you "found a different way" (ie discovered) to look at the
    relationship makes no difference to the relationship itself. As Perl teaches,
    "there's more than one way to do it", which implies there is no unique
    (therefore invented) way either.

    Some respondents have tried to take the argument to the computers themselves.
    Using first and second examples above, if machine A can do the first equation in
    its hardware but machine B has to do the second equation because it can only add
    and not multiply, is this invention? Only from the machine's side, it makes no
    difference to the programmer.

    Additionally if machine B prefers an equation of the form b = c / 2a because for
    some reason its faster than using addition to find c, is that invention? No, the
    maths is the same, its an issue of machine implementation.

    These issues came up often during the early years of Unix porting, and I'm sure
    you can find a hundred examples. In all cases, Unix on different hardware wasn't
    an invention. And this is the low-level stuff: if this cannot be invented, how
    can anything derived from it be?

    [ Reply to This | # ]

    What is "invention"?
    Authored by: Anonymous on Thursday, November 12 2009 @ 11:02 PM EST
    >>>Software is discovered as opposed to invented.

    >>>This is a consequence of the fact that software is
    >>>abstract and software is mathematics. You can't invent
    >>>abstract ideas and mathematics. You can only discover them.

    This assertion asks for the question, how are you defining invention? Edison
    demonstrated that invention is often search. 10,000 filaments to find the one
    that works.

    Invention is often just discovery, but one that follows a lot of search, and
    learning the parameters of the space, until you can start to make leaps of
    "intution" because you start to recognize patterns.

    The legal question is whether the results of this search should be protected to
    benefit the searcher so that other searchers will have the incentives to invest
    time into said search.

    [ Reply to This | # ]

    Complexity is the wrong direction.
    Authored by: BitOBear on Friday, November 13 2009 @ 12:15 AM EST
    I have said this before, and I'll say it again. Computer Programs are _not_
    math. Computers were, indeed, invented largely in the math departments of
    schools, and many of the words so used are from the world of math. But beyond
    the fact that all reality can be modeled mathematically if you know what you are
    doing, there is nothing more "mathish" to 'print("hello
    worldn");' than there is to "mow my yard!".

    The key is to really _listen_ to the words persons well versed in the art of
    programming used to describe their work.

    We _write_ software.

    Compilers _translate_ high level _languages_ into machine _language_.

    Useful routines are stored in _libraries_.

    The list is pervasive, and all the features of computer software tie back to
    _language_ theory far more than math. If the law would stop listening to the
    people trying to justify things for business purposes and just start listening
    to the actual words we use, then the truth would just sort-of fall into place.

    Software involves no invention, no discovery, not unless Margaret Mitchell
    _discovered_ "Gone with the Wind." And Software shouldn't be
    patentable unless Ian Flemming deserves a patent on the spy novel.

    The fundamental confabulation is that people insist on viewing
    function(software) as if it means function(math) instead of

    The first thing you must know and understand about software is that it is all a
    complete _fiction_. The letter "A" is, by convention, by standard,
    just the number 65, and "B" is 66, and so on. The fact that I can get
    "B" by doing "A" + 1 doesn't imbue "A" with

    Consider you Windows or Mac Desktop. The key word is "Desktop". A
    _metaphor_, just like "file" and "trash can". Email is just
    mail. Text Messages are just messages. Records are records.

    In fact, everything you can do in/with a computer you can already do on paper.
    Draw a picture? yep. Store medical records? yep. Bill a credit card? yep.

    None of the "computer theory" matters.

    Computer chips, and computers in general, are endowed by their creators with a
    finite and fully defined list of instructions. Everything the computer can do,
    it can do intrinsically.

    In the world of Reducto ad Absurdum, All computer chips either do electrical
    "and", "or", and "not" (usually as
    "and-not" or "or-not" chips since you don't need all three
    in any one theoretical design.)

    At the machine code level instructions are all verbs like add, move, exchange,
    compare. Each instruction takes a number of source and destination things, that
    is "nouns". Every useful instruction sequence is a sentence, or in the
    terms of the art "a statement".

    Higher level languages are just more-vague instructions that the compiler turns
    into very specific sequences. 'print("Hello world");' is put the
    address of the "H" into a register, move what that register points to
    into a temporary store, if it is zero then stop, otherwise cram that temporary
    store value into the output place, move the pointer to the next address, go back
    to the move instruction.

    Words. All words. Word salad. Creative use of those words is why two people can
    "write the same program" and end up with completely different results
    that run in different amounts of time. There are good programmers and bad, just
    as there are good authors and bad.

    The reason that software patents are bad is that they _don't_ describe the
    software, to come close a software patent would have to be at least as detailed
    as Cliffs Notes for a novel. But just as Cliffs Notes are not enough to really
    be the book, a notes version of a program still wouldn't even really describe
    the intricacy necessary to function. After all if you handed the Cliffs Notes of
    Gone with the Wind to dozens of authors you'd get dozens of different things,
    none of them the original, probably none of them even close.

    The reason that almost every programmer on the planet goes crazy when a software
    patent is issued is simple. We keep telling you. They are taking things from the
    real world, adding on the fact that they are doing it "in ow with a
    computer" and claiming it is a new thing.

    Contracts are not patentable. Novels are not patentable. Short stories are not
    patentable. Software is inherently not patentable.

    I draw a picture of cake on the screen, the cake is a lie, do I get a patent on

    [ Reply to This | # ]

    Editing Suggestion
    Authored by: m_si_M on Friday, November 13 2009 @ 12:24 AM EST
    Since it can't be ruled out that the article will be quoted in a legal document,
    I suggest the use of uppercase letters for all terms of art, like e.g. Effective

    As we have learned, some lawyers love to use quotes out of context. The use of
    uppercase letters would indicate that the meaning of the term is not susceptible
    to legal interpretation, but is clearly and narrowly defined in mathematics.

    [ Reply to This | # ]

    Hey, a bit of help from you maths geeks, please.
    Authored by: Ian Al on Friday, November 13 2009 @ 04:57 AM EST
    I need some help from you set theory geeks to solve a practical problem of data

    The universe in which the sets exist is not homogenious. It comprises serial
    domains of addressable groupings of binary, fixed length datum repositories. The
    boundaries of the domains have no relationship with the data stored. Don't worry
    about the data. It is symbolic in nature comprising binary words of length
    compatible with the granularity of the storage universe. The symbols could
    represent anything including other symbolsets.

