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An Open Response to the USPTO -- Physical Aspects of Mathematics
Sunday, September 26 2010 @ 10:32 PM EDT

An Open Response to the USPTO — Physical Aspects of Mathematics
by PolR, author of An Explanation of Computation Theory for Lawyers.

The USPTO has issued a request for comments on their new interim guidance, Interim Guidance for Determining Subject Matter Eligibility for Process Claims in View of Bilski v. Kappos [PDF]. They invited comment by Monday, September 27, from the public on three questions in particular:

1. What are examples of claims that do not meet the machine-or-transformation test but nevertheless remain patent-eligible because they do not recite an abstract idea?

2. What are examples of claims that meet the machine-or-transformation test but nevertheless are not patent-eligible because they recite an abstract idea?

3. The decision in Bilski suggested that it might be possible to "defin[e] a narrower category or class of patent applications that claim to instruct how business should be conducted," such that the category itself would be unpatentable as "an attempt to patent abstract ideas." Bilski slip op. at 12. Do any such "categories" exist? If so, how does the category itself represent an "attempt to patent abstract ideas?"

They ask in effect how to tell an abstract idea from an application of the idea. This article suggests answers to that question from the perspective of a computer professional.

The Supreme Court has reaffirmed the trilogy of cases Gottschalk v. Benson, Parker v. Flook and Diamond v. Diehr concerning the non-patentability of mathematical algorithms:
In Benson, the Court considered whether a patent application for an algorithm to convert binary-coded decimal numerals into pure binary code was a "process" under §101. 409 U. S., at 6467. The Court first explained that "'[a] principle, in the abstract, is a fundamental truth; an original cause; a motive; these cannot be patented, as no one can claim in either of them an exclusive right.'" Id., at 67 (quoting Le Roy, 14 How., at 175). The Court then held the application at issue was not a "process," but an unpatentable abstract idea. "It is conceded that one may not patent an idea. But in practical effect that would be the result if the formula for converting . . . numerals to pure binary numerals were patented in this case." 409 U. S., at 71. A contrary holding "would wholly pre-empt the mathematical formula and in practical effect would be a patent on the algorithm itself." Id., at 72.

In Flook, the Court considered the next logical step after Benson. The applicant there attempted to patent a procedure for monitoring the conditions during the catalytic conversion process in the petrochemical and oil-refining industries. The application's only innovation was reliance on a mathematical algorithm. 437 U. S., at 585586. Flook held the invention was not a patentable "process." The Court conceded the invention at issue, unlike the algorithm in Benson, had been limited so that it could still be freely used outside the petrochemical and oil-refining industries. 437 U. S., at 589590. Nevertheless, Flook rejected "[t]he notion that post-solution activity, no matter how conventional or obvious in itself, can transform an unpatentable principle into a patentable process." Id., at 590. The Court concluded that the process at issue there was "unpatentable under §101, not because it contain[ed] a mathematical algorithm as one component, but because once that algorithm [wa]s assumed to be within the prior art, the application, considered as a whole, contain[ed] no patentable invention." Id., at 594. As the Court later explained, Flook stands for the proposition that the prohibition against patenting abstract ideas "cannot be circumvented by attempting to limit the use of the formula to a particular technological environment" or adding "insignificant postsolution activity." Diehr, 450 U. S., at 191192.

Finally, in Diehr, the Court established a limitation on the principles articulated in Benson and Flook. The application in Diehr claimed a previously unknown method for "molding raw, uncured synthetic rubber into cured precision products," using a mathematical formula to complete some of its several steps by way of a computer. 450 U. S., at 177. Diehr explained that while an abstract idea, law of nature, or mathematical formula could not be patented, "an application of a law of nature or mathematical formula to a known structure or process may well be deserving of patent protection." Id., at 187. Diehr emphasized the need to consider the invention as a whole, rather than "dissect[ing] the claims into old and new elements and then . . . ignor[ing] the presence of the old elements in the analysis." Id., at 188. Finally, the Court concluded that because the claim was not "an attempt to patent a mathematical formula, but rather [was] an industrial process for the molding of rubber products," it fell within §101's patentable subject matter. Id., at 192-193.

In Benson and Flook the unpatentable algorithm is software. But it is recognized that an algorithm may take the form of hardware or software. In another case, In Re Alappat, the Federal Circuit had acknowledged the possibility that a patent on a circuit could read on a computer programmed with software and that this could make the algorithm unpatentable. They eventually ruled the circuit was patentable on the basis that they thought (among other things) the algorithm was not a mathematical algorithm in the sense of Benson, Flook and Diehr but this doesn't change the fact that they first considered the possibility of going the other way. (Also note the dissenting opinions in Alappat, which raised precisely the danger of patenting mathematical discoveries.)

The court in Diehr held an industrial process to cure rubber that uses an algorithm is patentable. This court made a distinction between the algorithm and a non-algorithmic process that comprises the algorithm as one of its elements. This article discusses a specific aspect of this issue. Could a patent on a circuit, as in Alappat, or a patent on physical computational activity, such as in Benson and Flook, be considered patents on an abstract idea? This is a notion that has a sound basis in mathematics. The purpose of this article is to explain this basis and show some possible problems that occur when one does not take it into consideration.

Physical Tools Are Necessary to the Practice of Mathematics

The relationship between mathematics and the physical world is often described, outside of the Supreme Court trilogy, with two notions:

  • Abstract ideas are disembodied thoughts.

  • Mathematics is often used as an abstract language to describe physical phenomena and laws of nature.

Taken together these ideas suggest that whenever we have a relationship between mathematics and the physical world, mathematics is always used to provide an abstract disembodied model describing the reality. This view, however, is not always accurate.

Take a pocket calculator. If one calculates the energy contained in matter with the formula E=mc2, does the formula describe the calculator? Of course not. The relationship goes the other way round. The calculator is a tool to do mathematics, not an application of the formula. I like to say the calculator is a physical model of the abstract mathematical calculation. Instead of doing the calculation in our head, we enter the numbers in the machine and watch what it does. The calculator shows us how the mathematics work.

The idea that mathematics can model the physical world is sometimes used to discount the use of mathematics in a patent claim. The argument is that it is not because an invention can be described mathematically that it is abstract and non-patentable, because otherwise everything is mathematics and nothing would be patentable. This is a fair point, but what if the modeling relationship goes the other way round? What if, like a pocket calculator, the physical entity is used to represent the mathematical abstraction? Isn't this the very issue raised by the Supreme Court trilogy? It is hard to see how the questions raised by Benson, Flook and Diehr can be answered without first getting an understanding of which direction the modeling relationship goes and what are the consequences.

So to summarize, when there is a relationship between mathematics and a physical entity this relationship may occur in two different ways:

1. Mathematics is used to provide a model of the physical reality. This is what happens, for example, when we use mathematics to describe the laws of physics.

2. A physical device of process is used to carry out an abstract mathematical operation. This is what happens, for example, when we use a calculator to carry out a calculation.

The task of sorting out whether or not a patent claim covers an algorithm requires that we understand which of these two alternatives is the correct one. These two situations are what I call the directions of the modeling relationship. In the first situation the direction is from the abstract to the concrete. In the second situation the direction is from the concrete to the abstract.

Any notion that the mathematics of computing are related to the computer in the same way as the laws of physics are related to physical phenomenon is erroneous1. In computation theory, the modeling relationship goes the other way round. This follows from common sense by observing a pocket calculator and the use of pencil and paper. This is also supported by established knowledge in abstract mathematics, computer science, and philosophy.

Another way to make the same point is to establish that the use of a physical symbolic representation is an essential requirement of the practice of mathematics with references to mathematical literature. Without some physical representation no mathematical activity can occur at all. If one is not careful, it is possible that a patent on a physical device happens to be a patent on mathematics. This is the outcome that Benson, Flook and Diehr are meant to avoid.

The simplest explanation of this point is that mathematics is a written discipline. One cannot use mathematical concepts without writing them down except in the most simple situations. The written text is necessarily physical. But mathematicians don't care whether the text is written on paper or is electronic signals such as bits. As long as there is a usable physical representation mathematical activity can occur.

But this simple explanation doesn't do justice to the actual mathematical truth. This is not just about the ability of text to express meaning. It is also about the ability to conceive. One cannot think of or use a mathematical idea in the first place if it cannot be written. There is a considerable body of mathematical research on the topic of the connection between what can be written and which mathematical truths can be understood by humans. There are some areas of mathematics that are inaccessible because the written language is insufficient even though its use is unavoidable.

I provide later in this article a selection of the main points with reference to mathematical literature in case lawyers would like authoritative evidence. But it is not the details of the mathematical research that matter, although some of them are legally relevant for purposes other than the main topic of this article. It is that the research has actually been made. We have factual evidence that one cannot think of or use a mathematical idea in the first place if it cannot be represented physically, and this is what gives Benson, Flook and Diehr a sound mathematical basis.

Application to Patent Law: Analysis of In Re Alappat

Although software is a recent technology it doesn't bring a new type of relationship between the abstract and the concrete. On the contrary, the same sort of relationship has existed since antiquity. The Antikythera mechanism is an ancient mechanical computer from the first century BC. The Romans used abacuses. More modern devices are the slide rulers and the differential analyzers. All these devices are physical models for abstract mathematical computational procedures. Although the range of applications and the economic consequences of software are arguably new2, the relationship of the physical world and the abstract mathematics is not.

