Symmetries Part I

Finding symmetries is often essential for solving a problem. However, it is necessary to specify what is meant by “symmetry”, to indicate how one can find them, and finally to show what they are used for.

There are several kinds of symmetries. I am only interested here in symmetries that are associated to a permutation of the unknowns of a problem: when each unknown is replaced by the corresponding unknown in the permutation, both sets of constraints are identical. For every solution, one has immediately a symmetrical solution when one gives to each unknown the value of its corresponding unknown in the permutation. Therefore, for each solution, one also has as many new solutions as there are symmetries. In CAIA surprised me Part II, we have seen that CAIA had found 47 symmetries for a magic cube problem: every new solution generates 47 symmetrical solutions. Now, we will see how finding symmetries is useful for solving problems.

The first reason is that it considerably decreases the number of solutions that one has to find: for the magical cube, the number of solutions we are seeking is divided by 48. Moreover, symmetrical solutions are usually no longer interesting when one has found one of them, the user considers them equivalent; one only indicates their number. However, when there is no evident geometrical interpretation, it is sometimes difficult to see that two solutions are symmetrical: giving them is not always a complete waste of time. Naturally, CAIA adds constraints to the problem formulation so that it does not generate its symmetrical solutions. This is useful since the more a problem is constrained, the easier it is easy to solve.

The second reason is that it makes the search for new constraints easier: when one has found a constraint, all the constraints generated by the symmetries are also true. For example, with the magic cube problem, the main step in its successful resolution was the discovery on one constraint with only three unknowns: F(13)+F(14)+F(15)=42. A combinatorial search is more efficient with constraints with three unknowns rather than with constraints with 10 unknowns, which happens for all the constraints on the definition of this problem. Therefore, it is possible to apply the 47 symmetries already found. Two of them immediately give a constraint: F(11)+F(14)+F(15)=42 and F(5)+F(14)+F(23)=42. Unfortunately, one has not 47 new constraints: all the other symmetries give one of these three constraints. Without this, it is not evident that the system could also find the last two constraints: it does not consider all the possible derivations and, in any case, that would require much more time.

The third reason was a pleasant surprise. I did not expect it: CAIA could solve some problems after finding their symmetries, while it did not find any solution when it did not search for them before. The explanation is that it happens that the only (or the easiest) way to prove the goal is using a proof by cases. However, difficult decisions have to be made in order to define the constraints that define subsets of the possible values of the unknowns. Fortunately, symmetries offer these constraints on a plate: they have been added so that CAIA finds only one symmetrical solution. Moreover, usually, for a proof by cases, one must successively consider all the cases; it is sufficient to consider one of the cases! I have no room to explain this here, but I will give an example in the following post: CAIA easily finds all the solutions of a particular problem when it has discovered its symmetries; otherwise, it finds nothing.

Unfortunately, besides these positive points, sometimes there is a drawback: too many symmetries. One can waste a lot of time for finding them. This may happen when a problem has billions of solutions; it may also have millions of symmetries. In that situation, finding symmetries have no interest; the difficulty is to find at least one solution.

Searching for symmetries is a small, but important, part of CAIA. This is a meta-problem, that is a problem helping to solve other problems. Therefore, it is an important step in the bootstrap: CAIA finds symmetries, and this improves its performances. It was not necessary to write new modules for finding symmetries. It was sufficient to define the problem of finding symmetries in the same way as the other problems submitted by the users.

Mathematics have disappointed me


Since Gödel, we know that mathematics have limitations: some statements cannot be proved, and cannot be refuted, that is one cannot prove their negation: for both, no demonstration exists. Many works have been made for this problem, and they show that, in some theories, some statements are true, although no proof exists. Some of these proofs are constructive, and they show statements in this category; usually, these statements use reflexivity. We know the Epimenides paradox, who is Cretan and says that all Cretans are liars. I was not embarrassed that so strange statements could not be proved.

However, for a recent post, I have considered a system that created new conjectures. It found several variants of Goldbach conjecture: every even number greater than two may be decomposed as the sum of two prime numbers. It had been a long time since I knew this conjecture, and I was not really annoyed that it had not been proved, although many mathematicians tried to do it since 1742. Indeed, I believed that it was a very difficult problem, that numbers greater than billions of billions had only one, two or three decompositions: it was ever likely that no decomposition existed. If so, it was possible that this conjecture was false; even true, it would be very hard to prove it.

Writing my preceding blog, I read the Wikipedia entry on Goldbach conjecture, and I saw that the number of decompositions is huge; moreover, it is strongly increasing with the value of the numbers. Therefore, with CAIA, I have made some experiments, I had only to write three rules. I was shocked by the results: large numbers really have a lot of decompositions. The greatest number with at most 1 decomposition is 12, with at most 10 is 632, with at most 100 is 11,456, with at most 1000 is 190,562, etc. I have stated a new conjecture:

The number of decompositions of an even number N, greater than 15,788, into the sum of two prime numbers is greater than the square root of N.

As a matter of fact, the greatest number for which this conjecture is false is 15,788: it has 125 decompositions, less than the square root of 15,788: 125.65….

