# Patents & Mathematics, by Attila Kimpan

It is my pleasure to present to you a guest contribution by **Attila Kimpan of Maiwald Patentanwalts- und Rechtsanwalts-GmbH**. Attila presented some of the below material on patents at the ECMI conference in Budapest in June this year.

**Patents & Mathematics**

Mathematics lies at the heart of modern industry. Call it “big data”, “digital” or, as the Germans do, “industry 4.0”. There is no shortage of different monikers with different degrees of aptness. But what is common is that it is “data” that seems to be grabbing all the attention. Data is hailed as the “Oil” of the new industry. See, for example, “The Economist” cover story on this in their 6 May 2017 UK issue. Indeed, the ready availability of large amounts of data undoubtedly opens up interesting options for businesses. In the wake of “datamania”, questions about ownership of data have been raised and new schemes are being sought to protect businesses in their commercial pursuits. Those are valid questions. And yet, it sometimes appears that the cart is put before the horse: because what literally drives industries – modern or old – is not the data. It is the mathematics! Mathematics in action are algorithms. And it is the algorithms that turn the data into something useful. It is the algorithms that extract, find, predict, secure higher yields or throughputs in industrial control processes, etc. In short, it is the maths that actually does things! Without algorithms, data is useless. It is mathematics that turns data into something “Big”. And if ownership of data is of concern, so should all the more ownership of the algorithms. But, unlike for data, there is already an answer: Patents! Patents can protect ideas, principles. Yes, mathematics can be patented, and businesses and academia should keep this in mind. You can patent the mathematics, if it has a technical effect in the context of solving a technical problem.

Patents confer a right in a technical invention to the patent owner. A patent is not a price (although it surely has one), but a commercial weapon: the right can be exercised to preclude others from using an invention commercially. To get a patent, one must convince a patent office that the invention is new and inventive over what is already known (“prior art”). Novelty is not the same as inventiveness. Something might be new, but not inventive: novelty is necessary, but not sufficient for inventiveness. As indicated, patents cost, and this should be balanced against the commercial usefulness of the matter one wishes to patent. Patents are a matter of national law but, to obtain patents, international mechanisms are in place such as the EPC (European Patent Convention). The EPC is administered by the European Patent Office (EPO). The EPC allows one to obtain, in a single grant procedure, a “bundle” of patents for a group of different countries. The EPO is not an EU institution, but the group of countries includes all EU countries (and more).

Those who thought mathematics is not patentable may be forgiven, as the facts seem disheartening at first. Let’s ask first what is mathematics? For most purposes, mathematics is a discovery of hidden relationships between entities. Symmetries for example are an important class of such relationships. Useful mathematics is a bag of tricks, exploiting these relationships. And for most practical purposes, mathematics is put to action as algorithms that run as software. Sadly, most patent systems exclude “mathematical methods”, “discoveries” and “programs for computer” from being patented. So there is, on the face of it, a triple whammy why mathematics cannot apparently be patented. Alas, not all is lost because there is – depending on which side you are on – a “small print” or “loophole” to those exclusions. The exclusions bite only if one attempts to patent “mathematical methods as such”. Same for discoveries and computer programs. The “as such” portion has been interpreted and developed through case law before the EPO’s Boards of Appeal. The approach that emerged may be summarized as follows: if the mathematics is used to solve a technical problem, then a method/ system/ device that uses this mathematics may in principle be patentable – provided, of course – it is also new and inventive. The use of the mathematics will most likely be in the form of a new algorithm implemented through software on a computing device. “Mathematical inventions” then are really a type of “computer-implemented inventions (CII)” for which a long history of settled jurisprudence exists, at least before the EPO, largely favourable to CII in most practical engineering applications. In short, if using the mathematics results in a better computing device for solving the technical problem, a case for patent can be made.

An example may be in order to illustrate the above: consider the task of computing the sum S of the first N integers: S=1+2+ …+N – like Carl-Friedrich Gauss did in around 1787 when he was a seven-year-old. As one of the famous anecdotes relates, a bored and tired Schoolmaster thought of this task in the hope that adding up all numbers up to 100 will keep the class busy for a while. How wrong he was! The young Gauss called out at the result within minutes, “5050!”, whilst his peers were still busy iterating towards the result by tediously adding, in sequence, one number to the next. Dead silence… What has happened? Gauss understood the symmetry that resides in the task and found a beautiful formula. There are N/2 partial sums that each yields N+1. He concluded that S= N/2 times N+1 (this approach works for N odd also).