To my knowledge, this is a fact that hasn't been acknowledged by the courts. This has led to erroneous analysis of the subject matter claimed in patents. A good example is the In re Alappat case. Its examination in the light of a correct understanding is instructive. The litigated patent is a simple one, and the mathematical aspects are clear cut. It is also a historically important case because it introduced the useful, concrete and tangible result test. This test was found to be unworkable for empirical reasons. After years of usage, it was found that this test allows patents that shouldn't be. When we are equipped with a correct understanding of the relationship between mathematical abstraction and the physical circuit, it is possible to see *why* the useful, concrete and tangible result test doesn't lead to a correct analysis of a patent.

The Alappat claim covers a rasterizer for an oscilloscope. The oscilloscope is a device for measuring electromagnetic signals and showing the waveforms visually on a screen. With modern digital electronics, the picture is drawn pixel by pixel on a rectangular grid. Without the rasterizer, the curved lines of the waveform have an awkward visual appearance because the pixels are drawn either entirely lit or entirely black and must be located at the intersections of rows and columns on the grid. These restrictions give the curve a discontinuous or jagged appearance. The rasterizer makes the curve appear smooth by averaging the luminosity of neighboring pixels. This makes the waveform more visually pleasing to the eye.

The patented claim is directed to a digital electronic circuit. The Alappat court quotes the claim as follow:

When independent claim 15 is construed in accordance with Section 112 Para. 6, claim 15 reads as follows, the subject matter in brackets representing the structure which Alappat discloses in his specification as corresponding to the respective means language recited in the claims:

A rasterizer [a “machine”] for converting vector list data representing sample magnitudes of an input waveform into anti-aliased pixel illumination intensity data to be displayed on a display means comprising:

(a) [an arithmetic logic circuit configured to perform an absolute value function, or an equivalent thereof] for determining the vertical distance between the endpoints of each of the vectors in the data list;

(b) [an arithmetic logic circuit configured to perform an absolute value function, or an equivalent thereof] for determining the elevation of a row of pixels that is spanned by the vector;

(c) [a pair of barrel shifters, or equivalents thereof] for normalizing the vertical distance and elevation; and

(d) [a read only memory (ROM) containing illumination intensity data, or an equivalent thereof] for outputting illumination intensity data as a predetermined function of the normalized vertical distance and elevation.

As is evident, claim 15 unquestionably recites a machine, or apparatus, made up of a combination of known electronic circuitry elements.

This circuit carries out a computation. The symbols are digital 0s and 1s. They are used to represent numbers that have real-world meanings, specifically the sample magnitudes of an input waveform, the computed anti-aliased pixel intensity data, and other numbers that are required in the course of the computation. The circuitry performs the computation according to mathematically defined rules. The rules are not stated in the patent claim although they can presumably be found in the patent specification3. Examining the components reveals their function is to carry out ordinary arithmetic. The two arithmetic logic circuits perform absolute value functions to determine distance of numbers. Barrel shifters multiply (or divide) by powers of two and the ROM stores a lookup table containing values of illumination intensity data that are precomputed according to a mathematical function. There is nothing in the circuit that is not doing an arithmetical calculation. However this arithmetic is thinly veiled by the mention of the meanings of the numbers in the claimed functions.

The issue in front of the court was whether this claim was patentable subject matter because it is a circuit or whether it is non-patentable because it would pre-empt some mathematics as per Benson, Flook and Diehr.

My view as a computer professional is that we can answer the issue in front of the court just by looking at the circuit. It does arithmetic calculations and produces numeric answers. Arithmetic and numbers are clearly mathematics.

The court didn't look at the circuit the way I would. They looked at the text of the claim. Upon analyzing the claim the court concluded the rasterizer is not doing a mathematical calculation with this reasoning. (emphasis in the original):

Given the foregoing, the proper inquiry in dealing with the so called mathematical subject matter exception to Section 101 alleged herein is to see whether the claimed subject matter as a whole is a disembodied mathematical concept, whether categorized as a mathematical formula, mathematical equation, mathematical algorithm, or the like, which in essence represents nothing more than a “law of nature,” “natural phenomenon,” or “abstract idea.” If so, Diehr precludes the patenting of that subject matter. That is not the case here.

Although many, or arguably even all, of the means elements recited in claim 15 represent circuitry elements that perform mathematical calculations, which is essentially true of all digital electrical circuits, the claimed invention as a whole is directed to a combination of interrelated elements which combine to form a machine for converting discrete waveform data samples into anti-aliased pixel illumination intensity data to be displayed on a display means. This is not a disembodied mathematical concept which may be characterized as an “abstract idea,” but rather a specific machine to produce a useful, concrete, and tangible result.

Please notice the use of the expression “disembodied mathematical concept”. This is a phrase that is inconsistent with the notion that an abstract mathematical calculation must be carried out physically in order to be used at all. Then the court concludes the circuit is not an abstract idea because it yields a useful, concrete and tangible result.

What is the useful, concrete and tangible result of the circuit? Again my answer as a computer professional is to look at the circuit. It carries out an arithmetic calculation and produces numeric answers in the form of bits. This is the result. They are the electronic equivalent of the symbols a mathematician writes on paper. The bits are electronic signals that are useful, concrete and tangible. And still they are numbers produced by an arithmetic calculation. When I look at the circuit I don't see how the useful, concrete and tangible result test separates mathematics from something that is not mathematics, because the symbols produced by any computation will always be useful, concrete and tangible.

The court didn't follow my approach. They didn't look at the circuit. They looked at the text of the patent claim. The court found that the display of the bits on a display is relevant. They also find that the meaning of the numbers as quoted in the patent text are relevant. The above analysis contains this text:

the claimed invention as a whole is directed to a combination of interrelated elements which combine to form a machine for converting discrete waveform data samples into anti-aliased pixel illumination intensity data to be displayed on a display means.

This is a very different from my look-at-the-circuit approach. Does the display really matter? When I look at the circuit I don't find any display. According to the claim, the rasterizer is comprised of four elements listed from (a) to (d), and the display is not among them. By itself the rasterizer has no capability of displaying anti-aliased pixel illumination intensity data on a display. For this you need an apparatus such as the oscilloscope where the rasterizer is connected to a display. But this claim, as I read it, is claiming just the rasterizer and not the entire oscilloscope. Do I misread something?4

Of course one may connect a rasterizer to a display as the claim preamble suggests. This is only an option because nothing in the laws and principles of electronics requires it. The rasterizer may as well be connected to a hard disk to record the result of the calculations if someone wishes. Or some other creative use could be found. The useful, concrete and tangible result contemplated by the court is one that may occur but will not necessarily occur. Its occurrence depends on how exactly the rasterizer is used.

Perhaps I am overly focused on the display. Perhaps they just point out that the bits being produced are not any random bits but they are bits representing pixel illumination intensity data. Perhaps the meaning of the bits somehow makes the bits a useful, concrete and tangible result in the eyes of the court. But then the correct understanding of the modeling relationships doesn't support the court conclusion. There are two modeling relationships as follows:

1. The rasterizer circuit is a physical model for the abstract calculation. This is a relationship from the concrete to the abstract.

2. The calculation is an abstract mathematical model (formula) for the illumination of pixels on a screen. This is a relationship from the abstract to the concrete.

These are two distinct modeling relationships going in opposite directions and their concrete elements are different. This court is using one modeling relationship to reach a conclusion about the other. But the two relationships are independent because the circuit doesn't have to be connected to a display and the mathematical calculation doesn't have to be about pixel illumination data. The court's logic doesn't follow from how the modeling works.

The court further explains (bold added):

The fact that the four claimed means elements function to transform one set of data to another through what may be viewed as a series of mathematical calculations does not alone justify a holding that the claim as a whole is directed to nonstatutory subject matter. See In re Iwahashi, 888 F.2d at 1375, 12 USPQ2d at 1911. Indeed, claim 15 as written is not “so abstract and sweeping” that it would “wholly pre-empt” the use of any apparatus employing the combination of mathematical calculations recited therein. See Benson, 409 U.S. at 68-72 (1972). Rather, claim 15 is limited to the use of a particularly claimed combination of elements performing the particularly claimed combination of calculations to transform, i.e., rasterize, digitized waveforms (data) into anti-aliased, pixel illumination data to produce a smooth waveform.

Furthermore, the claim preamble's recitation that the subject matter for which Alappat seeks patent protection is a rasterizer for creating a smooth waveform is not a mere field-of-use label having no significance. Indeed, the preamble specifically recites that the claimed rasterizer converts waveform data into output illumination data for a display, and the means elements recited in the body of the claim make reference not only to the inputted waveform data recited in the preamble but also to the output illumination data also recited in the preamble. Claim 15 thus defines a combination of elements constituting a machine for producing an anti-aliased waveform.

This court worked from the idea that the real-world meaning of the bits defines the actual computation. We can immediately see how this causes a problem. Please refer to the sentence in bold. How does this work? If an electronic engineer wants to build an apparatus employing the combination of mathematical calculations recited therein, he has to build the same circuit as the one described in the Alappat claim. Or in the alternative he needs some other circuit that has the same functionality and will be subject to the treatment patent law applies to functional equivalents. This is the very situation Benson is meant to avoid.