Naturally, I have not proven this conjecture, but I am pretty sure that it is true. When we consider large numbers, they have even much more decompositions than forecast. Moreover, when a number N has small numbers of decompositions, several of its neighbors have similar values: the curve representing the number of decompositions has no pick towards the bottom. CAIA has studied the first hundred even numbers from 100,000,000: the minimal value of the number of decompositions is 218,281 for 100,000,144 (we can notice that it is much larger than the square root of the smaller of these numbers, which is 10,000); among these 100 numbers, 12 have a number of decompositions between 218,281 and 219,000. On the contrary, the abnormally high values are often isolated from their neighbors, there are picks towards the top: in the preceding interval, the largest number of decompositions is 723,776, and the value of its immediate follow-up is only 595,554. It seems very unlikely that there exists a large number N with less decompositions than the square root of N; it is still harder to believe that there is an even number without any decomposition.

Goldbach conjecture has been checked until 4.1018 ; then, if my conjecture is true, every even number not yet checked has at least two billions decompositions! And mathematics cannot prove that there is at least one!!!! There would be no proof of a result which is ultra-true.

I do not believe that it is due to an inability of mathematicians; I have a tremendous admiration for the way they succeeded in developing mathematics. They are extremely competent, as long as they do not speak of AI. This is a weakness of mathematics: one cannot find a proof because it does not exist. And this happens for very simple statements, such as the Goldbach conjecture, that anybody can understand easily.

Then, one question arises: are there many conjectures of this kind, simple, true, with no existing proof? Some could say that the decomposition of a number as the sum of two prime numbers is an irrelevant problem. I do not agree: a similar problem is the decomposition of an odd number as the product of two prime numbers, and it is very important, especially for cryptography.

It is possible that the true and provable statements are only a small subset of all the true conjectures. In that situation, along with mathematics, we have to create a new field of research where one would find true (and sometimes ultra-true) conjectures for which no proof exists. Then, with all the power of mathematics, we would use them: the absence of formal proof must not deter us to do that. We know the importance of the Riemann conjecture; there are probably several other conjectures that would also be useful.

AI is ideally suited to this kind of research: a system can create and check a huge number of candidates, much more than any human being can do. It is not enough to build artificial mathematicians, whose performances would significantly exceed those of the best human mathematicians. Improving the investigation mentioned in a preceding blog, a new branch of AI will have to create conjectures very cleverly.

They are going to judge us!

Several AI systems currently have performances well above those of the best human experts. This allows the realization of systems that can assess the quality of human performances, much better than we could do.

Particularly, for many games, some AI systems are far better than the world champion: a Go program has recently won against two of the best Go players. Here, we will discuss of an outstanding study made by Jean-Marc Alliot on Chess; it was published in the first 2017 issue of the Journal of the International Computer Games Association.

For about fifty years, a method is used to determine the strength of chess players: an integer, his Elo, is associated with each player. It is computed from the result of every match that he has played (win, draw, loss), and from the Elo of his opponents. Now, Magnus Carlsen, the world champion, has also the highest Elo: 2857; less than 800 players have an Elo greater than 2500, most of them are International Grandmasters. It is difficult to evaluate Elo for the best chess engines: human players are not strong enough. Therefore, matches between human and computer have become very rare. Moreover, when a human agrees to play, he often requires to fight against a crippled engine for instance, without endgame table base or with odds (usually a pawn). As such, Elo for chess engines is mainly based on competitions between themselves. For the present time, the best ones are almost at 3400. With Carlsen, the difference is over 500; this indicates that the engine would win a game with a 0.95 probability.

Lacking anything better, chess players were content to use this rating system, although it can evaluate neither the moves, nor the quality of a game. Now, for the best chess engines, we can consider that they are playing the best move. If a human player chooses another move, one can evaluate its quality: it is sufficient to find the value given by the computer to the position after its move and the one after the human move. The value after the computer move is always greater or equal to the human move: if it was lower, it would not have played its move.

In fact, the author could not use a system with all the Elo difference that is theoretically possible: he cannot access a computer with all the processors which were used for the best performance; moreover, in order to limit computer time, the time allowed to a move was decreased. The system used for these experiments was a chess engine in the top three, STOCKFISH, which has also the benefit of being open source. With the restrictions on the computer speed, the Elo advantage on the world champion is now 300; the engine will still win, but only with a 0.85 probability.

Knowing the difference in value between the computer move and the human move allows to know the quality of each human move exactly. This could be useful to annotate a game, showing the good moves and the weak ones and, for each of these, to indicate what would have been the best move. However, this was not the goal of this paper, which has other purposes. The basic element is the construction of a matrix giving the probability that the move played by a particular player will change the value of the position; this probability depends on the value of the position before this move.

For each year of activity, this matrix has been computed for all the players that have been world champion, but not for those who played against a world champion, and never win: all the Ponomariov games have been considered, but not all those of Kortchnoi. Naturally, one considers only the games played at regular time controls, and in normal conditions: one does not keep blitz, blind, simultaneous, odds games. There is one matrix for playing White, and another one for playing Black. All in all, the system has analyzed 2,000,000 positions.

An element of the matrix is the probability that, when the value of the position is VA, the value of the position after the move has been made is now VB. The values are measured in pawns or in centi-pawns. For instance, we know that, in 1971, if Robert Fischer, playing White, is one pawn late in a position, after playing his move, he would still be one pawn late with a 0.78 probability, 1.4 pawn late with a 0.12 probability, and 1.8 pawn late with a 0.10 probability. The new value can never be better than the old one since we assumed that the machine is infallible.

