Bundy, Alan

English

We report the discovery of an unexpected connection between the invention of the concept of uniform convergence and the occurs check in the unification algorithm. This discovery suggests the invention of further interesting concepts in analysis and a technique for automated concept formation. Part of this technique has been implemented.The discovery arose as part of an attempt to understand the role of proof analysis in mathematical reasoning, so as to incorporate it into a computer program. Silver (1986)and Mitchell (1983) have investigated the automatic analysis of model proofs in order to extract and learn knowledge about controlling search,including the knowledge of new concepts. We focus on the analysis and correction of faulty proofs or 'poofs'. Especially where that correction involves the invention of new mathematical concepts.A classic example of where the analysis of a poof leads to a new concept is the invention, by Weierstrass, Seidel, Cauchy and others, of uniform convergence as a result of an analysis of Cauchy's poof that the limit of a convergentseriesof continuous functions is itself continuous. The correction consists of substituting in the theorem the new concept of 'uniformly convergent' for 'convergent'. We will investigate this example. The bug in Cauchy's poof is a violation of the occurs check. This observation suggests a technique for automatically correcting the proof and this leads to the concept of uniform convergence.The correction technique involves search. Other branches in the searchspacelead to alternative corrections and alternative concepts,one of which seemedinteresting to me and turns out to have been discovered by the mathematician, Ascoli

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     Edinburgh Research Explorer                                      Poof Analysis: A technique for Concept FormationCitation for published version:Bundy, A 1985, 'Poof Analysis: A technique for Concept Formation' Proceedings of AISB-85.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Author final version (often known as postprint)Published In:Proceedings of AISB-85General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015IG8Alan Bundy Department of Artificial Intelligence,Edinburgh University, Edinburgh EHI 2QL, U.K.ABSTRACTWe report the discovery of an unexpected connection between the inventionof the concept of uniform convergence and the occurs check in the unificationalgorithm. This discovery suggests the invention of further interesting con-cepts in analysis and a technique for automated concept formation. Part ofthis technique has been implemented.The discovery arose as part of an attempt to understand the role of proofanalysis in mathematical reasoning, so as to incorporate it into a computerprogram. Silver (1986) and Mitchell (1983) have investigated the automaticanalysis of model proofs in order to extract and learn knowledge about con-trolling search, including the knowledge of new concepts. We focus on theanalysis and correction of faulty proofs or 'poofs'. especially where that cor-rection involves the invention of new mathematical concepts.A classic example of where the analysis of a poof leads to a new conceptis the invention, by Weierstrass, Seidel, Cauchy and others, of uniformconvergence as a result of an analysis of Cauchy's poof that the limit of a* A 'poor, according to one of my mathematics lecturers, is a proof with something vitalmissing.Artificial Intelligence and its Applications, edited by A. G. Cohn and J. R. Thomas@ A. Bundy 1986, Published by John Wiley & Sons Ltd.52 ARTIFICIAL INTELLIGENCE AND ITS APPLICATIONSconvergent series of continuous functions is itself continuous. The correctionconsists of substituting in the theorem the new concept of 'uniformly con-vergent' for 'convergent'. We will investigate this example. The bug inCauchy's poof is a violation of the occurs check. This observation suggestsa technique for automatically correcting the proof and this leads to theconcept of uniform convergence. The correction technique involves search.Other branches in the search space lead to alternative corrections and alter-native concepts, one of which seemed interesting to me and turns out to havebeen discovered by the mathematician, Ascoli.1. INTRODUCTIONLakatos (1976) emphasizes the importance of poof analysis for discoveringhidden lemmas and hence correcting poofs, defining new concepts,generalizing theories, etc. For instance, he relates how the concept of apolyhedron was defined and refined by analysis of poofs of Euler's theoremand how the concept of uniform convergence was defined by analysingCauchy's poof that