    The sets are all special cases. There can be no unions between sets. The datum
    in the data sets must be kept in single file order to maintain data integrity so
    I will call these 'filesets'. There is just one superset per universe which
    contains pointers to complete filesets. I will call this superset the 'Fileset
    Access Table' for that is its function. For the contents of this table to be
    easily found, it is confined to a set of universe addresses which is
    standardised and cannot be used by filesets. It is essential that filesets can
    be added to, and copied and removed from the universe. This is the purpose of
    the FAT.

    I want the filesets to be distinquished by a naming syntax because real people
    will need to select specific filesets of data. To start with a modest number of
    filesets I am defining the naming syntax as a series of eight alphanumeric
    characters followed by a '.'. Three subsequent characters are added as a guide
    to which symbol set is used within the fileset. Because of the nature of the
    universe in which the FAT is stored, I propose to represent the fileset names by
    binary representations of the ASCII character values. We might as well stick to
    some sort of standard, here.

    OK, that's about it. Any ideas? When we have come up with the optimum workable
    solution this will be a handy addition to the storage of and access to symbolic
    data sets. Since it is all completely defined by your set theory and algorithms
    it should be easy to get a patent for it.

    Ian Al

    Linux: Viri can't hear you in free space.

    [ Reply to This | # ]

    Patenting a book
    Authored by: Anonymous on Friday, November 13 2009 @ 03:13 PM EST
    Maybe this will open some eyes.

    1)Software is patentable.
    2)Binary software files are a(large) number.
    3)Numbers can be encoded to correspond to symbols (Gödel).
    4)Books consist of symbols.
    5)Books are patentable.


    [ Reply to This | # ]

    Simple Rules For Understanding Patent Law
    Authored by: Anonymous on Friday, November 13 2009 @ 05:09 PM EST
    1) Whoever has the gold makes the rules.
    2) Whoever makes the rules can get the gold.

    Sad but true.

    I don't think fine points of mathematics are at issue here. Look at the reasons
    given for excluding basic things like laws of nature or mathematical truths. It
    is about how patenting them would be too broad, give too great a monopoly. It
    is their size and scope which led to the rule, nothing special about

    If there is any hope for reason or common sense I think the path for getting
    software out of patents is the observation of how they form such an impenetrable
    knot. Anyone actually doing work in software creates new things comparable to
    what the patent office is handing out patents on every day. Any useful bit of
    software steps on these patents unknown and at great frequency. Society is not
    gaining from disclosure, only creating a land grab. These are practical
    arguments unlikely to be aided by mathematics. That isn't to say all that
    theory from mathematics isn't interesting and useful. It surely is, but not I
    think to guiding policy.

    Once again we face a situation where Congress does not give us clear law. We
    have enormously important policy decisions belonging to the proper
    interpretation of the word "process". Congress should do that job,
    not the Court. However, if Congress does not then the Court must.

    My own view is patents are fundamentally flawed right at the core and this flaw
    extends to all fields of work. The flaw is preemption. It is not necessary to
    copy a patent to infringe it. You can break the law doing your own invention,
    your own original work. All it takes is someone getting to the patent office
    first. It is fundamentally unjust.

    I am aware of all the arguments about how it may be impractical to recognize
    independent work or how things become obvious once known. It is a point of
    human judgement to weigh these things. I regard them as inconsequential
    compared to the fundamental injustice of denying people the fruits of their own
    work. That is the ability to use one's own work, freedom as Richard Stallman
    would explain, as opposed to the power to forbid anyone else from using it.

    A just patent needs to pass the test of "We would not be able to do this
    had the inventor not taught us." Pass that test and patents don't take
    away from the public. Ignore that test and they are as close to extortion as
    two peas in a pod.

    [ Reply to This | # ]

    Lists of instructions
    Authored by: Jeays on Friday, November 13 2009 @ 09:54 PM EST
    At the risk of jumping in where angels fear to tread - software is just a list of instructions.

    As in a recipe - a list of instructions for making a given food item. They can be followed by a human being, and one could imagine a machine which would cook a meal given a recipe and a stock of ingredients. The machine might well be patentable, but not the recipes.

    As in a Jacquard loom, which has a list of instructions on punched cards for describing the pattern to be woven.

    As in sheet music - a list of instructions for what notes are to be played at what times. A human being can play the music, and one could imagine a machine to do the job, if it hasn't already been built. You could patent the machine, but would be limited to copyright to protect the sheet music.

    As in a CD player - the CDs hold a list of instructions about what sounds to make. Trying to patent a CD player that contains a CD for playing the "Bolero" as a "Bolero-playing-machine", and claiming this as a new machine, would stretch the definition of 'new machine' well beyond breaking point. It is the same machine, with a particular list of instructions.

    As in biology - DNA holds a digital description of complex molecules that are to be constructed. Genes are just data, and it is hard to see how they can be the subject of patents.

    As in software - a list of instructions for a computer to execute. There is something quite novel here, because a computer can treat its own list of instructions as data, and modify them, or write new ones. Therein arises much mathematical complexity.

    As in software for a new machine in the not-so-distant future - a universal fabricator. We already have limited examples, such as the 3-D printers that can make arbitrary objects out of plastics of some kind. It would be quite feasible to build a machine that can assemble arbitrary Lego models according to an externally-supplied description. Once such machines can make any object, including complete copies of themselves, the world will change dramatically. John von Neumann proved that this is possible, and the proof is based on the separation of the part that does the fabrication, and the list of instructions it requires. The list is used twice; once when it is executed, and again when a copy is made for the new machine. What universal fabricators will do for patent law is an interesting question.

    So to conclude - the paradigm of a list of instructions that can be executed either by a human being or a machine has been around for a long time, and is a fundamental idea. The machines may be patentable, but lists of instructions should only be protected by copyright.

    [ Reply to This | # ]

    An Explanation of Computation Theory for Lawyers
    Authored by: Anonymous on Saturday, November 14 2009 @ 12:19 AM EST
    From Linux Today

    For those that want to get to the nontech component of the argument, skip to
    "Software is Mathematics" near the end.