Thanks to these analyses, the author describes several interesting experiments; for example, it is possible to find the probability of the result of a match between two players, when the matrix is known for both. One assumes that the game is won by a player when he is at least two pawns better. As one has Black and White matrices for Spassky-1971, and the Black matrix for Fischer-1971, it is possible to compute the ten elements vector that gives the probability of the result of a game between these players in 1971. Here are some of these values: Fischer wins (0.40), Fischer’s advantage at the end is 0.6 pawn (0.07), perfect equilibrium (0.14), the final position is -1.4 pawn (0.01), Fischer loses (0.07). I emphasize that, at this step, the computer plays no move: it uses the vectors indicating the performances of each player. It only plays moves for computing the matrices.

These methods play a different role than the Elo, which evaluates a player for all its games against many players from only their result. Here, one is interested in the moves, and not by the result. Moreover, one does not define a vector against any player, but against one particular player in a particular year. With this method, it is possible to find the result of a match between Fischer-1971 and Fischer-1962! The paper gives the results of a virtual competition between the 20 world champions, taking for each one the year where he was the best, which is not always the one where he was the world champion. For instance, we learn that Kramnik-1999 had a 0.60 probability to win against Lasker-1907. In some cases, the result is analogous to Condorcet paradox: Petrosian-1962 wins against Smyslov-1983, who wins against Khalifman-2010, who wins against Petrosian-1962!

I cannot summarize a paper which contains many remarkable results from the analyses made from all the matches played by at least one past, present or future world champion. In his conclusion, the author plans to achieve this also for all the games in Chessbase where both players are above 2500 Elo.

This paper shows that it is extraordinary helpful to have an AI system that is well above the best human beings: one can very precisely appraise human behavior, and one can compare the performances of people who lived at different eras. Studying the capacities of an individual is therefore made with accuracy and completeness, incomparably better than with multiple-choice tests. Who knows, in some distant future, AI students will perhaps compare in their thesis mathematical geniuses such as Euclid and Poincaré!

An impossible task: to foresee the future of AI

Sometimes, we see predictions on an idyllic future of AI or, more often, on the catastrophes that it will cause. We love to scare ourselves, in that way we are sure to make newspaper headlines. However, given the vagueness of these predictions, it is impossible to see whether we could overcome these potential difficulties. Some even want to stop AI research; this is due to a distrust of science, and will leads to the catastrophe that they want to avoid. Indeed, to run efficiently a country or a large company is a daunting challenge; I believe that it exceeds the capacity of human intelligence. Refusing AI’s help will more surely lead to a catastrophe. In particular, some are worried that the robots would take power, and will enslave us. They believe that all intelligence must be similar to the human one; therefore, robots will be aggressive and overbearing, as ourselves. In fact, we have these characteristics because our intelligence is a product of evolution in a resource-constrained environment. However, Darwinian evolution is not the only path for creating intelligent beings; I even think that it is not the right direction in AI research: it requires too much time and too many individuals.

It is unrealistic to predict how AI will turn out for the long term. To be convinced, it is enough to look at what recently happened in Computer Science. Sixty years ago, who saw what it is now? I took my first steps in this domain in 1958, and almost all those who were in this area thought it would be useful for scientific computing and business management; nevertheless, in those days, no one was thinking about the Internet and the Web. Moreover, we did not think that the cost of computer time would decrease so fast. Only one hour of computing on the fastest machines was very expensive, it far exceeded a one-month salary. Their power seemed amazing: almost one Million Instructions Per Second for the IBM 704, the workhorse of many of the first AI realizations! We did not think that their power would incredibly increase, while their cost would incredibly decrease: c.1965, someone suggested (and I am not sure that he believed it) that, in the future, the computer plant visitors would receive the CPU as a key fob. We all laughed at this joke, how could such a precious component could become so cheap? Changing drastically the cost of computing has made it possible to realize applications that were unthinkable. This is the main reason in the mistakes that I had made in 1962, when I had written a paper describing the state of AI. Naturally, there was a section on its future; among my predictions, some were true, and some false. The main reason of my mistakes: I had not seen that the computer cost would go down so much, and that the computer power would go up so much.

However, it is possible to predict some specific achievements: for instance, there will be self-driving cars: this is the normal course of events for a research well under way. It is reasonable to think that with AI improvements, some professions will disappear, such as it already happened with computers. Unfortunately, many changes are impossible to foresee: they will depend not only on new research directions for AI, but also on progress in other domains, particularly with computers.

I strongly believe that all human activities, without any exception, could be undertaken by AI systems, and they would be much better than us. However, I do not predict that this will happen, even in the far future: human intelligence is perhaps not enough to reach his goal. Creating systems that create systems is an extremely difficult area, and evolution did not optimize our capabilities in this field.