Here is an example of how this problem would happen. Suppose our hypothetical engineer designs a new oscilloscope whose display draws the waveforms in yellow over a neutral gray background5. There is no change in pixel illumination intensity value between the background and the waveform. It is a change of color. The anti-aliasing is done by altering the color saturation value to make the yellow color more or less grayish. A fully saturated color is pure yellow and a totally unsaturated color is pure gray. If we use the same formula for the anti-aliasing, the rasterizer no longer computes pixel illumination intensity data. It computes pixel color saturation data. But the same circuit is being used.

This is a change from the overall function of the rasterizer, because the preamble says “converting vector list data representing sample magnitudes of an input waveform into anti-aliased pixel illumination intensity data”. This doesn't happen any more. This is also a change from element (d) which will turn out not to be present. This element says “means for outputting illumination intensity data”. This will not occur. The new rasterizer will output color saturation data.6

The arithmetic formula used to perform the anti-aliasing is the same. But the meaning of the bits is changed. In order to carry out this calculation, the same circuit is used. The question is, does it infringe? If the answer is yes, then how is it that “claim 15 as written is not 'so abstract and sweeping' that it would 'wholly pre-empt' the use of any apparatus employing the combination of mathematical calculations recited therein”? Conversely, if the circuit doesn't infringe, what exactly is being covered by the patent? Isn't it supposed to claim a circuit?

All the language about waveform data samples, anti-aliased pixel illumination intensity data and the likes imposes no limitations to the circuit beyond constraining the structure to represent the correct arithmetic calculation according to the intended interpretation of the numbers7. The fact that the numbers mean pixel illumination intensity data is not recorded with the bits. If the same calculations are required for another purpose the bits will be the same, the arithmetic operations will be the same, and the circuit carrying out the task will be the same. Therefore anyone who needs this calculation has no choice but to use the same circuit (or an equivalent).

The reason is this: the calculation is done with mathematical symbols. When I write 24, nobody can tell just by looking at the written digits whether this is the number 24 in the abstract, a count of 24 apples, or 24 monkeys. This is contextual information that is not recorded with the written digits. When someone adds numbers, the pocket calculator doesn't care if the numbers mean dollars in a bank account or something else. It just adds the numbers. The real life meaning of the numbers is not stored with the symbols, and it is not used by the computing process.

The same thing happens with bits. They are the electronic version of mathematical symbols. Whether they represent pixel illumination intensity data or or the quantity of jet fuel required by a plane at take off is not recorded with the bits. The bits just represent numbers, and the digital electronics components work just the same whatever the meaning is. If an engineer needs to carry out the same computation for another purpose, the circuit that will fulfill this purpose is the same circuit8.

Of course when an engineer uses the rasterizer circuit for another purpose, the circuit is no longer a combination of elements constituting a machine for producing an anti-aliased waveform as the court puts it. In this sense, it is no longer the machine claimed by Alappat claim 15. This doesn't change the fact that the same circuit is used.

I think that in addition to claiming an abstract mathematical computation, this claim is somehow faulty. I am not a lawyer. This is a hunch and not knowledge. My hunch is that Section 112 paragraph 2 might be applicable. (emphasis added):

The specification shall conclude with one or more claims particularly pointing out and distinctly claiming the subject matter which the applicant regards as his invention.

My hunch is that reliance on the real-world meaning does not distinctly claim the subject matter unless the real-world meaning is actually used by an element of the claim and not merely referred to. If I understand the law correctly, patent claims that do not conform to section 112 paragraph 2 are invalid and cannot be enforced.

A Note on the Machine or Transformation Test

Among the questions asked, the USPTO requests examples of patents that pass the machine or transformation test but are patents on abstract ideas. We have just seen such an example.

The Alappat rasterizer passes the machine or transformation test, because it is a patent on a specific circuit. But it is a patent on a mathematical computation as surely as the Benson patent is. Such a remark is applicable every time a specific circuit is built for the purpose of making a computation pursuant to a mathematical algorithm and there is no invention outside of the computation.

The machine or transformation test should work fine in most if not all circumstances where the abstractions involved are models for a real-world invention, like the application of the laws of physics to engineering. Then this test will provide a strong clue as to whether the patent is on the abstract model or on the real-world application. But this test doesn't work when the modeling relationship is reversed, because in such case the machine or process is the tool used to understand the abstract ideas. Then it is possible that a patent on the tool is effectively patenting the idea as was the case in Benson and Flook.

It is very difficult to make sense of the notion that a patent on a circuit is a patent on an abstract idea when we assume the modeling relationship always works from the abstract to the physical and never the other way round. The trilogy Benson, Flook and Diehr is best understood after the modeling relationship has been worked out properly.

Appendix: Mathematical References

Here are the references to published literature that establish that mathematics require a physical representation of the concept in order to be able to understand, use, or even conceive mathematical ideas. Physical tools impose fundamental limitations on the ability of human beings to reach mathematical truths. My hope is that lawyers can use these references to locate authorities to support this point in court.

The very first two paragraphs of the very first chapter9 of Introduction to Metamathematics10 by Stephen Cole Kleene are (emphasis in the original)

Before turning to our main subject, it will be appropriate to notice briefly Cantor's theory of sets.

A flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. Although the words for numbers have been used to state this truism on the printed page, the relationship to which we refer underlies the concept of cardinal number. Without counting the sheep or trees, one can pair them with each other, for example by tethering the sheep to the trees, so that each sheep and each tree belongs to exactly one of the pairs. Such a pairing between the members of two collections or 'sets' of objects is called a one-to-one (1-1) correspondence.

The first thing Kleene does is to use a concrete and tangible procedure to represent the abstract concept of cardinal number in Cantor's theory of sets. What is a cardinal number? On page 9 Kleene quotes Cantor:

Cantor describes them thus: “The general concept which with the aid of our active intelligence results from a set M, when we abstract from the nature of its various elements and from the order of their being given, we call the 'power' or 'cardinal number' of M.”

This is definitely an abstract concept. Cantor's theory of sets is a mathematical theory. Although this concept of cardinal number is described using English there is a more formal definition given mathematically by means of an operation of pairing which is called “a one-to-one correspondence” 11.

A numerically challenged shepherd may use pairing to keep track of his sheep. The shepherd may be unable to count past three, but he knows that if a tree ends up without a sheep, then a sheep is missing. It doesn't matter if the pairing is done “in the abstract” in the shepherd's mind or if it is done in the concrete by physically tethering the sheep to trees. Both ways of pairing count sheep.

In this little example the abstract theory of sets is not used as a model of the act of tethering sheep to trees. It goes the other way round. The act of tethering is a physical model for the mathematical theory of cardinal numbers. It allows the shepherd of the example to overcome his limited mind and perform an abstract operation he would otherwise be unable to do.

Most people are able to count normally. All of us are subject to ordinary human limitations. There are abstractions that cannot be conceived or handled without physical assistance. This is why we use pocket calculators or pencil and paper for all but the simplest calculations.

The ancient Greeks practiced mathematics, especially geometry. They used figures drawn in the sand or on papyrus to help them track the geometrical abstractions. As Plato puts it in book 7 of The Republic (around 380 BC):

You also know how they make use of visible figures and discourse about them, though what they really have in mind is the originals of which these figures are images: they are not reasoning, for instance, about this particular square and diagonal which they have drawn, but about the Square and the Diagonal; and so in all cases. The diagrams they draw and the models they make are actual things, which may have their shadows or images in water; but now they serve in their turn as images, while the student is seeking to behold those realities which only thought can apprehend.

Nowadays we use the written text to achieve the same result. Like the drawn figures, written text is a physical tool.

There is more at play than the ability of text to express meaning. The tools also enable procedures, the algorithms, for solving problems. The ancient Greeks carried out their mathematical algorithms mostly with compass and straightedge. Modern mathematicians use either pencil and paper or a computer. For example we all learned in school the algorithms to carry out additions, subtractions, multiplications and divisions by manipulations of decimal digits written with pencil and paper. Algebra, calculus and other mathematical disciplines provide methods for solving problems based on the written text. These methods can be automated by means of digital electronics devices like the pocket calculator and the computer.

The choice of the physical tool is important. This choice limits which abstractions we are able to use and determines how easy the computational process will be. If you have any doubt please try to multiply MDCCVIII by CCXXXIV with pencil and paper using roman numerals for all manipulations. Please don't cheat. No translation into decimal numbers is allowed.

Here is another example. Try solving x2 + 3x - 2 = 0 like you were taught in school but using only English words instead of the algebraic notations. You need to use sentences like “the square of the value added three times the value from which you subtract two yield nothing.” According to Kleene12, this is how mathematical problems were solved before the invention of the algebraic notation by Vieta and others. The discovery of new notations is one of the ways mathematics makes progress. Our ability to use algorithms that solve the problem at hand is contingent on the availability of the adequate physical tool.

This issue goes deeper than mere convenience and practicality. Some problems are fundamentally unsolvable with some categories of tools regardless of whether we may resolve the practical aspects. A famous example is the squaring of the circle problem in Euclidean geometry. As Howard Delong explains13, when discussing the power of modern algebraic methods when applied to geometry: (emphasis in the original)

Using this powerful method of analysis, nineteenth century scientists were finally able to solve the three famous outstanding geometric problems which the Greeks passed on to posterity: the duplication of the cube, the squaring of the circle, and the trisection of an arbitrary angle. The problem was to make an exact construction with the aid of a compass and straightedge alone. It was shown in each case that it is impossible to solve the problem under the condition stated. Impossible here means logically impossible. That is, given certain assumptions, it was shown that it is as impossible to solve these problems as it is to draw a square circle. The proofs are complex — especially the one for the squaring of the circle — but they nevertheless settle the problems once and for all.