    The computer is a physical contraption that allows us to see the net effects of
    the software/math (calculations and also side-effects like the painting of
    pixels on a screen). The computer has parts that are patented. Software cannot
    be patented.

    The hope of most software patent supporters is that loading up a computer with
    math and executing those instructions (including the driving of the visuals and
    sounds from connected peripheral devices on cue from the software/math) can be
    patented by way of calling these effects (that result from such software) a

    Thankfully, it's possible that the SCOTUS will not buy that line of reasoning
    (and instead use a stricter definition of "process" within patent
    law), though there are many sticky issues they face.

    Note a key distinction that PoIR tries to explain. Math is used to analyze and
    come up with physical contraptions that work. Many of these contraptions are
    patentable subject matter. The computer is one such contraption. There is much
    physics and math that is leveraged in building a computer that works.

    However, what we have with software is different. We are actually feeding math
    into the contraption. Software patent supporters want to patent this as a
    process: the physical beeps and blinks that can be cued directly by the math as
    the math is being evaluated automatically by a machine (eg, by a PC).

    The things the software patent supporters want to patent are the virtual
    creations that such an organized math allows us to perceive and utilize. They
    want to patent the abstract model components (like files and virtual desktops
    and "security systems"...) simply because things like a PC allow us to
    "touch and see" these abstractions.

    That was quite an effort by PoIR, btw. I noticed this person complain numerous
    times how the viewpoint of patent attorney Gene Quinn and perhaps others was
    missing the target.

    -- Jose_X

    [ Reply to This | # ]

    Discovery vs. Invention
    Authored by: Anonymous on Saturday, November 14 2009 @ 09:02 AM EST
    Another way to look at this is when people say they have invented a new 'process' that's not strictly true; what has happened is they discovered a process and invented a machine to implement that process.

    This is very true in IC design, where we forever are in search of more speed or smaller geometries; what really gets invented (and patented) is the machinery/materials that make the discovered process possible.



    [ Reply to This | # ]

    An Explanation of Computation Theory for Lawyers
    Authored by: grundy on Saturday, November 14 2009 @ 09:01 PM EST

    An excellent, very interesting and informative article.

    One of the worst things about patents is that they are written in a language that is never used for any other purpose and does not closely resemble either English nor ordinary Legalese, which means that the audience for this essay must be the Judges, not the patent lawyers.

    I expect this essay could be even more powerful if it were edited by an English Major or professional copy editor with very little or no knowledge of mathematics.

    [ Reply to This | # ]

    Reasoned response Counter argument
    Authored by: Anonymous on Sunday, November 15 2009 @ 07:45 AM EST
    If I were a Judge readying and analysing your paper, would be required to read
    and understand that are your primary arguments, facts and definitions to support
    your claims.

    My background is electronics engineering, and microprocessor design and
    programming, working in the fields of radio communications, computer systems
    engineering, SCADA and scientific instrumentation design and manufacture. I buit
    my first microprocessor system in 1978, and a 4-bit binary adder in about 1972
    using type 3000 telecom relays.

    I have been formally trained in microprocessor and computer design and
    programming, I presently work my own business as an engineering consultant.

    I fully understand the application of mathematical abstracts to describe
    physical machines or to evaluate, reduce and analyse designs in the abstract.

    First each point, as briefly as possible.

    <blockquote> <b> All software is data.

    Here the Judges are going to require and seek a more dininitive statement, if
    you say "All software is data".
    They are going to seek the definition of software and the definition of Data,
    they are also going to turn it around and as the opposite question is "All
    data software".

    What is data?
    Data is information, usually expressed in some abstract human understandable
    form. Data can be how tall you are in CM's but that data that number is an
    abstract definition and a form of notation that can be related to how tall you
    are. It's is NOT actually how tall you are.

    If I say im 163cm tall, that is an abstract statement that is the definition of
    my height.
    "163" and "I Am" and "Tall" are all abstractions
    that you apply mentally to order and evaluate "Data".

    If someone says to you "Im 163", that has no meaning, you can make an
    assumption that it's something, but you would have to guess.
    You do not have a complete data set for you to gain any practical information.

    Not until you tie the abstract decimal number to your height in CM does it make
    any sense.
    This is the differnce and the definition of "Data" it's information
    that you can act upon.

    <b> What is Software? </b>

    Software is a series of configuration instructions that turn on or off various
    switches within the CPU, memory and I/O that run on sequential logic, and the
    switches that configure the machine are transistors.

    Software is the series of configuration instructions, that may or may not
    perform or interact with data. software and data are two clearly distinctive

    If you are required to count the number of cars that pass a certain point, you
    can use an abstract and put a line on a piece of paper everytime a car goes
    past, that mark is an abstract symbol representing a car passing, it's not the
    actual car, but it's Data (information) in an abstract form of marks on paper,
    the abstract notation you use could be Roman Numerals, decimal numbers, a pile
    rocks that you add too each time a car passes, but all these are abstracts of
    the actual data.

    If you had to measure the number of cars passing a point on the road but were
    not able to use any abstractions, like numbers or rocks or marks on paper when
    how would you do it ?

    You would have to gather each car, so that when your boss turned up at the end
    of the day and asked you how many cars passed. You would point to the pile of
    cars and say that many.

    Remembering you are not allowed to use abstractions, like defining a number
    system and incrementing each time a car pased.

    Everything that is converted to the abstract can be manipulated within the rules
    and definitions of the abstraction, but does speak of the initial physical
    information apart from it being a definition and abstraction of reality.

    Anything that is real is not an abstraction, the cars passing the point on the
    road is real, the number you write on the paper is an abstraction of that real

    So Software and Data have to seperate definition, software like anything
    physical can be converted into the abstract, and therefore undergo mathematical
    rigor and analysis and reduction.

    <blockquote><b>All software is discovered and not invented.