Moreover, the research structure does not encourage what needs to be done. Many researchers do an excellent thesis but if they want to pursue a career, they must quit the kind of research that is interesting to the future of AI. New ideas come naturally when one develops a large system with a computer; to do that, for many years one must spend at least half of his time on it. This is impossible if one also has important responsibilities as a teacher and as a manager. Besides, the weight given to publications is not a favorable element in AI research: how can we describe a system that uses much more than 10,000 rules? It is almost impossible to do for a system with many meta-rules that create new rules. It is much easier to write a theoretical paper, which will be understood easily. It is no coincidence that several teams that recently achieved spectacular results were not from the university, but from the industry. However, the industry’s goal is not to develop research for the very long term: profitability is important for any business. I believe about the importance of bootstrapping AI, although this will take a lot of time and the results will be poor for long enough. This encourages neither the university, nor the industry to engage in this way.

I do not want to give precise predictions for the long term, mission impossible. Nevertheless, I am sure that if we succeed to bootstrap AI, the consequences will be immeasurable: intelligence is essential to the development of our civilization. However, we cannot conceive what could be a super-intelligence, in the same way that a dog cannot conceive what is our intelligence. And, finally, it is very possible that this goal will never be achieved because human intelligence is too limited for such a huge task; even so, it is worth a try.

A mathematician is more than a theorem prover

 I always believed that AI systems will become excellent mathematicians. Until now, AI focused its efforts on theorem proving, the most visible part of the mathematical activity. However, there are a lot of things to do other than proving theorems; one of them is to make conjectures, then he/she/it usually tries to prove it. This is not always the case, the main activity of EURISKO, developed by Douglas Lenat, was to make conjectures, which it did not prove. Ramanujan, a mathematician of genius, worked in an analogous manner: for many years after his death, distinguished mathematicians were still trying to prove some of his conjectures. Several conjectures are famous, such as Fermat’s last theorem that has been proved more than three centuries after it was stated. Goldbach suggested another well-known conjecture: any even integer greater than 2 can be written as the sum of two primes. Some conjectures, such as Riemann hypothesis, played a great role in the history of mathematics, theories have been developed in order to prove them.

In the February 2016 issue of Artificial Intelligence, C. Larson and N. Van Cleemput describe a system able to make conjectures in many mathematical domains. This system is not associated to a particular theory. Their work is based on a heuristic found and experimented since 1986 by S. Fajtlowicz. I did not know it, and this is not surprising, little was published about it: before this paper, it had never been referenced in Artificial Intelligence.

This heuristic may be used in many domains but not in all: the theory must include objects that have real number invariants, that are numbers linked to objects. For a graph, the invariants may be the number of vertices, of edges, the maximum degree of any of the vertices, etc. For a natural number, there are the number of factors, the number of distinct primes in its factorization, the number of its representations as sum of two primes, etc.

The system stores some objects of the theory under study; their number may be very low, for instance less than 10. For each object, it can find the value of all its invariants; therefore, it can check whether a new conjecture is true for all the given objects. The user indicates the name of an invariant I, for which he would like to conjecture the value of its lower or higher bound, using the other known invariants I1, I2, I3, …., In . The system knows a set of functions that can be applied to numbers such as addition, product, logarithm, etc. It will generate many conjectures such as C1 for I>I1+I2, C2 for I<=I1*I3, C3 for I<(I2-I4)², C4 for I>=log(I5+3),…The user may suggest other functions, especially those that are known to be useful in the present theory. In the different domains considered in the paper, it uses the following functions: frobenius norm, euler phi, mertens, next prime, and so on. This is the only situation in which the system is adapted to a particular domain.

When it has found a new conjecture, the system accepts it if it passes two tests:

Truth test: the conjecture is true for all the stored objects.

Significance test: for at least for one of the stored objects, the conjecture gives a stronger result than all the other known conjectures for the same object.

To clarify the interest of the second test, let us assume that we are looking for conjectures about the invariant G(n) when n is even: G(n) is the number of representations of n as sum of two primes. At the start, one has stored only two objects: 16 and 24. G(16)=2 (3+13 and 5+11) while G(24)=3 (5+19, 7+17, and 11+13). Conjecture C1 is: G(n) is greater than or equal to the number of digits in the binary representation of n less the number of its divisors. It is true for both objects: G(16)>=2 (5-3) and G(24)>=-1 (5-6). Now, conjecture C2 is: G(n) is greater than or equal to the number of digits of its decimal representation minus 1; this is also true since G(16)>=1 and G(24)>=1. However, C1 is stronger for 16 since it indicates that G(16)>=2 instead of 1 for C2; on the contrary, C2 is stronger for 24 where it predicts at least 1 instead of -1. Therefore, so far, one keeps both conjectures.

Naturally, finding a conjecture is not enough: one must prove it, and the conjecture must also be useful. For proving Goldbach conjecture (CG), that is G(n)>0 when n is even, one must find a new conjecture, which can be proved as a theorem, such that its value for G(n) is always at least 1. Even if C1 is true, it is not sufficient, since it only gives G(24)>-1. Therefore, it is certainly not a useful conjecture for proving CG, although it could give (if true) an interesting lower bound for some values of n. We are more lucky with C2: G is positive for both stored objects, and it has been also checked for many even numbers, but it has not yet been proved that the number of representations of n as sum of two primes is greater than or equal to the number of digits of n minus one. I give here only two very simple conjectures; the system found more of them, and checked them with much more than two stored objects. Perhaps, one day, it will find a conjecture predicting G(n)>0 that could be proved; then CG will also be proved.