the limit of a convergent series of continuous functionsis itself continuous. Poof analysis ensures that the new concepts it suggestswill be useful, because they will help change a poof into a proof. Thus apoof analyser provides information to guide the process of concept forma-tion, which is otherwise an explosive one. Such guidance can be providedby heuristic search based on 'interestingness' values (Lenat, 1982), buthuman concept formation is often problem driven, and poof analysis offersan alternative, problem driven, concept formation technique.Such a poof analyser could also be an important ingredient of an artificialmathematics teacher, i.e. a program which reads student's solutions anddiscovers errors. DEBUGGY, (Burton, 1982), is an example of such adiagnosis program in the area of elementary arithmetic.As a prelude to building such a poof analyser we investigate the examplediscussed by Lakatos-the invention of uniform convergence from ananalysis of Cauchy's poof. This lends itself to formal analysis more easilythan the poofs of Euler's theorem. We first give the informal poof and thentranslate it into a resolutionjparamodulation poof in order to locate thebug. This turns out to be a missing occurs check in a unification. We discusshow the poof can be fixed up by re-ordering quantifiers to remove thedependencies that caused the unification failure. The effect is to replace'convergent' by 'uniformly convergent' in the hypothesis of the theorem.We discuss alternative, systematic ways of fixing the poof leading to otherconcepts. Then we describe the partial automation of the process. Finally,we discuss how the technique may be used to disambigute the meaning ofa sentence during natural language understanding.2. THE INFORMAL POOFFirst the 'gist of Cauchy's proof' that the limit of a convergent series of'POOF' ANALYSIS-A TECHNIQUE FOR CONCEPT FORMATION 53continuous functions is itself continuous, quoted, slightly modified, fromLakatos (1976).Let f(x) = ~i:Ofi(X) be a convergent series of continuous functions and,for any n, define Sn(X) = ~i~oJ;(X) and rn(x) = ~i=n+1J;(X).Given any 8 > 0:there is a 0, such that for any b, if I b I < 0, thenISn(x+b)-Sn(x)1 <8(there is such a 0 because of the continuity of Sn(X»: (i)there is an M, such that I rn(X) I < 8 for all n ~ M(there is such an M because of the convergence of ~i:oJ;(x»: (ii)there is an M', such that I rn(X+ b) < 8 for all n ~ M'(there is such an M' because of the convergence of ~i:oJ;(x + b»: (iii)Hence we can infer that:If(x+b)-f(x)1 = Isn(x+b)+rn(x+b)-sn(x)-rn(x)1~ Isn(x+b)-sn(x)1 + Irn(x)1 + Irn(x+b)1< 38, for all b < 0 Q.E.D.This kind of semi-formal, 'mathematese', is typical of the notation used inhuman proofs. In order to locate the bug automatically it is necessary totranslate it into a formal notation which can be proof checked. We chosea resolutionjparamodulation type proof for this, although other typeswould be equally suitable.3. THE FORMAL POOF AND THE mENTIFICATION OF THE BUGIn this formal version we use the PROLOG convention that variables beginwith an upper case letter and constants with a lower case letter. WhenSkolemizing a variable we preserve the name, but demote the first letterfrom upper to lower case.The definition of convergence for 2:;i~O/;(X) is:coVXVE>O3MVN~MI ~ /;(x)I<E (iv)i=N+l(v)(vi)(vii)(viii)and the definition of continuity for f(x) is:vXvE> 0 311. > 0 vBIBI < 11. -+ I f(X +B) -f(X) 1< EThus the translations of (i), (ii) and (iii), to clausal form are:E> 0 -+ o(E, X, N) > 0E> 0 & I B I < o(E, X, N) -+ I s(N, X + B) -s(N, X) I < EE>O&N~m(E,X)-+ I r(N,X) I <E54 ARTIFICIAL INTELLIGENCE AND ITS APPLICATIONSE>0&N~m(E,X+B)-'1r(N,X+B)1 <E (ix)from which we can infer:E>O& IBI <o(E,X,N)&N~m(E,X)&N~m(E,X+B)I f(X + B) -f(X) I < 3.E (x)The goal is to prove f(X) continuous, which after negation and transla-tion to clausal form gives:-+ 8 > 0 (xi)11. > 0 -+ I b(11. ) I < 11. (xii)11. > 0 & I f(x + b(I1.» -f(x) I < 8 -+ (xiii)After cosmetic paramodulation with V = 3. (VI3), (xiii) resolves with (x)to produce0(8/3, x,N) > 0 &8/3 > 0 & I b(0(8/3, x, N» I < 0(8/3, x, N) &N ~ m(8/3, X) & N ~ m (8/3, x + b(0(8/3, X, N») -.(xiv)With the aid of U> 0 -.U/3 > 0, we can use (vi), (xi) and (xii) to resolvethe first three literals away leaving the rump:N ~ m(8/3, x) & N ~ m(8/3, x + b(0(8/3, x, N») -.