    This is a case of definitions and semantics, in definitions you are saying that
    a discovery and an invention are different things, and your right so you have to
    look at the definitions of "discovery" and "invention". A
    discovery is what ? it could be an observation, or the result of a measurement
    or expirement or a flash on intuition. It may be a usefull discovery or it may
    be a discover of a failure. You can discover that cheese does not taste that
    nice with peanut butter, or you may discover that a round rock rolls better than
    a squre one.
    The invention is in the taking of your discovery and making it into something
    unique and usefull, therefore a discovery can become an invention, but not all
    discoveries are able to be invented.

    In our world the difference between a discovery and an invention is if your
    discovery meets the innovation criterea for an invention.

    Discoving something that has allready been discovered is by definition not a
    discovery, even if done independly but AFTER the initial discovery. (it's the
    "no prize for second rule").

    When Edison invented the electric light he made many discoveries, and some that
    required inventions from other people. He tried many many different types of
    light falaments including horse hair, human hair, string by trying lots and lots
    of different things he found that a thin piece of wire glowed bright but would
    quickly burn out, a discovery was made what horse hair did not work, another
    discovery was made what thin wire worked but burn out in seconds.

    Edison then tried many other experiments along the way making variou discovies,
    sooner or later he placed the wire filament in a class chamber and tried
    different gasses, and pressures. He discovered that if you lowered the air
    pressure the light would last longer before it burnt out.
    So was the invention of the electric light a discovery or an invention?

    He did not invent electricity, or wires or glass or vacuum pumps, but he
    discovered that the right combination of these various components creates
    something unique and innovative the invention was born from a combination of
    discoveries and intuition, innovation, effort and the desire and effective

    The effective process in this case was to product a working light globe, this
    light clobe is subject to abstraction and analysis just like any other physical
    object or invention. It's not the underlying math or abstract notation that is
    patented it's the "proof" and the real aspect of what you have defined
    in the abstract.

    If you define something in the abstract it is and always will be an abstraction
    of the real.

    Software is REAL it's not in the abstract, software in configuration
    instructions that tell the CPU how to configure itself to perform a predetined

    <blockquote><b> All software is

    NO, software is the realy physical configuration information thats sets the
    configuration of the CPU due to predermined rules.

    It's the turning on and off of specific transistors within the memory and CPU to
    define a state. This is real, and by definition not an abstraction.

    Humans make abstractions, they replace a turned on transistor as a logical
    "1" and a turned off transistor as a logical "0". This is
    the first layer of abstraction.

    The the first time we step away from the real world to the abstract realm.
    1's and 0's do not move around the CPU or memory, physical switches open and
    close as a real representation of the abstract binary 1 or 0.

    It's a trivial for a human to convert into abstract terms, like symbols or
    numbers and to be able to derive rules and methods around those abstractions. In
    fast is so natural that we dont even think about it.

    But it's something we do all the time, convert to the abstract and assign
    symbols and rule to that abstraction.

    What is not an abstraction is something that is real, a car passing a point on
    the road you have to count is real, the car really did go past, you mentally
    convert that into the abstract when you assign a symbol to it and put a mark on
    a piece of paper, or hold a number in your head.

    Software is the REAL method of configuring a system to deal real functions /
    configurations on real information that is represented in the abstract form of a

    So your computer instruction (software) might be to say "every time a car
    goes past, put a mark on this piece of paper".

    this is a very low level of abstraction but it's IS an abstraction, the
    instruction "put a mark each time a car goes past" in an instruction,
    and relates to a physical act and a configuration, that configuration is the
    mark you make, that mark is an abstraction for a car that passes.

    The instruction is real, the mark is real but the mark is an abstraction for a
    car, and is data. The instruction is not an abstract as you really have to
    perform the instruction in the physical sence to perform the method.

    With knowing the exact definition of the instruction a pure number returned
    without an explination of the real aspect of the pure number does not take it
    out of the abstract into the real as we do not know what the question was

    If you say count the number of cars in a day, and you give him a pure number
    like 200 he may make assumptions that that numbers means there was 200 cars went
    past, but it might not, there might have been more, some might of been trucks,
    or he may of counted every second car so you would need to apply other more data
    or information to resolve.

    This can be done by carfully defining the question so there is no ambiguaty,
    with is using an "effective method" and being carefull to define your
    method of abstraction clearly. "EVERY time a car passes put a mark on this

    He could of said, "create a number system, and everytime a car passes the
    point increment your number system by one, and the end of the day explain the
    rules of your number system and we can evalute from the data how many cars

    But this requires incite on behalf of the human, and therefore fails the 4th law
    of your effective method rule.

    <blockquote><b> All software is

    It is true that mathematics can be used to convert software into the abstract,
    as you can convert car going past a point on the road into the abstract realm of
    math. Does not mean the cars are not real, but it does mean that your abstract
    number of cars or your abstract marks on paper do not represent something
    equally as real as the cars passing on the road.

    Everything from calling an ON transistor on a computer a "1" and an
    off transistor as "0" up to using the most high level programming
    language is an abstraction of the real functions a CPU and memory perform.

    To actually physically perform any operations on a general purpose computers,
    CPU and RAM/IO combination is to break the abstract definitions of the
    instructions and methods into REAL configuration information to allow the
    CPU/Memory to perform is predefined task in the real world, not in the abstract
    layers and world that humans have created to more easily understand and
    configure these specific sequential state machines. (computers).

    If you analyse the exact sequence of events of even the most simple of computer
    programs, you see no abstraction at all. You see a sequence of specific
    configurations that perform a predetermined function.

    The CPU has no concept of data, or memory or function it simply configures it's
    system to meet a predefined configuration, that has been designed to meet a
    basic physical function.

    2 + 4 is an abstract concept, bound by the rules of math and the decimal number
    system. The numbers "2" and "4" are an abstract for
    "something". It could me 2 cows or 2 trees or just 2 as a pure number,
    but its a human conceived abstract where the symbol "2" means
    "two of something" or just "TWO".

    Even saying "two cows" is an abstraction, it's abstract words
    describing a number of objects. (and a brand of beer).

    saying "within that memory location is the value 2"

    This is another abstraction, if you look in that memory location you wont see
    the symbol or value 2, if you put a meter of the bus or in the chip and looked
    at the voltages on each of the data lines you would read

    0v 0v 0v 0v 0v 0v 5v 0v

    so the CPU and has turned on all but one of the transistors in that memory

    we make our first abstraction and call 0v a logic ZERO and the 5V (on transistor
    in the configuration) and a Logic one.