The initial step in the research for a useful conjecture can be made with very few objects, then the system checks carefully that there is no trivial exception. For instance, after being discovered with few objects, conjecture C2 has been checked for n up to 1,000,000. After finding a useful and reasonably credible conjecture, a human or artificial mathematician has to prove it. Unfortunately, it may be impossible, even for a very clever artificial mathematician, to prove a conjecture because no proof exists although it is always true.

Anyway, I am impressed by the results from this system, particularly with conjecture C2. Firstly, this gives other ideas for proving CG; although it is much stronger, this could lead to consider new pathways. Moreover, if it is proved, it would be a satisfactory (although not very useful) result. For instance, C2 immediately states that:


which means that there exists at least 18 representations of this huge number as sum of two primes. With CG, we would only know that there is at least one representation. If I was a mathematician, I would be very proud if I could prove C2!

We can experiment this system, available at . Their program is implemented in the Sage open-source mathematical computing environment. Sage has a lot of built-in invariants for various mathematical objects. Therefore, it is easy to check with many instances, whether it is most likely that a promising conjecture is true.

This work does not solve all the problems linked to conjectures, but it shows that it is possible to add a new brick to the realization of what will someday be an artificial mathematician.

Algorithms to Live By

 This book, published in 2016, was written by a science journalist, BrianChristian, and by a specialist of Cognitive Psychology, TomGriffith; its subtitleis: The Computer Science of Human Decisions. AI researchers have always been very interested by human behavior: when we are developing a system that solves problems also solved by human beings, we try to find what methods we are using, and we include them in our programs. Here, the authors have a goal that is quite the opposite: they show that we, humans, can learn from the way computer systems solve problems, even for problems of our everyday life. They are not at all focused on AI: for them, every computer system may be interesting.

Sofar, I did not particularly think on this issue, but this is indeed an interesting idea. They are using many reallife examples of situations from everyday life, such as getting the best apartment, searching for a parking place, finding a restaurant, and so on. Some come from the authorspersonal experience.

I enjoyed their practical approach, but some academics may not like it, preferring to be abstracted from the real world. In the October 2016 issue of the AI Journal, ErnestDavis published an interesting review of this book; however, he does not like that way of presenting their views: «After reading about how Michael Trick used an algorithm to choose a wife, and how Albert McLay organizes breakfast for her children, and….., I started to have a craving for impersonal technical papers written in the passive voice.» For my part, I like this way of choosing situations easy to understand, without the need of a cryptic formalism.


I cannot consider all the topics examined in this book: when to stop looking, forget about it, when to think less, when to leave it to chance, the minds of others, and so on. The authors show that we could be more efficient if we borrow some of the methods used in computer systems.

For instance, in the chapter “Sorting-Making order”, they consider a problem that concerns all sports persons: how to order the best athletes. For tennis, the loser of a game is eliminated from the tournament. This provides the best player at the end, but produces little information about the strength of the others: the second best one was perhaps eliminated by the winner in the opening round. On grounds of efficiency, a race is much better than a fight: this gives a measure on the performance for all the participants. It is no longer necessary to fight again many players: at the end of a marathon, every runner knows his position compared to all the others.

Unfortunately, it is not always possible to eliminate fights everywhere. However, the solution chosen by the mammals for determine the alpha male is not very satisfactory because it often leads to too many fights. For their part, fish use a simpler method: the dominant is the biggest. With computers, sorting and ordering is made more quickly that fifty years ago, there are no longer unnecessary operations.

Tennis is not a perfect example: we are not so much interested in knowing who are the best players, we want many beautiful contests. If the ordering method is inefficient, we will watch more tennis matches! 150 years ago, Lewis Caroll showed that it was necessary to improve this method. Since that, there was some progress, particularly with the introduction of seeding; however, improvements are still possible. Personally, I think that one could use the difference in playing strength between the players during a match: the result was tight, one player was severely beaten, etc. A computer scientist knows that one can improve a result when an important information is not used.

Apart providing advices on how to organize ourselves better, this book also explains our behavior from understanding how computers work. For instance, it is well known that our memory deteriorates when we grow older. Wagging tongues discreetly hint to Alzheimer but, in the caching chapter, the authors explain that we have the same difficulties as computers to handle a lot of information. The more elements are in the memory, the more it takes time to retrieve what one is looking for: they call it “the tyranny of experience”. A ten years old child knows a few dozen friends; twenty years later, we know hundreds of people: in the towns where we spent some time, in the corporations where we have worked, the friends of our spouse, not to mention Facebook. Finding something can take more time if we stored it in an almost non-accessible area; we may even have decided to forget it in order to make room. When they have accumulated a substantial amount of information, old people and computers have the same problem: how to store it so that what will probably be useful will be quickly and easily accessible. Unfortunately, for everything remaining, one has to wait or to fail.

The authors essentially refer to Computer Science, although they mention several times “machine learning”, an AI subdomain. In my view, chapters inspired by AI could be added. Certainly, some AI methods cannot be used by human beings, mainly because our brain does not work fast enough: we cannot consider billions of Chess positions! However, in some domains, AI may be a source of inspiration, especially problem solvers that use heuristics.

