1. Graffiti’s vs. Human Conjectures in Chemistry and Mathematics 4 selected encounters

2. Acknowledgement I appreciate this opportunity to speak to an AI audience without any attempt by the organizers to intervene with the discussion of facts and topics of this lecture.

3. Impression/Expression Models Original idea of Graffiti: a program that Original idea of Graffiti: a program that computes invariants of certain objects (impressions) computes invariants of certain objects (impressions) Combines these invariants into formulae (expressions) Combines these invariants into formulae (expressions) verifies expressions against a collection of examples - a modest test of correctness. verifies expressions against a collection of examples - a modest test of correctness. uses a few heuristics to select interesting conjectures uses a few heuristics to select interesting conjectures

4. there were other claims of conjecture-making programs before but only some of the earliest attempts will be discussed but only some of the earliest attempts will be discussed here here

5. Artificial Mathematician and EURISCO It is not clear ( to me) what the relationship of well- known AM to I/E models is, but Lenat addressed the crucial issue of significance (interestingness) of conjectures, It is not clear ( to me) what the relationship of well- known AM to I/E models is, but Lenat addressed the crucial issue of significance (interestingness) of conjectures, leaving the criteria, however, to humans. leaving the criteria, however, to humans. Another important problem addressed in Lenat’s work was Another important problem addressed in Lenat’s work was the concept formation problem. the concept formation problem.

6. Bacon according to its authors, rediscovered Newton’s, Kepler’s according to its authors, rediscovered Newton’s, Kepler’s and other classical physical laws. and other classical physical laws.

7. out of interest in fully automated programs, I was trying to discuss with Professor Simon the concept formation problem but eventually … his answer was : “ Didn't even God need some mud to make Adam …” one different possibility was proposed in: “On one different possibility was proposed in: “On Conjectures and Methods of Graffiti”, infeasible at the Conjectures and Methods of Graffiti”, infeasible at the time, but probably not any more, because time, but probably not any more, because of new versions of the program, not more powerful computers. of new versions of the program, not more powerful computers. The proposal was for concepts, not Adam, although not because of Searle’s objection that programs are purely syntactic because of Searle’s objection that programs are purely syntactic

8. automated versions (apart from concept formation)  there are now algorithms for fully automated versions of Graffiti, apart from concept formation  as well as some implementations,  primarily in educational versions, but see the 3 rd encounter below – for perhaps one of Graffiti’s best conjectures.

9. full automation is of marginal interest here; but since man vs. machine debate goes to the marrow of the bone of this subject. it is necessary to say here that some drastically false claims, and even actions were taken to confuse this fairly simple issue, namely it is necessary to say here that some drastically false claims, and even actions were taken to confuse this fairly simple issue, namely what is a conjecture-making program what is a conjecture-making program The degree of automation may be of course still debated for long time The degree of automation may be of course still debated for long time

10. AI was always controversial, but AI was always controversial, but Hansen, starting with a DIMACS “discovery” meeting (of which he was a director) began to make specific, obviously false, still not retracted claims (startlingly defended, in the published DIMACS volume) according to which, authors of very well known graph-theoretical programs, one preceding Graffiti by perhaps as much as 10 years, (once!!) (once!!) said that their programs were making conjectures. ( “ Towards Fully Automated Fragments of Graph Theory” )

11. goal of this lecture comparison between a few selected human acts and Graffiti’s conjectures, or inspired by this program but primarily the former to show what can be expected of computer programs without much, and even without any, human intervention.

12. 1 st encounter, early nineties 1 st encounter, early nineties Professor Simon acknowledged that Graffiti made mathematical conjectures, but emphatically denied the possibility that the same can be done in physical sciences. Professor Simon acknowledged that Graffiti made mathematical conjectures, but emphatically denied the possibility that the same can be done in physical sciences. Remarkably, as far as I know, this was the only occasion on which he argued against computers being capable of performing certain Remarkably, as far as I know, this was the only occasion on which he argued against computers being capable of performing certain ( scientific ) tasks. ( scientific ) tasks. In fact, Herbert Simon may have been the most outspoken of In fact, Herbert Simon may have been the most outspoken of all the proponents of the so called strong AI point of view. all the proponents of the so called strong AI point of view.