Here is another example. There are real numbers that can't be defined by any mathematical formulas. The reason is that formulas are countably infinite while real numbers are uncountably infinite. This is a limitation of the algebraic notation as physical symbols written on paper.

Countably infinite means an enumeration like the natural numbers 0, 1, 2, 3 … If an infinite set can be enumerated in such a way each member can be associated with a natural number, then it is countably infinite14. Formulas can be put in lexicographical orders, this is something like a, b, … z, aa, ab, ac, … zz, aaa, aab etc. except that the order must be defined on the mathematical symbols rather than the Latin alphabet. You group formulas by their length first, and then you sort the formulas of the same length in the lexicographical order. The result is an enumeration that could be numbered15. Formulas are countably infinite.

It is proven that real numbers are not countably infinite. The proof is given by the Cantor diagonal method16. Assuming you have an ordering of real numbers you prove there is a real number not of the list as follows.

  1. 4.35447646 …
  2. 2.45790209 …
  3. 0.75890000 …
  4. 5.98237515 …
  5. 2.83425897 …

You write the infinite decimal expansion of each real number in the enumeration as in the above example. If some real number doesn't have an infinite expansion then you pad it with infinitely many zeros (like the third number in the example). Then you take the first digit of the first number, the second digit of the second number and you continue like this over the diagonal. You make a new number with the rule: every time the diagonal digit is 5, you make the new number a 6, otherwise you make it a 5. With the example this gives:

  • 0.56556 …

The result is a real number guaranteed not to be in the list. In other words, there can be no enumeration of all the real numbers, because given any enumeration, we can always find at least one real number that is not included.

The consequence of the Cantor diagonal method is that there are real numbers we cannot define and comprehend because there is no formula able to define them. This is a limitation of relying on a mathematical language made of written symbols.

Physical tools do more than representing meaning and computing solutions to problems. They are also the embodiment of the rules of logic. This is a topic that has been studied by philosophers and mathematicians since the nineteenth century. The name of the discipline is symbolic logic. Its most prominent features are propositional calculus and predicate calculus. They are synthetic languages made of written symbols whose purpose is to provide a logical foundation to mathematics. The formulas and expressions in these languages must follow a rigidly defined syntax, like a programming language. This formalism allows to subject the language itself to mathematical study. This is called metamathematics, or mathematics about mathematics17.

Some of the greatest mathematical discoveries of the twentieth century arose from metamathematics. For instance see this series of theorems18.

  • Gödel's first incompleteness theorem: in every formal system of mathematics that is powerful enough to represent the arithmetic of natural numbers there is a formula that is true but cannot be proved.

  • Church theorem: consider a formal system that represents pure ordinary mathematical logic (this is known as predicate calculus), there is no algorithm that will decide whether or not a formula in this system can be proven as a theorem.

  • Skolem theorem: if a formal system is powerful enough to represent the arithmetic of natural numbers then it simultaneously supports an alternative interpretation where there are “unnatural numbers”, that is number other than those in the familiar sequence 0, 1, 2, 3… to infinity.

I have translated into plain English the mathematical terms of arts and notations in order to make their meaning accessible to laymen. Please go by the mathematical text for any use that requires mathematical accuracy.

These theorems and others like them are called limitative theorems. They establish fundamental epistemological limits to what can be known by man. This is a fascinating topic of deep philosophical consequences. This is an area of knowledge where mathematics, philosophy and computer science overlap. Delong, philosopher and author of A Profile of Mathematical Logic, describes these epistemological limits as follow19: (emphasis in the original)

What the limitative theorems represent then is the discovery of an abstract structure which is of such a sort that it is impossible for any human to make systematically complete and correct assumptions about it. No matter how hard we try, we cannot talk about just the natural numbers, but must always talk about the unnatural numbers, too. In other words, just as our sensory perceptions have limits which can be extended but not eliminated by certain technique of science (for example, the microscope), so our abstract conceptions are limited and the methods of mathematics which are intended to extend them (for example, mathematical induction) are at best partial in their effect. Our powers of conceptual discrimination are no less limited than our powers of perceptual discrimination.

There doesn't seem to be any way around these conclusions. But it should be noted that the above argument depends on what is known as realism (or platonism) in mathematics. That is, it is assumed that the abstract structure of arithmetic exists independently of human conceptions about it. The assumption is embedded in classical mathematics, but we might question it.

Then Delong proceeds to examine what happens when we question this assumption. The topic is quite fascinating, but I don't want to stray too far away from the legal issues. This quote will be enough for our purposes.

It is not so much the theorems that are relevant but the way they have been derived. Howard Delong explains20: (emphasis in the original)

[It can be argued that: “I object,] the impossibility of squaring a circle or trisecting an angle is not philosophically interesting. It is perfectly possible to do it, but not with just a straightedge and a compass. Similarly, when Gödel proves that there are undecidable formulas, or Church that there is no decision procedure, or Skolem that a calculus doesn't categorically represent the natural numbers, all this comes to is that using the means they have selected (just as Euclid selected a straightedge and compass), their conclusions follow. There is no philosophical import to this: it just suggests that mathematicians and logicians must look for other means.”

This objection would have a great deal of sting were it not for one circumstance: There do not appear to be any other means. To see this, let's consider briefly the requirements we make in order to claim deductive knowledge. We first require that all the “symbols” which are used be publicly and effectively recognizable. The word “symbols” is in double quotes because it is not required that they be symbols in the usual sense of the word, that is, marks on paper or on a blackboard, etc. They might be sounds or colors or electric charges. The requirement includes not only being able to tell the difference between a symbol and something else (for example we should be able to tell the difference between a parenthesis and a broken fishhook), but being able to effectively recognize repeated occurrences of the same symbol type [for example, '(' is of the same symbol type as '(']. The 'all' in our requirement means that the number of symbols be finite (for example, a zillion symbols) or, if infinite, that there be an ordering (called the alphabetic order above) such that only a finite number of symbols precede any given symbol in that ordering.

Second, we require that the length of any given sentence be finite and that a sentence be effectively recognizable as such. We make the same requirement of rules of inference. As for the axioms, we require that they be either finite or, if infinite, that there be an ordering such that only a finite number of axioms precede any given axiom in that order. For the theorems, we do not require that there be an effective method to recognize them, but only that there be a constructive program which, if carried out, will produce any given theorem. Finally, we require that for any given sentence we be able to effectively recognize a proof of it if one should be presented. The above are all syntactical requirements. There is only one semantical requirement: that the theorems all be true under every interpretation which makes the axioms true.

The point is that fundamental limits on the extent of mathematical truths that can be known to mankind follow from the physical tools that are necessary to the practice of mathematics.

Metamathematics gives a mathematically exact definition to each element identified as a requirement of mathematics by Delong. Then mathematical methods are applied to these definitions to prove the limitative theorems and other results. This is an inquiry on the very nature of the tools required to gain deductive knowledge, essentially the symbols and the various syntactical elements that are made with them. This inquiry goes well beyond the basic observations that the written text has a meaning, like is done in the printed matter legal doctrine. It extends the analysis to computational and deductive processes which exist in the real world as manipulation of symbols and syntactic elements.

This is the sort of thing the law should look at in the context of Benson, Flook, and Diehr. It is not possible to fully understand the relationship of mathematical abstractions and physical reality without knowing what is found in this part of mathematics. It not only relates to why a patent on a circuit may be a patent on an abstract idea, it also relates to why software is speech because it involves the study of the relationship between text, computation, and meaning. Its very topic is the ability of mankind to reach knowledge by means of written symbols.

Among the features of metamathematics there are objective definitions of what is a mathematical proof21. These definitions define 'proof' by syntactic means. It is the organizing of the formulas as syntactic elements in the proper sequence that makes a mathematical proof valid. Semantics, in the sense of looking at the meaning of the formulas and verifying their truths, plays no part in these definitions. A proof is correct when it satisfies syntactic rules. The consequence is that mathematical reasoning is amenable to machine processing. Kleene observes22:

[C]omputers may be applied in metamathematics to seeking proofs of theorems, or to checking proposed proofs etc.

This is known as automatic theorem proving. This is the automation of operations of abstract logic. Quoting from the linked page: (emphasis and bold in the original)

Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement (the conjecture) is a logical consequence of a set of statements (the axioms and hypotheses). ATP systems are used in a wide variety of domains. For examples, a mathematician might prove the conjecture that groups of order two are commutative, from the axioms of group theory; a management consultant might formulate axioms that describe how organizations grow and interact, and from those axioms prove that organizational death rates decrease with age; a hardware developer might validate the design of a circuit by proving a conjecture that describes a circuit's performance, given axioms that describe the circuit itself; or a frustrated teenager might formulate the jumbled faces of a Rubik's cube as a conjecture and prove, from axioms that describe legal changes to the cube's configuration, that the cube can be rearranged to the solution state. All of these are tasks that can be performed by an ATP system, given an appropriate formulation of the problem as axioms, hypotheses, and a conjecture.
This is an example of using the real-world device to model abstract logical processes. The linked page discusses more, pointing to actual programs and applications.