(xv)These almost resolve with -.Y ~ Y, but the presence of N on the right-hand side of the second literal causes the occurs check to fail. In fact,counter-examples can be found to the second literal.4. CORRECTING THE POOFNow to fix up the poof. The occurs check would not fail if N were not con-tained in m(s/3. x + b(o(s/3. x. N») and it would not be so contained if mwere a unary Skolem function dependent only on s/3. Locating the in-troduction of m in the definition of convergence of f. we see that m wouldhave the required argument structure if the quantifier order were changedto:(xvi)«IVE>O3MVXVN~MI L:; h(X) I <Ei=N+lbut this is the definition of uniform convergence. We will call this them unary' solution.Now the rump goals are:N ~ m(s/3) & N ~ m(s/3)which will resolve with --Y ~ Y to prove the theorem.'POOF' ANALYSIS-A TECHNIQUE FOR CONCEPT FORMATION 55However, m is not the only Skolem function surrounding the offendingoccurrence of N, there are also 0 and b. Similar corrections can be madeby making 0 a binary function, independent of its third argument, N, or bymaking b a constant, independent of 0(8/3, x, N).5. ALTERNATIVE CORRECTIONS-MAKING b A CONSTANTThe Skolem function b arises from the definition of continuity.vXvE> 03.l > 0 vB I BI <.l-' I f(X + B) -f(X) 1< Ewhen it is negated and Skolemised as the conclusion of the theorem. Therequired correction involves putting the vB before the 3.l to create aconcept I shall call weak continuity, because it is a weaker property thancontinuity, i.e.vXvE>OvB3.l >OIBI <.l-'lf(X+B)-f(X)1 <EHowever, this is a trivial property that holds for all functions. In particularthe square wave (the normal counter-example to Cauchy's proof). This isbecause we can choose .l to depend on B. For instance, we can choose .lto be I B 1/2, but anything guaranteed less than or equal to I B I will do. NowI B I < .l is always false, so weak continuity is always true.In contrast to the make m unary case, the rump subgoals.N~ m(s/3, x) &N~ m(s/3, x+ b)-'are not identical and cannot be resolved away with -.Y ~ Y. However,they can be resolved away with -.max(Y, Z) ~ Yand -.max(Y, Z) ~ Z.Hand checking indicates that this version of Cauchy's proof is legal,although the theorem it proves is trivial, since its conclusion is always true.6. ALTERNATIVE CORRECTIONS-MAKING 6 A BINARYFUNCTIONThe Skolem function () arises from the lemma that s(N, X) is continuousfor all N, i.e.vNeZvXvE>O3d >oVBIBI <d-+ fS(N,X+lJ)-S(N,X)1 <EThis makes () dependent on N, X and E. We can remove the dependence onN by shifting vN after 3d to create a concept called equi-continuity byAscoli (Simmons, 1963, p. 126), i.e.vXvE> 03d > OvNeZvBIB! < ~ -+s(N,X + lJ)-s(N,X) I <ETracing this lemma back in the proof we see that the family of Ins must alsobe equi-continuous.VXVE>O3d>OvNeZvBIBI <d-+ I --~f(N, X + B) -f(N, X) I < E56 ARTIFICIAL INTELLIGENCE AND ITS APPLICATIONSThe proof of the rump subgoals,N ~ m(e/3, x) & N ~ m(e/3, x + b(tJ(e/3, x»)-+is similar to that in the make-b-constant solution, i.e. they are resolvedaway with -+ max(Y, Z) ~ Yand -+ max(Y, Z) ~ Z.Hand checking of this version of Cauchy's proof suggests that it is legal.This version of the theorem seems to be a genuine and interesting additionto the standard one suggested by the make-m-unary correction. It involvesa different, but independently interesting, concept: equi-continuity. I havebeen unable to find any reference to it in the literature. Its proof is similarto parts of the proof of Ascoli's theorem. (Simmons, 1963, pp. 124-128),but Ascoli's theorem itself is different. This version of Cauchy's theoremseems too obvious not to have been previously discovered by someone, andI would be grateful for any help in tracking it down.7. THE AUTOMATION OF POOF ANALYSISWhat are the requirements for automating this process?(a) Assuming that the initial representation of the poof is in mathe-matical English (or 'mathematese'), the program must comparepredicate calculus definitions like (iv) and (v) above with themathematese statements like (i), (ii) and (iii) , to produce predicatecalculus versions like (vi), (vii), (viii) and (ix).(b) the program must fill in gaps in the informal proof, e.g. derive thelemmas (xiv) and (xv).(c) The program must recognise failure, as at step (xv), and the reasonfor it, i.e. occurs check failure.(d) The program must be able to fix up failures by modifying thetheorem, e.g. by reordering the quantifiers in the definition ofconvergence.