13. by then, there were many papers published, some by superb mathematicians inspired by conjectures of Graffiti Noga Alon Noga Alon Bela Bollobas Bela Bollobas Fan Chung Fan Chung Paul Erdös Paul Erdös Daniel Kleitman Daniel Kleitman Laszlo Lovász Laszlo Lovász Janosz Pach Janosz Pach Yuri Razborov Yuri Razborov Paul Seymour Paul Seymour Joel Spencer Joel Spencer

14. but to appreciate Professor Simon gesture but to appreciate Professor Simon gesture one should be aware that 10 years later, DIMACS director Roberts refused to include on the website of the 1 st mathematical conference dedicated to automated chemistry conjectures some of the fullerene findings of Graffiti, giving as a reason that he couldn’t do this without permission of the conference directors, one of whom - Fowler, see below – was a fullerene expert. one should be aware that 10 years later, DIMACS director Roberts refused to include on the website of the 1 st mathematical conference dedicated to automated chemistry conjectures some of the fullerene findings of Graffiti, giving as a reason that he couldn’t do this without permission of the conference directors, one of whom - Fowler, see below – was a fullerene expert.

15. it should be mentioned here that apart from a false claim concerning conjecture-making programs being displayed anonymously for years on the DIMACS website, the same site, presumably with the consent of the conference directors, was asserting that computers will replace humans in the process of scientific discovery. even Professor Simon might object to it. I did, but Roberts refused to remove this claim from the DIMACS website. (“Fragments, I and II”, “Postscript”, “dimacs.events”)

16. Independence-stability Hypothesis The classical stable fullerenes (i.e., those with at most 84 atoms) tend to minimize their independence numbers. The classical stable fullerenes (i.e., those with at most 84 atoms) tend to minimize their independence numbers. The corresponding mathematical conjecture of Graffiti was easy to prove (alpha < a/2 – 1) The corresponding mathematical conjecture of Graffiti was easy to prove (alpha < a/2 – 1) the hypothesis was my interpretation of Graffiti’s display of its sorting patterns. the hypothesis was my interpretation of Graffiti’s display of its sorting patterns. Fowler’s initial reaction was as critical, as it had been on a previous occasion Fowler’s initial reaction was as critical, as it had been on a previous occasion

17. See “Toward Fully Automated Fragments of Graph Theory” for a discussion of the previous stability-expanding hypothesis conjectured by the same method, and misquoted in my joint paper without my knowledge for a discussion of the previous stability-expanding hypothesis conjectured by the same method, and misquoted in my joint paper without my knowledge

18. Then, the same day, at the conference dinner, Fowler told me after checking his data, that the icosahedral C60 – still the most stable fullerene isomer found so far, is the unique 60- atom fullerene minimizing the independence number among 1812 mathematically possible fullerenes. Fowler told me after checking his data, that the icosahedral C60 – still the most stable fullerene isomer found so far, is the unique 60- atom fullerene minimizing the independence number among 1812 mathematically possible fullerenes. Fullerene C60 ball and stick created from a PDB using Piotr Rotkiewicz’s iMOL

19. And then, that according to his data, the icosahedral C70 – the 2 nd most stable fullerene isomer is the only 70 atom fullerene minimizing its independence number among 8149 of mathematically possible fullerenes. the icosahedral C70 – the 2 nd most stable fullerene isomer is the only 70 atom fullerene minimizing its independence number among 8149 of mathematically possible fullerenes. Fullerene C70, Polish wikipedia, author probably Pawel T. Jochym

20. 1.5 th encounter this event – Fowler’s evidence, after he checked his data the very same day, is documented in this event – Fowler’s evidence, after he checked his data the very same day, is documented in

21. a joint paper in Chemical Physics Letters  statistical evidence for this and another one of Graffiti’s conjectures, years before the DIMACS volume “Graphs and Discovery” was published (more than 4 years later after the conference.)  data was obtained by Larson running Fullgen, and after having slightly modified my program (based on Tarjan- Trojanowski’s algorithm) for the independence numbers.