Too often, teachers do not explain their students how they could solve a problem, and many students believe that there exist a special math skill (which they do not have) which is a prerequisite for solving mathematical problems. Let us take an example from Gelertner‘s Geometry Theorem Proving Machine: it must prove that an isosceles triangle (two equal sides) has two equal angles. Here is a machine proof:

It is assumed that triangle ABC has two equal sides: AB=AC. Triangles ABC and ACB are equal because their three sides are equal: AB=AC, AC=AB and they share BC. Therefore, the angle ABC of the first is equal to the corresponding angle ACB of the second.

If I had to write a Geometry program, I would have introduced a rule such as: if one must prove that two angles are equal, consider two triangles, each one containing one of these angles, then prove that these triangles are equal. It does not always succeed, in the present situation five triangles containing the angle ACB can be compared to triangle ABC: ACB, BAC, BCA, CAB, et CBA. The last four are not working, but the first one leads to the solution given before. With this rule, the student can understand how this solution has been found, and he can use it for other problems.

When an AI system makes many trials for finding the solution, it can indicate why it has to consider them. Even when they fail, it is interesting to show them because, in other situations, they may be successful. If the students could see how an AI system has found a solution, including its unsuccessful attempts, this would certainly allow them to perform more effectively afterwards.

This book is a most enjoyable read, but it could be completed by chapters adding Artificial Intelligence to Computer Science as a source of inspiration.

Bootstrapping CAIA Part II: Choosing a new domain

 There are many ways to increase the range of problems that CAIA could solve. For instance, one can introduce continuous variables, or define unknowns with an infinite number of possible values. It would also be interesting to consider meta-problems, that can be convenient to CAIA when it is solving a problem, for instance to monitor the search for solutions. Meta-problems are particularly useful in a bootstrap; however, when I made my last choice, I was afraid that it was too early to include such meta-problems: I didn’t have enough experience for monitoring the search for a solution. For this reason, I chose to consider problems where the unknowns may have an infinite number of possible values.

Firstly, this allows to have a large number of problems, mainly in arithmetic. More importantly, this necessitates to find new methods, different from those used for solving constraint satisfaction problems. Indeed, for these problems, the combinatorial method always gives all the solutions; it may be practically unusable, especially when the problem is intractable. However, this is a starting point, which can be improved.

Naturally, when an unknown may have an infinite number of possible values, one can no longer use the combinatorial method. Other methods must be used, some of them similar to theorem proving. This does not prevent to develop a tree, for instance one can consider several possibilities for an unknown: x<0, x=0, and x>0, or also x is even and x is odd. For many problems, looking at all the possible values, even very cleverly, is not always the best method; CAIA is obliged to experiment with other approaches in this new domain.

When the system stops, the result from these problems may be:

The proof that there is no solution: find two integers a and b such that 4a+3b²=34.

The discovery of all the solutions: find three integers a, b, et c, greater than a given integer k, such that the three remainders of the division of a.b by c, of b.c by a, and of c.a by b are all equal to k. For instance, if k=12, one solution is a=293892, b=1884, and c=156; in all, there are 792 solutions.

The discovery of one or several families describing an infinite number of solutions, which include all the possible solutions: find two integers a and b such that 4a+3b²=36 (any integer value for x, a=9-3x², b=2x).

There is a solution for every combination of the possible values of the unknowns: find n such that 2006 divides 2005n-1887n-1954n+1836n ; this constraint is true for any positive integer value of n.

No solution, or a finite, or an infinite number of solutions, have been found, but one has not proven that there is no other solution. This happens when one is stuck for at least one leaf of the tree.

For the combinatorial search in constraint satisfaction problems, sometimes one has not all the results because the resolution may require too much time, especially when the problem is intractable. Here, there is a new possibility: one does not see what could be done, all the known methods have been unsuccessfully used. In the tree, we have not only as value for the leaves: solution, or contradiction, or not enough time, but also: one does not know what to do.

 When a system results from a bootstrap, too often it does not find some solutions, although it believes that it has completed its task. When CAIA finds a solution, it is always correct; however, as the search is not as systematic as when the set of possible values are finite, sometimes it erroneously concludes that there is a contradiction, or it wrongly eliminates a possible value for an unknown. When a system builds itself its method for finding solutions, a misguided modification in the knowledge that creates these methods can improve its performances in some situations, but may introduce particularly vicious mistakes in parts that led before to excellent results. I have experimented with systems where I gave knowledge, and with others where I also gave meta-knowledge creating knowledge. Undeniably, developing the second approach is tremendously more difficult. One reason is that, if AI systems are brittle, systems that result from a bootstrap are much more brittle. In another post, we have seen the problems coming from meta-bugs.

Naturally, to proceed further, it is necessary to introduce new kinds of problems, until it can solve any problem. As of now, I do not want to consider other families of basic problems, such as games or various mathematical theories. I prefer to add families of meta-problems that will be useful at the meta level, where one has to build new methods for solving problems, or to monitor efficiently the search for solutions. It is much more necessary for quickly moving forward in this bootstrap, and particularly to have a less brittle system: experience has shown that, when CAIA does something for me, it does it better than myself.