    So our first level of abstraction is the definition of an on transistor on a
    specific bit at a specific memory location is turned ON, and that from that
    transistor being turned on you make the abstraction that that byte of data at
    that memory location on which that bit is turned on represents the decimal value
    of "2".

    This is your abstraction, the computer, memory and CPU do not work in the
    abstrat but in the real and physical.

    All software is data.
    (Data is gathered, and can be an abstract representation of a physical event, it
    can be the number of cars that pass a point in the road, or the current
    temperature but it is abstract in nature, the Nuber 34Deg C is an abstract of
    the a temperate expressed in specific units. It's DATA.

    Software is the real and specific sequence of configuration instructions that
    allow the manipulation of information by means of it's absolute configurations
    and the specific order of the software instructions, and the specific
    configuration of the hardware.

    All software is discovered and not invented.
    There are clear distinctions between discovery and invention, you can invent
    discoveries if they are new and unique and practical and meet the requirements
    for patents.
    There is no distinction between how you invent something regardless of if it's
    from invention or discovery.

    A discovery need not lead to an invention, and a discovery does not preclude
    something from not being an invention.

    An invention is the application of an idea or discovery in a new and practical


    You then go on in great detail to explain first how hard computation theory is
    and how you base your argument that high levels of abstraction, and mathematical
    analysis including Lambda calculus is showing how software is math.

    We have allready established that math is an abstraction, you even say so
    So the fact you can break down algorithms into effective methods that can run on
    a turing machine (another abstraction) does not make the actual software or
    function of the algorithms any less real.

    All software is abstract.
    Software is real, it is an exact set of instructions that direct the CPU and
    memory to enter specific states, those states are used in combination and in
    sequence to perform the operation of the function.

    This series of instructions is what turns a general purpose device into a
    special purpose device.

    Being able to convert that real set of instructions into abstract concepts and
    apply abstract rules to those abstracts does not make the original system itself
    abstract, the real configuration information that is called software exists in
    the phycials world in the form of turned on and OFF transistors.

    High level and any level of programming a computer that does not directly
    involve the configuration settings are an abstraction that has to be converted
    into real configuration information in software (also real) for it to function.

    A computer does not have 1's and 0's running through it, it has states or ON and
    OFF transistors

    "All software is mathematics. "

    All mathematics is an abstract representation of some physical or pure entity,
    the number / symbol "2" is an abstract for the number two, the word
    "number" is an abstract for a quantity of something.

    These are all abstractions.

    if you say you have 4 apples on a table, and you really do have 4 real apples
    on the table, Saying "I have 4 applies on a table" is an abstraction,
    the number 4 is an abstract for the value "four of something" and is a
    description of something physical, there is another abstract assigned to
    "apple" the word apple is not a REAL apple it's an abstract definition
    of an apple.

    Software is in itself not abstract but the definition and methods of
    development, construction and design can be performed in the abstract world to
    achieve the non-abstract (ie Real) sequence of configuration instructions that
    make up a software program or the set of instructions that make up an

    <b> Inventing and patenting a new bridge design </b>

    Consider you make a discovery and with inventive and innovative initive and by
    your own logical reasoning you invent the design for a new type of bridge, one
    that is much lighter and cheaper to make, and can span any distance.

    You can use pencil and paper and by using language, math and diagrams and
    intuition you can create an abstract model of your bridge, you can then apply
    the laws of physics, and the abstract of math the analyise your design and
    invention, you may invent specific construction methods or design methods that
    are unique and could be patented.

    But you cannot build the bridge, and take the patent office to the bridge and
    tell them to allow a patent on your invention.
    You have to transform your design and invention from the physical to the
    abstract, you have to reduct your design to math and abstract terms and
    descriptions including possibly diagrams.

    You may have to take the results of your abstract analysis and build test models
    to ensure your abstract design and analysis is real world correct.

    Or you may be able to use the abstract rules of analysis to refine and improve
    your initial desgn.

    But applying analysis, and abstract idea's to a real phycial object does not
    change the fact that the object of you're abstract analysis is no less real.
    Or it does not exist in the physical world.

    The only thing that does not live in the physical world is what lives in the
    abstract, and the abstract can represent the physical or it can be pure

    So yes, software or a bridge can be expressed in abstract form, but in
    themselves they are not abstract.

    Abstractions are what humans need to do to make thigs more understandable and to
    allow manipulation of idea's and Data/information in specific ways.
    But those abstractions are just that abstract, the number / symbol "5"
    does mean anything real or abstract.

    computers do not deal in the abstract, or in numbers, they deal is specific and
    exact known states and configurations.

    All abstractions are to simplify the understanding and configurations in manner
    more easily understood my humans.

    Our highest levels of abstraction is human intuition, your own mind is very good
    and abstraction it does it without effort and automatically. Our own speech in
    an abstraction of the real world, if you hear the word "tree" in
    instantly relate that abstract of the word "tree" to a real item of a

    But the word "tree" and the mental picture in your mind of a tree is
    an abstraction and may or may not represent a REAL tree. Dont have enough
    information to make that establishment between the abstract and the real.

    Computers cannot handle abstractions, and software is not therefore by an
    definition abstract.

    The actual code is absolutely real, physical configuration information used by
    specific CPU and memory configuration to perform a special function. Turning a
    general purpose computer or machine into a special purpose devive that is
    special purpose based in the unique configuration and sequence information.

    Not even the abstract of pure numbers or binary numbers are known to computers,
    there are no 1's and 0's running in your CPU, that is the first layer of
    abstraction humans attribute to the real as a means of easier understanding.

    A programming language like assemby or C or fortran are further levels of
    abstraction further isolating you from the underlying real instruction sequence
    that is necessary to make your CPU and memory perform a specific function.

    The C programming language is a layer of abstraction it allows humans to more
    easily manipulate the configuration information that are directed to the CPU and
    memory, and as another layer of abstraction makes the process of defining and
    creating and documentating your method.

    But it does not change the underlying innovation in your orignal design.