22. a referee, of my other paper listing these facts and statistics  wrote that all that it took to conjecture the independence-stability hypothesis was to compute the independence number of all classical fullerenes.  I wouldn’t, and I couldn’t do this:  I wouldn’t, because I would consider this act to be about as challenging as breaking into a piggy-bank, and following Graffiti’s example, I believe in the principle of the strongest conjecture.  ( according to the 1 st version of Hansen’s DIMACS submission, I mislead readers of my Graffiti papers, because this program was not proving its conjectures.)

23. but I also couldn’t do it because the only program capable of this task was, and probably still is, Fullgen, which was not available to me until well after the DIMACS meeting. It is also relevant to the issues discussed here, that Hansen made many suggestions for improvement of Graffiti and other programs, that at one point included enhancing AGX with a procedure for computing the union of two graphs. “Toward Fully Automated Fragments, II”

24. before Graffiti’s first chemistry conjectures trying to invent a counter-argument against Professor Simon claim, I considered a possibility of refuting it without writing actual programs, (which I did not anticipate at the time.) actually, I still did not write some programs, that I thought after reading “Scientific Discovery” and discussing with authors, of the book, the “discovery = problem-solving” hypothesis.) Eventually, my counter-claim was based on the:

25. program-accelerators argument: one can write a program (call it Mach = Mach(P)) capable of making conjectures about any program P one can write a program (call it Mach = Mach(P)) capable of making conjectures about any program P (including herself ) on the basis of which Mach may accomplish the task of P faster than P. (including herself ) on the basis of which Mach may accomplish the task of P faster than P. later, although not very much later, I realized that in a sense, Graffiti is already such a program, and then later, although not very much later, I realized that in a sense, Graffiti is already such a program, and then almost at once, it occurred to me that I can test the feasibility of my argument on a very short notice. almost at once, it occurred to me that I can test the feasibility of my argument on a very short notice.

26. From the Previous Slide From the Previous Slide alludes, of course, to the concept of self - much debated in AI literature and, thus (similarly as Professor Minsky, who stated this in his UH lecture) I also do not consider writing programs that make conjectures about themselves a great AI obstacle. Indeed, Graffiti is capable of making conjectures about itself.

27. first attempt to test the idea of program accelerators: P: input i:=0;sum:=0; n: integer; repeat sum:=sum + i; i := i + 1; i := i + 1; until i := n; Mach (P) Mach (P) And I ran the program P, for n:=0, 1, … (within Graffiti), running at the same time conjectures about it.

28. after a few rounds Mach printed: Mach printed: sum= 1 + 2 + … n = [n/(n+1)]/2 sum= 1 + 2 + … n = [n/(n+1)]/2 and stopped (Graffiti stops after accomplishing her goals) and stopped (Graffiti stops after accomplishing her goals)

29. 2 nd Encounter Next day, I realized, that in a sense Mach (P) reincarnated the famous story about Gauss. Before this talk, I reran the test again in Red Burton – an educational version of Graffiti well suited for repeating this test - with the identical result. Next day, I realized, that in a sense Mach (P) reincarnated the famous story about Gauss. Before this talk, I reran the test again in Red Burton – an educational version of Graffiti well suited for repeating this test - with the identical result.

30. Gauss (Small Wonder) “won” the 2nd Encounter Gauss (Small Wonder) “won” the 2nd Encounter Indeed, he figured-out a brilliant, at least for his age (which may have been close to Graffiti’s at the time ) explanation of the formula for the sum. Indeed, he figured-out a brilliant, at least for his age (which may have been close to Graffiti’s at the time ) explanation of the formula for the sum.

31. Mach (P) is an Universal Turing Machine Additionally, it is capable of improving the performance of its input program P. Additionally, it is capable of improving the performance of its input program P. Of course, as anything based on conjectures, the performance of Mach is subject to errors. Of course, as anything based on conjectures, the performance of Mach is subject to errors. This issue is related to the “Emperor’s New Mind” dispute This issue is related to the “Emperor’s New Mind” dispute

32. perhaps, not surprisingly, as a Universal Turing Machine Mach (P) leads to a new kind of halting problems

33. Related Penrose’s Arguments “Computer programs cannot have human mathematical intuitions.” “Computer programs cannot have human mathematical intuitions.” For a computer, a “belief” would have to be synonymous with a proof. For a computer, a “belief” would have to be synonymous with a proof.