Bootstrapping CAIA Part I: The initial domain

I can’t just say “one must bootstrap AI”, how to do it must also be explained. Naturally, I will use my own experience in CAIA’s development, started in 1985.

I shall quickly go through the first step, where I defined a language and knowledge for translating itself into C programs. This is well known by those who write a compiler in the language of this compiler. It was interesting to define a new language rather than using an existing language for two reasons:

1. This language must change over time, so that it becomes more and more declarative. At the beginning, in order to facilitate the compilation, it has many procedural aspects. They are gradually replaced by more declarative possibilities. Declarativity is essential because it is easier for the system to create declarative rather than procedural knowledge; it is also easier to study its own knowledge when it is given in a declarative formalism. Very important elements in this language are sets and bags, which do not imply an order to be followed, contrarily to the lists. Expertises are sets of rules, and rules have sets of clauses.

2. It is important that CAIA and myself could thoroughly examine any module of CAIA when it is executed. An unrestricted access to the present state of CAIA and its knowledge by CAIA itself is essential if we want to give it a kind of consciousness. It is easier to have something that suits us when the system is specially built for this purpose. Black boxes are great foes of intelligence: they restrict consciousness since one does not know what happens when one executes them. There are still two main black boxes for CAIA: the operating system and the C compiler.

Using its knowledge, CAIA translates all its knowledge, either given in the initial formalism or in more declarative formalisms that I later introduced. Since thirty years, CAIA does not contain a single line of C that I have written. All in all, there are 500,000 lines of C, and 13,500 rules. Many rules have not be created by myself, but from rules that create rules.

AI essential goal is to create a general problem solving system; all the human activities are, in fact, problems that we have to solve. The most important problem for AI researchers is to realize a system that could solve every problem, including this last one. It is foolish to begin with the most difficult problem, it is better to consider simpler problems, then to extend this domain gradually. This is one of the main directions of a bootstrap, the other one being to improve the performances.

It is important to choose the initial domain well. CAIA started to solve problems defined by a set of constraints. Firstly, this domain is interesting because many problems may be defined in that way; but it is even more interesting because some of these problems may be used by the solver itself. Since a long time, I have added two such problems:

1. To find symmetries from the study of the formulation of a problem can be stated as a constraint satisfaction problem. This is useful because it reduces the size of the search space, it facilitates the search for new constraints, and it enables to find a decomposition of the search space so that proofs will be easier in each area. I will explain this last point in a future blog, I completely underestimated it: it was a pleasant surprise.

2. For experimenting with a system, it needs many problems. Often, there are not enough problems in the literature, or I have to give them myself; moreover, there is a lack of very, very difficult problems, which are far too hard for human beings. However, finding new problems is a problem that can be defined with constraints; therefore, many CAIA’s problems have been found by CAIA itself.

In the next blog, we will see how a first step has been made to extend this initial domain. When CAIA will be able to solve any problem, one direction of the bootstrap will be completed; however, it will also be necessary to solve these problems more and more efficiently.

Everything but the essential

One Hundred Years Study in Artificial Intelligence is the title of a very interesting report that has just been published by a group of prominent researchers in AI. One of their goals is that «the report can help AI researchers, as well as their institutions and funders, to set priorities».

The report considers a large variety of domains; it shows that AI could be very helpful in situations where it is difficult to find an adequate staff. This is especially the case for health care. In particular automated assistance for the clinicians, image interpretation, robotics, elder care, etc. are very promising directions of research in this domain. Self-driving cars and home robots will change our day-to-day life. With intelligent tutoring systems, the students will get help adapted to their needs. This report provides an overview of many activities where AI systems will be able to help us in the next fifteen years.

It seems that the authors have begun to explore what the future needs will be; then, for each one, they have carefully examined what AI techniques could be useful. For that purpose, they did a wonderful job: if we make the required effort, many applications will be in widespread use throughout the world in 2030.

Curiously enough, the authors have completely forgotten a domain with really high needs, which they should be familiar with: to help the development of AI research!

It is very difficult to conduct AI research, especially if we just don’t imitate human behavior, where we improve our own results thanks to powerful computers. This vision of intelligence is anthropomorphic: other forms of intelligence exist. Unfortunately, building a super-intelligent system is perhaps too complex for human intelligence with all its limits: we have been shaped up by evolution only for solving the problems of our hunter-gatherer ancestors.

Serendipity allows us to produce acceptable results in new domains, such as computer science, mathematics, management, etc. However, they are necessarily limited: evolution lacked time so that we can adapt to their specific requirements. Our geniuses are probably not as good as they think, in the kingdom of the blind, the oneeyed is king. We are awfully handicapped by our working memory, with only 7 elements, and by our reflective consciousness, which ignores almost all the events that occur in our brain. Therefore, we will not get far without AI help. If we want to increase more and more AI performances, we must call upon AI assistance.


This has been completely ignored by the report, where the word “bootstrap” does not even occur. However, this is a technique well adapted to the resolution of very difficult problems; moreover, developing AI is an intelligent activity, therefore it depends on AI. The authors only say: «No machines with self-sustaining long-term goals and intent have been developed, nor are they likely to be developed in the near future.». This is almost true for the existing state of AI, but it is imprudent to predict what will happen in the next 15 years: progress is very slow in the beginning of a bootstrap then, suddenly, things move quickly. Naturally, if this kind of research has no priority, this is a self-fulfilling prophecy: we will be at the same situation in 15 years. Nevertheless, if this kind of research receives a small part from the funds rightly allocated to the applications described in the report, the results will likely surprise us.