Another feature of metamathematics is the precise definition of what is computable. This is what Benson, Flook, and Diehr refer to as a “mathematical algorithm”. This is knowledge which should be used by the law. This definition follows from the same type of epistemological inquiry as the limitative theorems. Delong explains23 (emphasis in the original, the occurrence of patent statute section numbers is pure coincidence):

We begin by imagining some human who is faced with a specific computational problem. It might be some such problem as computing the sum of 101 and 102 and 103 and . . . and 198 and 199, where the three dots indicate one occurrence of all the natural numbers between 103 and 198. We assume that he is working according to a finite set of rule which have been fixed before the problem was given and that he is using pencil and paper. We also assume that after a finite amount of time he stops with the correct answer.

If we examine this paradigm case of computing with a view toward eliminating inessential features, a number of such features comes to mind, such as the use of pencil and paper or the particular computational problem chosen. However the most striking one appears to be the human: the computer might as well be a machine.

Here too we see the prominent position of the use of symbols in mathematical processes. The analysis of how a human computer can be replaced by an automatic device has been carried out by Alan Turing. The outcome is the Turing machine24.

John von Neumann was familiar with Turing's work and used it to derive the stored-program architecture which forms the basis of nearly all modern programmable computers25. Jack Copeland reports on The Turing Archives for the History of Computing:

In 1944, John von Neumann joined the ENIAC group. He had become 'intrigued' (Goldstine's word) 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 (Julian Bigelow in personal communication with William Aspray, reported in the latter's John von Neumann and the Origins of Modern Computing Cambridge, Mass.: MIT Press (1990), pp. 178, 313).

Digital computers relate to written text in a way analog technologies implementing computations don't. Digital information is made of symbols, the bits, the 0s and the 1s which represent the boolean values False and True respectively. By design, the same class of computational processes that could, in principle, be carried out by means of pencil and paper are automated by the computer. This is in accordance to the analysis of computation done by Alan Turing and the engineering decision made by John von Neumann and his colleagues.

This is more than enough evidence that physical devices and computations have the ability to represent abstract ideas. Their use is essential for human beings to even be able to conceive, let alone use, these ideas. In particular the computations described by algorithms must be carried out physically in order to be used at all. If you patent the physical means to carry out the computation then you may end up patenting the computation itself if you are not careful. This is the mathematical basis for the non-patentability of algorithms as per Benson, Flook and Diehr.

References

Cases

Bilski v. Kappos [PDF] [text] The Supreme Court opinion

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

Gottschalk v. Benson, 409 U.S. 63, 71-72 (1972)

In re Alappat, U.S. Court of Appeals Federal Circuit, 33 F.3d 1526 July 29, 1994

In re Bilski the Court of Appeals for the Federal Circuit decision

In re Gulak, 703 F.2d 1381

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

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.

[Delong 1970] Delong, Howard. A Profile of Mathematical Logic. Addison-Wesley Publishing Company. 1970. Reprints of this book are available from Dover Publications.

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

[Kleene 1952] Kleene, Stephen Cole, Introduction to Metamathematics, D. Van Nostrand Company, 1952. I use the 2009 reprint by Ishi Press International 2009.

[Kleene 1967] Kleene, Stephen Cole, Mathematical Logic, John Wiley & Sons, Inc. New York, 1967. I use the 2002 reprint from Dover Publications.

[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]

Biographies

Biographies are from the MacTutor History of Mathematics Archive

Kurt Gödel

Stephen Kleene

Alan Turing

Footnotes

1 See Donald Knuth's 1994 letter to the US Commissioner of Patents and Trademark: “An algorithm is an abstract concept unrelated to physical laws of the universe.”

2 One shouldn't underestimate the importance of the written text for the storage, communication and processing of information. Before the advent of computers written text was as important as computers are nowadays. It still is. The social, scientific and economic issues raised by software are not nearly as new as students of software patents may think. Computers represent an expansion of existing capabilities and not the creation of new capabilities that didn't exist before. This expansion is formidable, I will agree, but an expansion of existing capabilities no matter how formidable is not the same thing as the creation of new capabilities.

3 The text of the court ruling contains a description of the mathematical formula.

4 The phrase “for converting vector list data representing sample magnitudes of an input waveform into anti-aliased pixel illumination intensity data to be displayed on a display means” implies that the act of displaying occurs after the rasterizer has produced its output. It appears to me that the display is explicitly excluded.

5 Whether or not the engineer has an awful taste in matters of color is beside the point.

6 This design may have an amusing twist. In this particular setup the blue component of the video signal happens to have a well defined mathematical relationship with color saturation. When the blue component has a high value it indicates the pixel corresponds to the gray background because pure yellow doesn't have any blue component. And conversely a zero value of the blue component indicates pure yellow. What if the engineer modifies the display to monitor the blue component and adjust the pixel illumination intensity inversely proportional to the blue component? Low blue value raises the intensity to the brightest yellow while high blue values drop the intensity to pitch black. Then we display the waveform as a yellow line on a black background just as the Alappat rasterizer might do. The color saturation rasterizer would control pixel illumination intensity without computing pixel illumination intensity data. I am bringing up this point just for its amusement value. This scenario is beside the point of this article. Lawyers may wish to discuss whether this modified display makes a functional equivalent of the Alappat circuit and whether this will lead to infringement.

7 If the point needs further hammering, such calculations can be done by a standard off-the-shelf pocket calculator when only the human being operating the calculator knows the meaning of the numbers.

8 Had the patent been on the process manipulating the bits then by the same logic assigning a different meaning doesn't make a physical process distinct from the base mathematical computation.

9 See [Kleene 1952] p. 3.

10 [Kleene 1952] in the reference section.

11 See [Kleene 1952] p. 9 where the exact mathematical definition is given.

12 [Kleene 1952] p.61.

13 [Delong 1971] p. 69.

14 Kleene [1952] pp. 3-6.

15 [Kleene 1967] p. 178. This is assuming there is a finite number of allowable symbols in a formula. If the number of allowable symbols is infinite (not a practical possibility) then the formulas are still countable but the ordering requires the use of Gödel numbers. See [Gödel 1986]

16 Kleene [1952] pp. 6-8, [Kleene 1967] pp. 180-183 or [Delong 1971] pp. 75-76.

17 This the the topic of [Kleene 1952] which is titled Introduction to Metamathematics. Chapter IV to X are especially applicable. This is also covered in [Kleene 1967] chapters I to III and in [Delong 1971] chapters 3 and 4.

18 [Delong 1971] p. 195.

19 [Delong 1971] p. 204

20 [Delong 1971] pp. 193-194.

21 This is known as proof theory. See for example [Kleene 1967] pp. 33-58, 107-134. There are several proof theories depending on the system of logic being used. Each of them is mathematically unambiguous.

22 [Kleene 1967] p. 201.

23 [Delong 1971] p. 197.

24 Turing's analysis of the use of symbols by humans is found in section 9 of [Turing 1936]. His thesis is that all computations done with pencil and paper can be replicated by Turing machines, therefore Turing machines are a mathematically accurate description of what is computable.

25 This history is told in [Davis 2000] chapters 7 and 8.


  


An Open Response to the USPTO -- Physical Aspects of Mathematics | 284 comments | Create New Account
Comments belong to whoever posts them. Please notify us of inappropriate comments.
Corrections
Authored by: PolR on Sunday, September 26 2010 @ 10:37 PM EDT
If any are needed

[ Reply to This | # ]

OT Here
Authored by: PolR on Sunday, September 26 2010 @ 10:39 PM EDT
And make clickies if you know how. Hints are in the red text below the comment
box.

[ Reply to This | # ]

News picks here
Authored by: PolR on Sunday, September 26 2010 @ 10:40 PM EDT
Please refer to the news title in your comment title so we know which one you
talk about.

[ Reply to This | # ]

COMES contributions here
Authored by: PolR on Sunday, September 26 2010 @ 10:42 PM EDT
The COMES work need to go on. A sincere thank you to all the contributors. We
can't make it through without you all.

This is a first for me, to get the first four comments on my own article.

[ Reply to This | # ]

hmm I will re read this later instead of skimming it :-)
Authored by: SilverWave on Monday, September 27 2010 @ 03:01 AM EDT
Thanks for producing this.

---
RMS: The 4 Freedoms
0 run the program for any purpose
1 study the source code and change it
2 make copies and distribute them
3 publish modified versions

[ Reply to This | # ]

Physical Aspects of Mathematics and Severable Parts
Authored by: Anonymous on Monday, September 27 2010 @ 05:54 AM EDT
Another excellent article, thanks!

So, some early and not yet though out reaction. I think
you've got a critical issue here, however I'm not sure that
this is the only issue that needs to be considered. It
would be great to devise some bright-line test broadly
applicable, but that my be asking a bit much. A workable
test may need more than just your analysis of the nature -
or direction - of the model.

Consider a completely mechanical analogue computer. You
identified one such, the slide rule. (BTW, one still sits
on my desk!). I think it would be hard to argue for a
patent on the slide rule. On the other hand, consider fire
control computer from the pre-electrical analogue days.
Some of these were very complex, and contained some rather
ingenious mechanical innovations. For all their delightful
mechanical aspects, they still are just model abstract
mathematics. Were they legitimately patentable? They
certainly were not 'general purpose computing machines'.

Then consider electrical analogue. Still not general
purpose? Maybe, certainly potentially more so than the
fire-control computer. What about the totalizer?

What about a device built using modern microcomputer
components, which has no removable media, no network
connection, and not general purpose data/program entry human
interface, built to perform a specific purpose? You could
solder on such components, and then presto, it's a general
purpose computing device, once again.