34. 3 rd encounter, a counter-argument against Penrose’s claim? Graffiti’s conjecture: f = r + 1 Graffiti’s conjecture: f = r + 1 where f is the number of faces and r - the number of repetitions in the coding sequence (i.e., the list of edges) of a planar Eulerian graph. where f is the number of faces and r - the number of repetitions in the coding sequence (i.e., the list of edges) of a planar Eulerian graph.

35. # faces = # repetitions + 1. 1 4 32 121324234 121324234 0 1 1 2 2 3 3 4 4 5 5 6 r= f=

36. Graffiti’s “proof” of an almost 400 year old formula v – e + f = 2, is one of the simplest, perhaps the simplest (400) because Descartes discovered it about 100 years before Euler. (400) because Descartes discovered it about 100 years before Euler. Euler’s, 1736, birth of graph theory Euler’s, 1736, birth of graph theory Graffiti’s discovery f = r +1, automatically implies Euler’s formula for Eulerian graphs, (since r = e – v + 1.) Graffiti’s discovery f = r +1, automatically implies Euler’s formula for Eulerian graphs, (since r = e – v + 1.)

37. Prediction vs. Understanding I/E models may make correct predictions, but as I was always freely admitting, the understanding of conjectures is a different issue. The evidence, (as opposed to a proof) for the discovery of f = r +1 is so convincing that a human in Graffiti’s shoes might is so convincing that a human in Graffiti’s shoes might form a belief (!) and claim understanding of this form a belief (!) and claim understanding of this phenomenon on the basis of this argument. phenomenon on the basis of this argument.

38. the 4 th encounter the 4 th encounter

39. Within a few years after the “encounter” with Professor Simon, Graffiti’s conjectures in chemistry led to the independence and the expanding stability hypotheses about classical fullerenes, for which (particularly for the independence-stability hypothesis) there is considerable statistical evidence. the independence and the expanding stability hypotheses about classical fullerenes, for which (particularly for the independence-stability hypothesis) there is considerable statistical evidence. several papers about graph-theoretical independence, an invariant hardly ever, if at all, studied in chemistry before, in spite of (see below,) the almost certain chemical significance of this invariant. several papers about graph-theoretical independence, an invariant hardly ever, if at all, studied in chemistry before, in spite of (see below,) the almost certain chemical significance of this invariant.

40. The Theorem of Horst Sachs Correct conjecture about benzenoids proved by Horst Sachs, and extending a classical mathematical chemistry result from the fifties. Correct conjecture about benzenoids proved by Horst Sachs, and extending a classical mathematical chemistry result from the fifties. Similar (still open) conjecture about diamondoids Similar (still open) conjecture about diamondoids

41. New Representation and Characterization buckminsterfullerene as a truncated cube buckminsterfullerene as a truncated cube

42. Portrait of buckminsterfullerene as a truncated cube and a truncated icosahedron

43. unexpectedly strong supporting statistical evidence for the independence-stability hypothesis at the DIMACS meeting, I was promised more data within days, but I never received it, until it was computed by my student at the DIMACS meeting, I was promised more data within days, but I never received it, until it was computed by my student The next slides shows that out of 8 classically observed fullerenes, seven minimize their independence number The next slides shows that out of 8 classically observed fullerenes, seven minimize their independence number

44. Ranking by Homo – Lumo, Separator, and Independence AtomsHomo-LumoSeparatorIndependenceTies60111 7012691 761111831 78.124137141(3) 78.221201342 78.31313991(3) 84.124201(17) 84.22411(17)

45. Not longer than a year, I think, after the the independence hypothesis, I understood that the conjectured relation is valid for benzenoids – close relatives of fullerenes that the conjectured relation is valid for benzenoids – close relatives of fullerenes

46. The independence-stability hypothesis is true for benzenoids  According to classical chemical theories going back to 1 st half of the 20 th century, the most stable benzenoids maximize their matching number and thus  according to a classical graph- theoretical results, also from the same period they minimize their independence number  The number of atoms in a benzenoid equals to its matching number + the independence number.  n = m + i, for every bipartite graph  And since m is never more than n/2, i is at least n/2 and it is minimum iff it is = n/2 iff the benzenoid has a Kekule structure iff it minimizes it independence