The importance of bootstrapping AI is seen since its beginning: in 1959, Allen Newell and Herbert Simon considered the possibility for their General Problem Solver to find the differences necessary for GPS itself. The bootstrap has two major drawbacks: it is slow, and hard to achieve. However, many achievements of our civilization come from a bootstrap: we could not design our present computers if they did not exist.


Few AI researchers are presently interested in this approach. I suspect that many AI researchers even think it is impossible, while most of the others believe it is much too difficult to embark on this path, taking into account the current state of development of our science. Stephen Hawking is really a genius: although he is not an AI researcher, he has seen the strength of such an approach. It is unfortunate that he wrongly concluded on the dangers of AI.

What is in this report is excellent, I am only critical of what is missing, which is essential for the future of AI. In the end, it lacks coherence: the authors rightly say that AI will significantly help the resolution of many important problems in many domains. However, they do not think to use AI for the most important task: to advance AI, which is central to the success of all the other tasks considered in this report!



Ladies and Gentlemen Mathematicians, now it is your turn

A turn for what? Well, I have seen twice the same succession of events. Firstly, outstanding specialists of a domain assert, without any justification, that an AI system could never outsmart the best human experts of this domain. For many years, they are right, and they laugh at the poor results obtained by the first programs. Then, suddenly, everything changes: artificial systems outperform the best humans.

This happened for Chess, and recently for Go. I believe that the next domain where this sequence of events will occur is mathematics, a propitious domain for several reasons. Firstly, as games, mathematics is far removed from reality; the researcher has not to confront the many constraints coming from the real world, such as it is the case for driving a car. Secondly, games have been created by humans, based on what they can do best: from the start, we, humans, are in a good situation. On the contrary, mathematics has not been developed to solve problems that are not too difficult, but to solve useful problems. It is not obvious that mathematics is as well suited to our capacities as games; therefore, artificial beings are in a much better situation than for games: they could use all their qualities, in situations where they are essential, and where we are not very good. Our intelligence results from the importance for our ancestors of hunting and fighting for surviving and breeding. The fighting aspect is important in games, but not in mathematics. There is no reason why human intelligence would be well adapted to mathematics. It is simply the case that some capacities developed for other goals are somewhat helpful for mathematicians.

For the present time, no limit is known that would prevent AI systems to outperform human mathematicians. The well-known incompleteness theorems restrict what can do any cognitive system, human or artificial. Nobody could ever prove a result if no proof exists for this theorem in this theory.

Fist-class mathematicians have claimed that machines could never be very strong in mathematics, but they never justify this assertion. For them, it is not necessary, it is evident. However, a good mathematician should know that, when there is a mistake in a proof, it is often when one says that something is evident. Probably, they are so confident because they do not see how this could be made. This is normal: they are mathematicians, not AI researchers, this is not their job. In the same way, Chess and Go players’ job was not to write Chess and Go programs: they could not foresee what happened.

Mathematics is also a very interesting domain for AI because their results are extremely helpful. It is natural to invest in artificial mathematicians: their discoveries will be more useful than those of a Chess program. Naturally, their role will not be only to prove conjectures: they will have to build new mathematical theories. Pioneering works, such as those of Douglas Lenat, have shown that an artificial mathematician could discover concepts it did not know. For instance, for a function, it is interesting to consider the situations where the values of two arguments are the same: from addition one discovers the concept of double, and from multiplication, the concept of square. As extreme situations are often remarkable, considering them leads to the concept of number with few divisors, the prime numbers. Surprising its author, Lenat’s program studied also the numbers with many divisors; then Lenat discovered that Ramanujan had already studied them. Naturally, a lot of work is necessary for developing systems with many more possibilities that this old system, but this is certainly not impossible.

CAIA has a powerful method for solving problems where it cannot use a combinatorial method because there are too many possibilities, and even sometimes an infinite number. With the meta-combinatorial search, one does not consider all the possible values of a variable, or all the legal moves; instead, one considers many methods for trying to solve the problem. One can waste a lot of time, with methods that fail; however, at the end, one has an excellent solution, as one keeps only the methods that are necessary for justifying it. When one has a good set of methods, one may find sometime surprisingly elegant solutions. As computers are much faster than human brain, they can perform a meta-combinatorial search wider than those made by human mathematicians.

To date, AI has progressed rather slowly in the Universities, which are certainly not the right place for developing very complex systems quickly. Since recently, industrial firms have heavily invested in AI realizations; it is no coincidence that IBM succeeded for Chess and Jeopardy!, and Google for Go. They moved to the upper category very fast: before AlphaGo, I did not think that an AI system would win again one of the best professional Go player before ten years.

In view of the importance of mathematics, it will be tempting to create a competent team, with substantial resources: they could rapidly have results more valuable than would be expected. Human mathematicians would help us to understand the results of their artificial colleagues, so that they could be used efficiently and effectively. Naturally, they will also continue to do mathematics for the fun, in the same way as Chess and Go players are always playing their favorite game.