    If I design a device that uses combinational logic (AND, NOR, OR gates and so
    on) that performs a specific function, and that function is new and innovative
    and ableto be patented. If I perform a conversion from the real circuit and
    create that circuit (before I build it) in the abstract, I might start by
    drawing a circuit diagram of my design, this is an abstract definition of a real
    physical device.

    Once i have the design and abstract circut worked out, I might apply a set of
    equations to my design I might use a form of calculus and math to define my
    design in the abstract, then I might raise the level of abstraction and convert
    my circuit diagram full of logic gates to boolean truth tables and boolean
    expressions a further level of abstraction.

    Then I might use my knowledge of the abstract boolean algebra to reduct my
    design down to the minimum number of gates and still achive my function, (AND NO

    So you have invented a ciruct that does a function, you have converted it into
    the abstract of a circuit diagram, and mathematical equations, you have used the
    abstract math and circuit diagram to refine your invention but maintain it's

    Then you can take your abstract design, and result of your analysis and
    reduction and build a real physical device.

    This process is exactly the same is the invention of a method or algorithm, or
    in the invention and creation of software.

    It does not matter how you achieve the end result whether by abstraction,
    invention or discovery, what matters is the real and physical end result or
    product/innovation/invention that is what you consider as the final
    "proof" in your analysis.

    The analysis of an abstract does not make the physical and real item any less

    The method of achieving innovation has no reflection on the reality of that
    innovation or invention.

    If you invent something in the abstract or by using an abstract method it may be
    able to be converted to the real world.
    This is why C code instructions which are abstract representation and a more
    human readable form of abstract instructions can be compiled on different CPU's
    and memory configurations.

    The final compiled code for each system is differnt as the CPU's and memory
    configuration are different, the same real and raw computer configuration
    instructions for one CPU are different from another.

    But the high level abstract definition of the algorithm is abstract and not
    real, it has to be converted into the very specific and real configuration
    instructions for the particular CPU and memory, to allow the CPU to enter a
    specific configuration, with certain transistors either turned on or off.

    Real computers do not work in the abstract, the abstract has to be converted
    into absolute and precise configuration which means the exact combinition of ON
    transistors and OFF transistors, to create specific real voltage levels.

    Our first level of abstraction is in defining the presence of an ON transistor
    and the 5 volts on the bus to represent the abstract of the binary logic level
    The highest levels of abstraction moving AWAY from REAl is assembly, first gen
    programming languages, second gen programming languages, and may evolve to
    levels on abstraction where you can just tell your computer to calculte the
    following two numbers.

    But before a computer can actually perform any operation it has to take those
    abstracts and convert them to real configuration / information settings and the
    correct sequence of turning ON and OFF transistors to achieve a specific

    lets look at a very simple computer program (software) to add two numbers, oh
    sorry a number is an abstract term and a computer has no concept of the
    So in computer terms what do we need to do to achieve a calculation of 2 + 3.

    Start with understanding the term 2+3 is a pure abstract as numbers themselves
    are abstract and can be a representation of a physical entity or not if it
    represents a physical entity it can be called data.

    So you want your cpu to give you the result of the abstract calculation of 2+3.

    The computer does not have any concept of the symbols "2" or
    "3" and "+" or that the term 2+3 means "take the
    decimal abstract number 2 and "add" it do the decimal abstract number
    3 and return the result".

    The computer deals with these abstract concepts by converting it into
    "real" objects, such as a transistor being ON or OFF.

    So well use abstraction (are using abstractions) here in our description.

    Lets look at how a simple CPU performs our abstract equation.

    Well make it a bit simplier and assum a level of abstraction of say an ON
    trasisitor and 5 volts on a bus represents a logic 1 and 0volts and transistor
    OFF represents a logic zero. (keeping in mind this is also an abstraction).

    There is no instruction for a CPU or computer that says "add 2 and 3
    together and return the result of that calculation".
    Computer have no concept of the abstracts of "numbers" just as they
    dont can any concept of "words". they only know how to set up it's
    internal configuration based on a specific set and rules that it performes in a
    sequence of events.

    So now you have to invent and design a method from going from the abstract of
    math to the real of transistors beign turned on and off and in translating that
    abstract however methods are employed is the basis for real computer programs
    derived from the abstract and theoretical.

    <b> create a computer program what add 2 and 3 abstract numbers and return
    a result </>

    First: you need to create an abstract model of what you are trying to achive,
    then you have to realise that abstrct concept into real terms that your CPU can
    configure itself to achieve.

    The invention of your design is how you go about achieving your function or
    algorithm within the confines of the functions you have.

    So you have a cpu and some memory connected to it. you have to create a set of
    configuration information that if performed in the correct sequence with result
    in performing your calculation.


    so in a location of your available memory you set up a configuration of ON and
    OFF transistors that represents the binary abstract word of 0000 0010 with for
    you represents the abstract concept of the decimal number 2 as defined by the
    abstract symbol "2".

    so there are two layers of abstraction from the real ON/OFF transistors and the
    absence or presence of 5 Volts and you abstract to a logical "1".
    which is a real binary value and in our case we use that logical 1 ON transistor
    presence of 5V to on the second bit of the 8 bit word at a specific memory
    location to mean the Decimal symbol abstract of "2".

    Next to place the configuration of On/Off transistors in another specific memory
    location the abstract binary equavilent to the Decimal value abstract number
    (again just a combination or configuration of on/off transistors).

    Remember (2+3), ok now we have to perform the abstact function of addition, what
    we are doing is not addition at all, it's the logical AND'ing of the bits in one
    specific memory location with another specific memory location providing the
    results in a third location.

    The CPU and memory have no concept of "numbers" or
    "addition" or math, it simple performs a dumb operation according to
    specific instructions.

    Now the contents of the 2 memory locations (dec 2 and dec 3 in their abstract

    so what you tell the CPU to do is not an instruction saying "add the two
    number together" that is a human abstraction and not within the realm of
    understanding by a computer.

    So now you have the two "numbers" fixed in specific locations within
    memory, you now have to "add" the two numbers, as numbers and ADD does
    mean anything to the memory, and numbers mean nothing to the CPU and ALU
    (arithemic and logic unit).

    you instruct the CPU in very specific terms to enter a preset configuration, you
    may state (in form of a machine instruction whish happens to be in the same for
    as the "data" in memory).