The points I'm trying to make are two. First, it seems to
me there is a continuous spectrum of device. If so, then
any endeavor to produce a bright line test is doomed. That
does not mean that we should not strive to produce a
workable test.

If there is no bright line, but there is a workable test,
then it likely will take more than one abstract dimension of
the problem into account. This leads to my second point,
that of severability and intent - which was triggered by
your analysis of Alapat. The four circuits in that case
_were_ severed, as you pointed out. As such, they were a
general purpose computing device; general in the sense that
they could be used for any purpose where the mathematical
operation they embodied was useful. On the other hand, a
mechanical analogue fire control computer is very limited -
it's pretty hard to think of another use, other than fire-
control, for such a device. When someone tries to assert
claims under various tests by tying a program to a GP
computer, these parts are clearly severable.

What about my hypothetical special purpose device I
mentioned. There is a remotely controlled video camera I
know of, that has a general purpose SoC (system on a chip)
running uCLinux. A skilled technician could in principle
add components to this to make it a GP computer. But that
was not the _intent_ of the device.

So I haven't reached particularly clear conclusion, or even
a clear statement of my case, but I did say this was my
initial reaction!

Thanks for the great article.

[ Reply to This | # ]

ThrPilgrims Perl Script to create every computer program ever
Authored by: ThrPilgrim on Monday, September 27 2010 @ 08:29 AM EDT

#!/usr/bin/perl

use strict;
use warnings;

use Bit::Vector;
use File::MimeInfo::Magic qw(extensions mimetype);
use IO::Scalar;
use File::Spec;
use File::Path qw(make_path);

# Set up a Bit::Vector to count from 0 to +infinity
my $vector = Bit::Vector->new(1);

# Get the current working directory
my $dir = '';
my $cur_dir = File::Spec->curdir;

# loop for ever
while (1) {
# Convert the vector into hex digits and
# Split into groups of for to create a directory
# Name based on the contents of the vector
my $hex = reverse $vector->to_Hex;
my @dir = ();
while ( $hex=~ s{ A (.{4}) (.) }{$2}msxg ) {
push @dir, $1;
}
$dir = File::Spec->catdir($cur_dir,@dir);

# Split the vector into 8 bit bytes
my $data = join('', map {chr} $vector->Chunk_List_Read(8));

# Calculate the file extension based on the vectors
# contents
my $mime = mimetype(IO::Scalar->new($data));
my $extension = (extensions $mime)[-1];
$extension ||= '000';

if (! -d $dir) {
make_path($dir);
}

# Write out a file containing the data
local $"=', ';
print "$hext$mimet$extensionn", $vector->Size, "n";

open my $file, '>:raw', File::Spec->catfile($cur_dir, @dir,
"$hex.$extension");
print $file $data;
close $file;

# The important bit. Add one to the vector
my $carry = $vector->increment;
if ($carry) {
$vector->Fill;
$vector->Resize($vector->Size() + 1);
$vector->increment;
}
}



---
Beware of him who would deny you access to information for in his heart he
considers himself your master.

[ Reply to This | # ]

An Open Response to the USPTO -- Physical Aspects of Mathematics
Authored by: philc on Monday, September 27 2010 @ 09:18 AM EDT
I have had a lot of difficulty with software patents from the first I heard of
them. Much of the difficulty has to do with how software development is
practiced.

First off we write software to solve problems. We use math extensively. Most
programming languages are just glorified math. We rely heavily on libraries of
functions that perform various operations. We envision various solutions, make
drawings of of the designs, which is a creative artistic activity. We code up
the design and get it to work. But mostly we take what exists and change it
around to solve the next problem.

All this is to solve a problem.

The biggest problem I have with software patents is the requirement to not be
obvious. If you decide to do something and succeed it doesn't mean that any
number of others wouldn't be able to do what you did. Actually there are a lot
of bright engineers that can solve the same problem, maybe they can come up
with an even better solution.

We almost always take some working code and hardware put it together in a
different way and customize parts to come up with the next "big
thing". Many people could do the same if they were inspired to solve the
same problem.

Even the example above of solving a raster problem is much the same. A raster is
fundamentally a sawtooth wave form that is driven into a magnetic coil to
generate a magnetic field that can move an electron beam. All of this is
non-linear so corrections need to be applied to make the beam track properly
across the screen. There are a lot of design teams that have solved this
problem. Its not all that hard. One team got a patent on it. The other teams may
or may not have know about the patent.

Actually, when independent teams that, unknown to each other, solve the same
problem, that should in itself make the an applicant fail an obvious test and
invalidate a patent.

They mentioned electronic calculators. These are mathematical in nature. There
are thousands of ways to mechanize math. None should be patentable.

I think patents should be reserved for the few really impressive new things that
come along that solve really difficult problems. Anything that uses theme and
variation on something that exists should be excluded as obvious.

In short it should be difficult to get a patent and there should be few issued
per year.

[ Reply to This | # ]

"For the purpose of" patents
Authored by: BitOBear on Monday, September 27 2010 @ 10:12 AM EDT
I have said here many times that my problem with software patents is that once
you have removed what isn't being patented and what are capabilities already
built into the machine, you are left with a string of statements reading
"(deleted) for the purpose of (normally abstract thing)."

That is, the one click patent doesn't invent the computer, network, browser,
catalog, credit card, image, text, or main-page text "button", nor the
pushing of same. It patents pushing the button for the purpose of having the
entire sale transaction take place in response to one (final) click."

In the physical sense, some guy is not patenting the tee-shirt, the sewing
machine, embroidering, or the automatic embroidering of particular images; but
instead said guy is patenting an automated process for embroidering a flower on
a tee shirt to make someone happy.

In the siplest terms, the computer was, during its creation, imbued with various
powers of computation, memory, and production (display, printing, reaming vague
chunks of metal into particular shapes, etc). The patentability of the machine
is complete with the production of the machine. No particular combination of
operations on that machine should then be patentable because those instructions
operate the machine entirely within its design.

Being able to patent a process executed by the machine is being able to
double-dip on the machine's existing function "for a particular
purpose".

Even in concrete machining, I can have patents on the fabricating machine(s),
and I can have patents on the results of that fabrication. That is, I can have
patent a new kind of router, and I can have a patent on a new kind of eccentric
lever-arm that improves the breaks on your car. I am not, or at least should
not, be able to come along and patent the specific guide plate for the router
that produces the lever-arm piece. The patentability of the fabrication was
exhausted by the invention of the fabricator.

These are the same arguments being made above. One guy gets a patent on the
invention of the calculator, or the slide-rule (not "slide rulers" by
the way), but nobody gets to patent using that calculator for the purpose of
finding specific values of E for corresponding values of m in E=mc^2.

The rule should be simple: Any piece of software, operation, or process that
runs on, or can be run on, a general purpose (commodity) computer shall be held
to be unpatentable; and implementation of any existing patent running on, or
which can be run on, a general purpose (commodity) computer shall be presumed,
and held to be, non-infringing of that patent.

[ Reply to This | # ]

Software As Art
Authored by: sproggit on Monday, September 27 2010 @ 10:45 AM EDT
Apologies if this is slightly off topic, but with your indulgence I'd like to
suggest a slightly different way of looking at software - as an Art form.

Let's narrow down this field of comparison to painting only, just to try and
make it more precise.

We could reasonably break painting down into sub-categories and genres in a
number of ways. For example, if we consider materials used - oil, acrylic,
watercolour - even aerosol cans - may be analogous to writing software in
different languages.

Similarly, the subject matter might be analogous to the nature and purpose of
the software. More specifically, we might classify portraits, landscapes, still
life, abstract and other styles as being analogous to different aspects of
software development - Operating Systems, Applications, Databases, and so on.

Dropping to a more specific level, we could say that two paintings of, for
example, the Golden Gate Bridge in San Francisco Bay, might be similar to seeing
two piece of word processing software - inasmuch as the subject of the painting
is exactly the same, although the *interpretation* of the subject is unique and
different.

Finally, we could even show that two paintings of the same subject by the same
author would be unique, because at a "micro" level, there would be
subtle differences between the individual copies, even though the artist may
have used broadly the same techniques and approach in more than one example.

Thus it is with software. Software is, by nature of it's purpose in our lives, a
*representation* of thoughts, ideas, concepts, strung together in a creative and
coherent whole where "the whole is greater than the sum of the parts".


So even though Microsoft produce a C++ Compiler for Windows, they don't (to my
knowledge) claim that every program compiled using their tool becomes their
property. Similarly, because it is *entirely reasonable* for two programmers,
skilled in the art, to deduce a process that is more-or-less similar to solve a
common problem, this means that in most cases software fails a test of
"non-obviousness to someone skilled in the art".

After all, just as a painting consists of many brush strokes, so software
consists of a series of basic primitives lovingly assembled to serve a purpose.



I can't - and won't suggest that I am inherently right and the USPTO is
inherently wrong. However, I think that the US needs to urgently look at the way
that the patent system has been usurped and peverted by a relatively small
number of companies and is now being used for a variety of non-competitive
business practices.

If the first person to produce a portrait in oils had slapped a patent on it,
just think how much poorer the art world would be as a result... The same is
true of software.