Playing a time traveller’s game, and assuming he had cared, would Gauss have gotten a patent for this using the EPO’s effect-based approach? Let’s walk through this example. The closed formula for S is, at this stage, although impressive, a mathematical discovery “as such”. There is no patent for this. But one can see that rather than computing S for each N by iteration, as it had been done before, the Gauss formula is much quicker. We need fewer computational operations than in iterating. Now the mere mathematics turns into something more palpable, as technical effects emerge from what was once a mathematical discovery! And assuming there is an engineering problem (probably there isn’t), for example, a signal or image processing task, that requires one to compute S for different N’s, then a case for a patent could be made. Incorporating the formula for S in such a computing device could be patentable because we end up with a better, quicker, computing device for the task. Interestingly, although the difference of the Gauss formula over iterating is purely mathematical, there is nevertheless a technical result of quicker computation! A difference that is purely mathematical, and hence in isolation would be excluded from patenting, may nevertheless be patentable if it unleashes useful technical effects when practiced in a technical context. One such useful effect is producing results more quickly, as in this example. It is interesting that the EPC and many other patent systems do not actually say what “technical” actually means. Unlike mathematics, this is no exact art. But this is on purpose: for one thought it better to keep the term woolly enough, so it can morph alongside industrial evolutions through jurisprudence. This is how most of the field of computer engineering eventually came to fall under the term “technical”. At present, “technical” will cover most engineering applications but it would not normally cover, for example, business methods or administrative tasks.

In general, if the mathematics, when used in a technical context, produces results quicker, uses less storage, provides more accurate results, etc, then this may indicate that the mathematics may be patentable in this context. At least there is then probably no outright exclusion from patenting. But you still need to get over the other two main hurdles (there are others): novelty and inventiveness (better known as “inventive step”).

For those who think the above Gauss example is artificial, it probably is. But it has the merit that it may also illustrate in simple and plain sight the inventiveness hurdle: for why would anyone tasked with working out S get the idea of dividing things into two? The factor “N/2” in Gauss formula appears odd, given the problem. This is how inventiveness “feels” like. Do something that goes against what one of “average skill in the art” would do to solve the problem. But unless you discover the symmetry in the data, as Gauss did, the formula may be out of reach. However, this is an illustration and in practice there are more factors to consider specific to a case and the conclusion on inventiveness depends on all the information of record. And, of course, you do not have to produce “Gaussian” scale mathematics to get a patent.

If the above excursion is too much of a toy example, do not fret! Real examples of patented “mathematical inventions” abound, including the Diffie-Hellman key exchange for public-key cryptography, or the “MP3”-compression method. Perhaps one of the most useful mathematical inventions of all times, is the Fast Fourier Transform (“FTT”). It drives data munching, big and small, in practically all electrical engineering as we know it. It propels an otherwise sluggish and unwieldy number tinkering to jaw-dropping speeds, based on yet another discovery of symmetry in the data that becomes apparent only when mapped into the complex plane! FTT delivers a lighting fast runtime in the order of o(N*log*N), down from a slumbering o(N2). Such are the technical effects mathematical patents are made from! The reason FTT was not patented when it was discovered in the US in the mid-1960s had nothing to do with patent exclusion (or “patent eligibility” as it is called in the US). It had to do with FTT being, in fact, a re-discovery. Others have gotten there before, including no other than Gauss himself at around 1805, when trying to get his head around astronomical data. At least, inventiveness was doubted, given what was already known at the time. James W. Cooley, one of the FFT inventors, wrote engagingly about this. J.W. Cooley, “The Re-Discovery of the Fast Fourier Transform Algorithm”, Mikrochim. Acta, Vienna, 1987, III, pp. 33-45, in particular page 36.

And this brings us back to Big Data, but not only. The field at present is largely dominated by developments in machine learning algorithms for all things autonomous. These algorithms operate on vast amounts of data. Being able to see symmetries in the data or in the data processing, making judicious simplifications and shortcuts that still produce useful results, coming up with potent models to explain data, all these are likely discoveries which may result in powerful algorithms in Big Data and beyond that outclass the competition, are faster, more accurate, etc. It is all well and right to explore new concepts around data. But businesses should be jealously seeking to ensure that these powerful algorithms are protected. And the single best tool to achieve this are patents.

Attila Ferenc Kimpan

European Patent Attorney

Munich and London