    CPU place the binary word in the form of voltages or not voltages that addresses
    the location of the memory where the data I have to manipulate is stored.

    This tells the CPU the output a binary word of a specific value on the address
    bus, in fact it's a specific configuration the CPU can enter. (and usually
    derived by a circuit called a program Counter.

    But before the computer can manipulate data that represents the number 2, it has
    to know what to do and exactly how to do it.

    This is what computer code, or software is and does.

    the computer will start and will jump to a predefined start vector, when means
    the address bus will set up a specific address, and at that address will be the
    first instruction of your program. (software not data).

    The software or instruction is encoded in a form that the CPU can understand, in
    this case it's a binary word that is a state of memory, and again transistors on
    or OFF.

    The CPU when started will read this software, this instruction and configure
    itself based on it, it's not data, it's a clear instruction to do a specific

    So the CPU start with our program to add 2 and 3

    CPU starts, CPU output a specific address word on the adress bus, CPU initiates
    a "read" command to the read/write line of the memory and the CPU
    reads the instruction as a binary word that represents a specific configuration,
    in our case we want to increment the address bus by one and load the contents of
    that address location into the a register in the CPU, or the accumulator.

    the abstract definition of the instruction might me

    MOV 0002,A ; move contents of memory location to the accumulator register.

    still the CPU and memory have no concept of what is "in" that memory
    location, but in our cast it represents the abstract decimal number
    "2". but that is our abstraction a human abstraction, and the memory
    and CPU cannot deal in abstractions.

    next the cpu moves to the next memory location, and it's knows within that
    location there is a specific instruction the CPU knows it's an instruction
    because that is what it's expecting, if it's the wrong instruction the program
    will crash, thats how computers work, it performs an instruction completes that
    instruction and then looks for or waits to be told to perform another

    so you have the number "2" abstraction configured in the accumulator
    register of the ALU within the CPU you have set the CPU into a specific
    configuration. Now the CPU clock will cycle, and the program counter will
    increment, as it just performed a process on data and completed that
    configuration exercise, it seeks out it's next instruction, the PC or program
    counter is incremented by the clock and the address bus value will be
    incremented by one, the data bus returns another binary word, and the CPU reads
    that word as an instruction.

    In our case it may be "output the address for the memory location of the
    second 'number' is located and connect that data from the data bus via logical
    AND gates to the contents of the accumulator and leave the result of that
    function in the accumulator."

    IF the memory location of the accumulator and the corresponding memory location
    contain your abstract of the number "2" and if the memory location you
    have defined for your abstract number "3" are present and correct and
    IF the CPU and memory have taken that configuration information and performed
    exactly the tasks you have asked it to do you will achieve the result of your
    abstract equation. But you have had to do this by converting your abstract
    concepts into real concepts.

    you also cannot reduct the real, you need to convert the design or idea into
    abstract idea's that humans can understand and manipulate as required.

    All the Lambda calculus or other levels of abstraction still need to be reduced
    into real world functions to be completed in the same real world.

    Doing something in the abstract does not make it real, doing something real can
    be readily converted into the abstract and back to real. but to do actual work
    you need to operate in the realm of real and not abstract.

    Inventions and concepts and idea's can be represented in the abstract, in terms
    of functions, numbers, notation, words diagrams, math and functions, in the
    abstract you can manipulate those abstracts as you see fit or guided by specific
    rule, samantics, logics, words and diagrams.
    These are all abstract tools, to design, model and analyse the real in abstract

    To to analyse the abstract in abstract terms.

    "All software is data"

    No data is data, software is specific configuration information and data is
    gathered information that may be an abstraction of a real or an abstraction of
    an abstraction.

    the Number "2" is an abstraction, just as the value E in E=MC2 is an
    abstraction or Energy in jules it could just as well be an abstraction for
    "Eggs" and M might me an abstraction for "mice" but that is
    a different abstraction to what is commonly know what E=MC2 is and means.

    software can be expressed in the abstract, and high level programming languages
    are just that, they are abstract languages that make it easier for humans to
    apply these abstracts to end up with a real and working function.

    again, the highest level of abstraction is either spoken or written plain
    english or your native language, or the universal language of mathematics, but
    in either case they are stall abstract concepts that are human aids, that have
    to be converted from the abstract to the real and practical to be real and

    you can program a computer without the aid of abstraction, but you have to work
    on the real level of the computer/hardware/software to achieve real results.

    Or can you can program a computer using abstraction and rely on the standards
    methods of de-abstractification that you can apply automatically or manually to
    achieve your function.

    Ie, you can use a high level programming language to create your sequential
    configuration information for your CPU and memory, but as some point the high
    level language has to be converted into non-abstract specific configuration
    information for your CPU and memory.

    [ Reply to This | # ]

    Obviousness test silly; also there has to be a better way
    Authored by: Anonymous on Monday, November 16 2009 @ 02:29 AM EST
    Using an "obvious" standard is ridiculous when you consider how many
    very complex analytical things get produced yearly by mathematicians and by many
    others (yes, it takes "ingenuity" many times to formulate useful
    theorems and to prove them). Are poems patentable? These can involve tremendous
    amounts of creativity as well as knowledge of and command over a given language.
    How about a well crafted piece of law or musical score or play script or film?
    Do achieving the latter not require lots of hard work and skill? Aren't many
    nonobvious things created daily in these fields?

    Many things that aren't obvious can be discovered after days of work and
    certainly wouldn't be considered "brilliant" by many. And many things
    that are brilliant aren't patentable, and if they were patentable, many might
    conclude 20 years was way too long.

    I think the novel and nonobvious test is a bit.. misguided.

    Are we giving monopolies, that is, abridging the rights of everyone on the
    planet and destroying a potentially ridiculously gargantuan amount of
    collaboration and further discovery, because someone concluded something
    brilliant (or simply nonobvious) and filed the paperwork first? Is stopping
    everyone else (including many who likely helped lead to that "spark")
    for twenty years any way to promote the progress of science and useful arts? ..
    and to do so when the described "brilliant" (aka nonobvious) invention
    is described in claims so incredibly broad?

    Do patents serve any use that helps society (after opportunity costs and
    abridgment of rights are factored in)?