[ Reply to This | # ]

I'd appreciate comments on this draft response to the USPTO
Authored by: mjscud on Monday, September 27 2010 @ 01:44 PM EDT
To the patent office:You asked for comments on [Docket No. PTO–P–2010–0067]
Interim Guidance for Determining Subject Matter Eligibility for Process Claims
in View of Bilski v. Kappos

I am commenting in particular on the wisdom of providing patents on algorithms,
from the point of view of a practicing computer programmer.

The U.S. Constitution empowers the congress, and by delegation the patent
office, "To promote the Progress of Science and useful Arts, by securing
for limited Times to Authors and Inventors the exclusive Right to their
respective Writings and Discoveries."

For the patent office to fulfill its constitutional mandate, its policies should
promote the progress of science and the useful arts. In the case of patents,
this particularly means to promote investment in inventions and along with this
discourage the use of trade secrets, so that inventions are created and widely
shared.

If the policies are instead restricting the widespread use of inventions that
would be created anyway, they are unwise policies and should be changed. This is
especially true if the usual effect of a category of patents is for patent
owners to restrict the activity of competitors who have made independent
discoveries.

In these cases, trade secrets would have been useless; the incentive to create
was sufficient without the statutory monopoly conveyed by the patent. Therefore
granting the patent was a loss to the economy as a whole.

If patents on algorithms were serving their purpose, it would be common practice
for programmers to search existing patents in search of solutions. That is, when
confronted with a difficult problem which they needed an algorithm to solve,
they would search first expired patents for algorithms which they could freely
use, and then patents still in force, looking for a less expensive alternative
to developing their own algorithm.

From my own experience and from questioning other programmers, including some
who have patents on algorithms, this is not done. It is a ludicrous idea.

In most cases, a suitable algorithm can be created by the programmer without
searching for prior art. In addition, there exists in academia a large
literature of freely available algorithms. These algorithms are explained with
the intent of being easily understood and reused. The motivation for publishing
these algorithms is not to gain licenses for their use, but rather to gain
recognition for inventing them. The patent applications do not begin to approach
the usefulness of this literature. Therefore the patent literature is ignored.

In fact, when the patent literature is actually searched, it is searched for a
reason in direct contradiction to what our country's founders in their wisdom
intended. Instead of being used to help find ways of doing things, they are
searched to avoid ways of doing things, out of fear that a programmer's
independent inventions will expose them to litigation. This reduces, not
enhances, the productivity of programmers and the progress of the useful arts.

Respectively,
Michael Scudder

p.s. I am using the Creative Commons (BY) Copyright on this.

---
Even a fool, when he keeps silent, is considered wise. Proverbs 17:28

[ Reply to This | # ]

An Open Response to the USPTO -- Physical Aspects of Mathematics
Authored by: jeleinweber on Monday, September 27 2010 @ 02:02 PM EDT

My position is that there should be no thoughtcrimes. No permanent transformation of matter, no patent. A computer controlled method for curing rubber would still be patentable, but mere algorithms or business processes, not.

---
-- Jim Leinweber (Madison, WI)

[ Reply to This | # ]

the test they should be using
Authored by: LaurenceTux on Monday, September 27 2010 @ 03:23 PM EDT
Given a group of fairies or ompa lompas or asguard could they implement the
algorithm by hand??

I would say that if the algorithm requires the use of say fairies (excluding the
use of ompa lompas or asguard) then the patent should be issued LIMITED TO THE
FAIRY PLATFORM.

this way if somebody figures out how to do the same thing with asguards then
they can continue (but can not patent due to the prior art).

So an algorithm that uses some feature of a Windows Intel computer system should
be able to be patented ON THAT PLATFORM (so the next inventor can do the same
thing on a Windows AMD or Linux or Macintosh platform)

[ Reply to This | # ]

Arrrh...My brain hurts.
Authored by: Anonymous on Monday, September 27 2010 @ 05:25 PM EDT
When formulating scientific or mathematical laws, scientists and mathematicians
try to reduce things to their simplest form. Hand things over to patent lawyers,
and they make the simplest things incredibly convoluted and complicated. Just
reading the above interpretations of case studies makes mt brain hurt.

[ Reply to This | # ]

Automated Theorem Proving
Authored by: Anonymous on Monday, September 27 2010 @ 05:42 PM EDT
Given that this is aimed at judges, it might be nice to give them something they can actually see with respect to automated theorem proving. Metamath would be a good fit for that because they encode all the basic rules of mathematics (we call that system 'ZF' after the mathematicians who came up with it) as software.

In particular, for people who can't make much sense of anything else, I would refer you to their proof that 2+2=4 using only basic axioms.

The list above was produced by typing the commands "read set.mm" then "show trace_back 2p2e4 /essential /count_steps" in the Metamath program. By the way, the complete proof of 2 + 2 = 4 involves 2,452 subtheorems including the 150 above. (The command "show trace_back 2p2e4 /essential" will list them.) These have a total of 25,933 steps — this is how many steps you would have to examine if you wanted to verify the proof by hand in complete detail all the way back to the axioms.

[ Reply to This | # ]

Too much philosophy, and a specific mistake
Authored by: Anonymous on Monday, September 27 2010 @ 07:06 PM EDT

Like your previous article on this subject, this article gets so lost in philosophy and metaphysics, that it looses the connection to the realities of the subject matter of distinguishing between patentable inventions and unpatentable math.

Anyway, your article also contains a more specific mistake: In your critique of the ruling In Re Alappat you read the ruling as if it got the direction of modelling wrong. I do not think that it did so, the ruling (as quoted by you) clearly understands that the circuit models the math, then allows the patent because it doubly restricts its coverage in two ways:

  1. It restricts itself to a specific physical use of the formula, namely to produce an antialiased rendering of a curve, thus not claiming other uses, such as computing taxes or producing an antialiased rendering of the letter "W". That it does not restrict the choice of pixel display method or color is much less important than that.
  2. It restricts itself to one particular physical representation of the formula, which is clearly not the only circuit that can be built to do so. For instance it does not claim a circuit where the lookup tables are replaced by circuits computing the desired values on the fly (no ROM), nor a circuit with less (or different) computations (e.g. more ROM), nor circuits that operate on the signed differences rather than absolute values. It took me just a few minutes to think up the principles of at least one alternative circuit (using adders and digital comparators, plus a few output gates).

Thus the ruling was that this patent did not try to cover all uses or implementations of the mathematical formula, just a specific implementation for a specific useful purpose.

It is worthwhile to compare such patents to pre-computer patents on obtaining some result by doing something physical in accordance with a formula, such as the Wright brothers patents on how to compute the best shape of an airplane wing, in order to make it fly.

It is also worthwhile to compare to patents on particular ways to build a calculation device, such as a microprocessor or a slide rule (I am too young to have had formal training in those, but I actually used one in place of my calculator while taking a written exam once, just for the fun of it).

While it is probably too late in the day to send new comments to the USPTO, a much better example of passing "machine or transformation" while still failing the math exception is to consider if the patent application or a subset of its claims effectively tries to cover *any* machine embodying the formula in any way. Many dubious patents do just that, formally reciting the generic characteristics of computing machines as if they were magical incantations to circumvent the law, thus making the patents more difficult to read for both examiners and practitioners, without actually changing that which is claimed beyond the trivial, obvious, known, excluded or otherwise unpatentable.

[ Reply to This | # ]

Bilski-Related appeals at the U.S. Patent Office
Authored by: macliam on Monday, September 27 2010 @ 08:09 PM EDT

Recently I started skimming Final Decisions of the Board of Patent Appeals and Interferences again (which I hadn't done since soon after In re Bilski hit Groklaw) to get an idea of how the U.S. Patent Office were applying the Bilski v. Kappos decision.

This has involved going to the webpage making available to Final Decisions of the BPAI and searching for recent decisions including the search term 'Bilski'.

Here is a recent recent final decision rejecting an IBM software patent. This decision employs forms of words that I have seen in rejections of similar patent applications as being drawn to abstract ideas. Typical of this sort of rejection, by my observation, are the quotes from In re Nuijten, In re Ferguson and In re Warmerdam. The last of these supplies the following principle:

A claim that recites no more than software, logic or a data structure (i.e., an abstraction) does not fall within any statutory category.

Also In re Nuijten is often cited in these sorts of rejections, since it holds that signals do not fall within any statutory category, and thus claims that can be interpreted as encompassing electrical signals are unpatentable.

The decision I have linked to includes the following paragraph. I have noticed this paragraph, or similarly worded paragraphs, in recent decisions rejecting certain software patent applications as being drawn to non-statutory subject matter.

Consistent with our earlier-noted invention statement taken from Appellants’ Summary of the Invention in this application, the disclosed and claimed invention is directed to software per se, abstract ideas, abstract concepts and methodologies and the like, including various data structures and named entities, such as various labeled modules and processes and various abstract, logical relationships and functionalities associated there with, an abstract model, software components/applications, and abstract intellectual processes associated with them within the claims on appeal.

It seems to me that, rather than making the case for unpatentability of software on the grounds that computer programs etc. are equivalent to mathematical algorithms and structures, and relying on precedents establishing that mathematics is unpatentable per se as an abstract idea, it might be noting that the Patent Office seems to be rejecting certain sorts of software claims as being drawn to abstract ideas, without needing to address the question as to whether they are unpatentable as being essentially mathematical. The paragraph quoted above gives a list of features of certain types of software claims that the Patent Office itself currently considers as non-statutory. Anything else to add to this list?

And how does the BPAI deal with the machine-or-transformation test? In a number of recent decisions, they note that it was described as an important test in the Bilski v. Kappos decision, apply it, and, if the claims fail the test, they note that this in itself is not conclusive proof that the claims are drawn to non-statutory matter. But they must next consider whether the claims are drawn to an abstract idea. Then they cite Gottschalk v. Benson, and say that the claim at issue is of a similar nature to the process rejected as unpatentable in Benson, is therefore itself not patentable. I have come across a number of recent decisions that follow this structure.

[ Reply to This | # ]

Awesome work, but pseudonymous
Authored by: Anonymous on Monday, September 27 2010 @ 10:36 PM EDT
I have immensely enjoyed both of your articles. I'm not in
a mathematics-related profession, but was a sort of math
prodigy in school and have been an amateur math enthusiast
ever since.

Anyway, I have to wonder if your fine work would be accorded
more weight by our policy makers if it were authored under
your real name rather than just an online pseudonym. I
realize there could be factors that preclude this.

Either way, thanks immensely!

David Bruce

[ Reply to This | # ]

An Open Response to the USPTO -- Physical Aspects of Mathematics
Authored by: Anonymous on Tuesday, September 28 2010 @ 12:57 AM EDT
Patents seem to assume the normal development/manufacturing/life of a product.
If you make machines made of cast iron widgets the life of a patent may seem
reasonable. OTOH software life cycles are far less, 1-5 years may be typical.
Perhaps this is where a disjoint occurs? Also, how many, err cough, 'inventions'
would be better served by copyright being a clear and particular implementation
of an idea, rather than a distinct 'invention' of an idea?

Tufty

[ Reply to This | # ]

Untimely - why wasn't this posted in July?
Authored by: Nagle on Tuesday, September 28 2010 @ 02:16 AM EDT

This was posted one day before the closing date, even though it was published in the Federal Register back in July.

Not helpful.

[ Reply to This | # ]

Slicing and Dicing.
Authored by: Ian Al on Tuesday, September 28 2010 @ 04:35 AM EDT
I have used POiR concepts and methods (neither of which can be protected by
copyright - are you listening, SCOG?) to take an alternative look at some of our
favourite patent cases.

I draw from the court opinions the requirements they put on processes and
machines when considering patents. From this I develop a short set of
requirements for process patent claims and machine patent claims. I try these
out on the older cases and see if they align with what the courts found.

I claim no expert knowledge of patent law and how it is applied. I have only
used common sense to use the information given in POiR's legal references. PJ
will confirm the madness of applying common sense to predict court judgements!
After much thought I decided that my understanding of math was less than my
understanding of patent law and you will find few references to goats and
groves.

It is a large comment and so it appears as a sibling post.

---
Regards
Ian Al
SCOG, what ever happened to them? Whatever, it was less than they deserve.

[ Reply to This | # ]

Blatant Math Patents
Authored by: Epicanis on Tuesday, September 28 2010 @ 05:25 PM EDT
Check out US Pat#6253162: "Method of identifying features in indexed
data", issued in 2001.

If I'm reading that right, the claims are purely mathematical in nature.

"What is claimed is:

A method of identifying a feature in indexed data of responses, comprising the
steps of:

(a) selecting a subset of indices having a beginning index and ending
index;
(b) computing a measure of dispersion of said subset of indices using a
subset of said responses corresponding to said subset of indices as histogram
frequencies; and
(c) comparing said measure of dispersion to dispersion critical value;
wherein said dispersion critical value exceeds a background level of
dispersion.

2. The method as recited in claim 1, wherein said beginning index and said
ending index are advanced at least one index for computing a second measure of
dispersion.

3. The method as recited in claim 2, wherein said dispersion critical value
exceeds a background level of dispersion.

4. The method as recited in claim 3, wherein a noise is identified as data
corresponding to weighted measure above said dispersion critical value and a
signal is identified as data corresponding to weighted measure below said
dispersion critical value.

5. The method as recited in claim 4, wherein said weighted measure below said
dispersion critical value is a plurality of weighted measures having consecutive
indices.

6. The method as recited in claim 1 repeated for at least two replicate data
sets for a sample.

7. The method as recited in claim 6, further comprising the steps of obtaining
an estimate of expected value and an estimate of uncertainty of the data from
said at least two replicates for both the index and the response corresponding
thereto.

8. The method as recited in claim 7, further comprising displaying said estimate
of expected value and said estimate of uncertainty together.

9. The method as recited in claim 1, wherein said data is selected from the
group consisting of spectral data, chromatographic data, time series data, and
combinations thereof.

10. The method as recited in claim 1, further comprising computing a weighted
statistic of the index for characterizing said feature."

[ Reply to This | # ]

Abstract == information processing
Authored by: jbb on Tuesday, September 28 2010 @ 07:43 PM EDT
Note: PolR kindly asked me to repost this here so that's what I'm doing.


I think your analysis of in re Alappat is spot on. I also thought the ruling in the Diehr case was spot on. The court said that the invention was patentable because even if all the computational parts were considered prior art, when those parts were combined with the rest of the apparatus then the machine as a whole was patentable.

This concept is easily generalizable and gives a very clearcut way of avoiding patenting abstract ideas. For any patent application, take the parts that are pure information processing and put them in "black boxes"[1] that are considered to be prior art. If the remaining (physical) parts of the invention pass all the other tests then the invention is patentable but everyone is free to use the stuff that was inside the information processing "black boxes".

This idea is very simple. It defines "abstract" to mean pure information processing. It prevents patenting any parts of an invention that are pure information processing and only allows patenting of the physicals parts of the invention. The majority opinion for in re Alappat wanted to have their cake and eat it too. As PolR highlighted, the court said:

Indeed, claim 15 as written is not “so abstract and sweeping” that it would “wholly pre-empt” the use of any apparatus employing the combination of mathematical calculations recited therein.
The problem is that they just created a huge muddle that we are now trying to sort out. The muddle was created by the court not explicitly removing the abstract parts of the invention from the patent claim. In the highlighted passage above, the court said other people would still be allowed to use the abstract ideas in the patent claim but it was perfectly unclear (as PolR demonstrated) when those abstract ideas could be used freely and when their use would infringe the patent.

My suggestion completely clears up this ambiguity. People would always be free to use the parts that have only information going in and only information coming out. Judges and lawyers would no longer have to have Ph.D.s in computer science and information theory in order to figure out what is abstract and what is not. If you can reduce it to pure information processing then it is abstract and you're allowed to use it. This is a simple and straightforward delineation that almost everyone can see without having to invest a fortune in lawyers. Even if it is not perfect, it removes the minefield of uncertainty that is currently hindering innovation.

The in re Alappat minority dissent was striking:

A vigorous dissent by Chief Judge Archer, joined by Judge Nies, argued that the majority's opinion will have "untold consequences" by making "mathematical functions" patentable.

... The dissent was concerned that Alappat's claim could be read broadly to encompass its use in "computer monitors, televisions, laser printers, mechanical printing devices," as well as oscilloscopes.

In other words, the dissent was concerned that the abstract ideas in Alappat's claim were not tied down to any particular physical input and/or output device. My black box approach solves this problem explicitly by only allowing the physical input and output parts to be patented.

As I have mentioned before, one of the features of the simple delineation I suggest for what is abstract and what is not is that as more and more once patentable processes become completely computerized, fewer and fewer of them will remain patentable. This is a feature not a bug. Once people are grown up we no longer pat them on the head and put their finger paintings on the refrigerator. Likewise once our society evolves technologically to the point where a process can be completely computerized, it should no longer be patentable because granting a patent monopoly on abstract information processing only hinders progress and innovation. Things on the path toward full computerization should only have physical input parts and physical output parts that are patentable.

Back when patents were first invented, there were no "black boxes" that dealt with abstract information. The inventions were entirely physical. They encompassed the physical input part and the physical output part. This is key because when an invention is tied to its physical input part and its physical output part, it has a completeness, an end-to-endness, a specificity, that is lacking in pure information processing. What I mean is that a complete physical invention is unlikely to be used as a building block for other inventions the same way information processing ideas are used as building blocks to create new software. Sure, physical patentable inventions can be small parts of a greater whole. The difference is that a physical invention is tied to a particular physical input and output while abstract ideas and information processing are not.

It is already impossible to write a significant software project that doesn't violate hundreds if not thousands of patents. The same is not true of physical machines. This is because many (probably most) software patents are way too broad and way too obvious but our patent and court systems are technically incapable of figuring this out. Any solution that depends on these two systems discerning obviousness or over-broadness in software patents is bound to fail.

Information processing is an abstraction pretty much by definition (I'm tempted to say that information and information processing are the embodiment of abstraction but that might be very confusing). I believe this is a much stronger and more direct argument than the roundabout way of saying that all software is math therefore all software is an abstraction because math is an abstraction.


[1] A "black box" is a engineering term that means we know what goes into the "black box" and what comes out but we don't care (or perhaps don't know) about the details of what goes on inside the "black box".


---

[X] Ignore DRM Restrictions

[ Reply to This | # ]

PoIR, what is your next target?
Authored by: reiisi on Saturday, October 02 2010 @ 09:25 AM EDT
I'd like to see what you do with, for instance, the four classes of mathematical
grammars and where natural language fits in vs. where most physical machines fit
in.

(And why being context-free is not the freedom we should be protecting with the
Constitution.)

[ Reply to This | # ]

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