    Aren't there alternative ways to promote innovation that do less damage?

    If we aim to protect the "little guy", can't we simply condition
    patents that way (ie, they can only be used to guarantee a "little
    guy" some market share against a "big guy").

    Can't we give tax credits or other monetary grants as a reward for someone
    taking significant risks (with money) trying to innovate? Surely, getting extra
    cash and having to pay less in taxes should help drive investors to the company
    (not that "driving investors" should be the ultimate goal). And if
    this isn't enough, example because of monopoly abuses, then doesn't it suggest
    that some other laws are broken elsewhere and perhaps those should be fixed?

    Do we really want to encourage an extravagant wasting of resources simply
    because patents are so powerful that they can allow huge cost to be recouped and
    huge profits to be made? Open source development has shown that lots of R&D,
    manufacturing, distribution, and other costs can be very low and/or otherwise
    distributed among many entities voluntarily cooperating, perhaps off the
    promises in various copyright licenses.

    -- J

    [ Reply to This | # ]

    An Explanation of Computation Theory for Lawyers
    Authored by: CraigG on Monday, November 16 2009 @ 03:05 AM EST
    Portions of this seem to prove rather too much. In particular, with symbolic systems we need to distinguish between denumerable and non-denumerable infinities; basically the difference between integers and reals. We can in principle sort every possible integer into a strict order, though the list is infinitely long. We can't do the same with reals, because for every two putatively adjacent values in the ordering, there exists a value between them (an infinite number of values, in fact).

    Now, the number of potential well-formed sentences in any language is non-denumerably infinite. I would expect the same thing to be true of mathematical theorems.

    But in any case it is true of computer programs. The number of possible novels, for example, is non-denumerably infinite, and I can write a program

    printf(%s, "It was a dark and stormy night. Cobol the butler was roused from the pantry by a knock ...")

    So I'm not sure what the whole enumeration and discovery vs. creation business means.

    [ Reply to This | # ]

    Positive Integers vs Natural Numbers ...
    Authored by: kjhambrick on Monday, November 16 2009 @ 08:13 AM EST
    PolR --

    A question ... You say:

    Kurt Gödel was working on the Hilbert program. He was working on a formal system called "Peano arithmetic" that defines the arithmetic of positive integers. This is the numbers 0, 1, 2, 3 ....
    Is not {0,1,2,3,...} the set of Non-Negative Integers while {1,2,3,...} is the set of Positive Integers ?

    -- kjh

    [ Reply to This | # ]

    Error in Kurt Gödel's Recursive Functions
    Authored by: Anonymous on Monday, November 16 2009 @ 03:29 PM EST

    The explanation of recursive functions is flawed. Author gives the "two rules" for recusive functions, then shows an example with several steps. Step number six is:

    (0''''')''' = 0'''''''' because the parentheses can be removed at this point

    Yet there is no rule that tells you when parentheses can be removed, or even a rule explaining what parentheses mean.

    [ Reply to This | # ]

    Digitalization: Revolutionary Access
    Authored by: Anonymous on Tuesday, November 17 2009 @ 10:20 PM EST
    >> .. with the fact that every single change of state the CPU goes through
    is describable by a math function.

    A distinguishing characteristic is that the "state" of the CPU and of
    associated memory is precise to the highest degree. Digitalization has made the
    state of electronics modeled precisely. The machine fits the abstract model (the
    abstract machine) and vice-versa perfectly for all intents and purposes.

    This contrasts with the mathematical description of material objects which is
    much less precise. We have a limited knowledge and control over nature..

    ..but we have absolute control and knowledge of a computing machine, at least as
    far as analysis goes with respect to normal functioning (since all machines do
    break down).

    Digitalization means the all-important information (the "product") is
    the abstract software which can be transfered at lightning speed across the
    world at essentially zero costs.

    Truly, the Internet and the PC has given ordinary folks the ability to create
    significant products for mankind. This contrasts sharply to the wonderful work
    done by factories, as most people have very limited roles in factories since
    it's extremely costly to retool (reprogram if you will) a factory. Access to
    factory retooling is essentially denied to all but a handful. And the retooling
    is frequently very costly to enable.

    Yes, there is much on the line when it comes to software patents' overstepping
    and abridgment of rights and of newly-gained accesses by society.

    -- Jose_X

    [ Reply to This | # ]

    Patenting free markets away
    Authored by: Anonymous on Wednesday, November 18 2009 @ 11:16 AM EST
    By removing the incentives for most people to pursue an area (the area covered
    by the extremely broad patent), each new patent serves to slow down the pace of
    innovation faster and faster as more patents are taken out until a saturation
    point is reached. That saturation point depends in part on how aggressively the
    patent holders decide to use the patents.

    There are many inefficiencies in such a system -- obviously -- with the huge
    number of removed rights and incentives.

    For software, the opportunity costs are that much larger than say for an area
    related to industrial manufacturing because of the very large number of creative
    minds that could otherwise participate in practice (only in the software/PC

    Slowing down the pace of growth lowers the risks that incumbents will get
    displaced or seriously challenged or that their relative handful of employees
    will have to worry much at all about losing their jobs to the large number of
    (figuratively) hungry folks outside.

    And, sadly, the question should not be about losing a job among a lower number
    of jobs (something unlikely to happen if the job holder can be political). The
    big problem is that more jobs won't exist because of all the inefficiencies,
    loss of incentives, and many new illegalities.

    This model approaches the Communist approach (where the number of paths to
    success are very limited and in most cases there is a large number of people
    ahead of "you" in line for the positions).

    Too many removal of rights will lead to a lot more law-breaking, forcing the
    government to move in the direction of ruling with an iron fist or else allowing
    a mockery to be made of the law.

    And the Internet and computers will go severely underutilized (at least in
    relative terms).

    BTW, in the early stages, there will be a temptation to think that the patent
    system is creating incentives. But eventually, the chickens will come home to

    Well, gloom and doom aside, those that brought patents into the Constitution
    were careful and left a way out for all of us: [Article I, Section 8:] "To
    promote the progress of science and useful arts...."

    Hopefully, we will be able to identify what hurts us from what helps us.

    [ Reply to This | # ]

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