(d), above, has been implemented, by the author, in a Prolog program,SEIDEL. The program takes a description of an occurs check bug in a poof,and produces all those modifications of the original theorem which correctthe poof. The description of the occurs check bug consists of the variable,Y, and term, f(Y), on which the unification attempt should have failed inthe original poof. SEIDEL finds a skolem function, sk, in f(Y), and marksthe argument place of sk which contains Y (suppose this is the nth place).It recovers the formula, Fm, whose skolemization gave risk to sk andmodifies Fm to remove the dependence of sk on Y. SEIDEL searches, depthfirst, for all such modifications.'POOF' ANALYSIS-A TECHNIQUE FOR CONCEPT FORMATION 57In the current version of SEIDEL, Fm has to be in prenex normal form,i.e. a list of quantifiers, Prefix, followed by a quantifier free formula,Matrix. Prefix must be of the following form:...QX, ..., R Sk, ...where Q is the quantifier governing variable X, and R is the quantifiergoverning variable Sk. Sk is the variable which skolemizes to sk, and X isits nth argument. If Fm is skolemized positively then R will be an existentialquantifier and Q a universal one. If Fm is skolemized negatively, then viceversa.The idea of the modification is to move the quantifiers in Prefix so thatQ X no longer appears to the left of R Sk. This will remove the dependenceof sk on its nth argument place and thus on Y. We try to do this in a waythat minimises the disruption to Fm, i.e. so that there is no unnecessarymodification.Fm preserves its meaning under any permutation of similar quantifierswithin Prefix, i.e. two adjacent existential or two adjacent universal quan-tifiers can be exchanged without affecting the meaning of the formula. Thisdevice is used to mini mise the subsequent disruption to Fm. That is, R Skis moved as far leftwards as possible within its group of R-type quantifiers,and Q X is moved as far rightwards as possible within its group of Q-typequantifiers.At this point R Sk and Q X may be adjacent. This happens in all theexamples in this paper. They may then be exchanged to remove thedependence of sk on x. Otherwise, there are two minimal solutions: R Skmay be moved leftwards to immediately left of Q X, or Q X may be movedrightwards to immediately right of R Sk.SEIDEL works correctly on the Cauchy poof, producing the concepts ofuniform convergence, weak continuity and equi-continuity.This completes the description of the current state of SEIDEL. Furtherwork required includes: the removal of the restriction to prenex normalform; the addition of other bug types and fixes; and the addition ofmechanisms for (a), (b) and (c) above.-(a) looks feasible-it requires fairly straightforward natural languageunderstanding, perhaps using semantic grammars geared tomathematese.-(b) could be done using theorem proving with some length of proofbound, say.-(c) could be tricky-how do you know when failure is absolute andnot some limitation of the proof bound? Perhaps counter-examplescould be brought to bear here. If the poof comes with a globalcounterexample then this can be used to test gaps-a la Gelernter(1963).58 ARTIFICIAL INTELLIGENCE AND ITS APPLICATIONS8. AN APPLICATION TO NATURAL LANGUAGEUNDERSTANDINGThe purpose of this section is to show that poof analysis, and even theparticular technique of quantifier swapping described above, may findapplication outside of mathematics. We will try to demonstrate a use forthe technique in a very different area: disambiguating the meaning ofsentences in natural language understanding.A classical ambiguous sentence in English isEverybody loves somebody. (xviii)This can be represented in Predicate Logic as:3XvY(person(X & (person(Y) -+ 10ves(Y,X») (xix)orvY 3X(person(X) & (person(Y) -+ 10ves(Y, X») (xx)Suppose (xix) was intended by a speaker but (xx) was understood by thehearer.The speaker might go on to sayTherefore, someone loves themself. (xxi)This does indeed follow from (xix) but not from (xx). The hearer may thendetect the problem by realising that (xxi) does not follow from his/herunderstanding of (xviii). However, by relaxing the occurs check s/he cangenerate a poof of (xxi) from (xviii).In clausal form the hearer's version of the poof is:-.person(x(Y))person(Y) -'loves(Y, x(Y))person(Z) & loves(Z, Z) -.person(x(Y)) -.(xxii)(xxiii)(xxiv)from (xx)from (xx)from the negation of (xxi)from (xxiii) and (xxiv)from (xxv) and (xxii)The faulty step is the unification of loves (Y, x(Y» and 10ves(Z, Z), sincethis violates the occurs check-one is not allowed to unify Yand x(Y). Thepoof can be corrected by removing the dependance of the skolem functionx on the variable Y. Using the technique of section 7 this is done byreversing the order of the quantifiers in (xx), which gives (xix). Thus usingpoof analysis to correct the faulty argument reconstructs the correct transla-tion of the ambiguous sentence.This example has been run successfully on SEIDEL.'POOF' ANALYSIS-A TECHNIQUE FOR CONCEPT FORMATION 599. CONCLUSIONConcept formation is an important area of research in Artificial intelligenceand a vital ingredient of a program for modelling the cognitive process in-volved in mathematical reasoning. New concepts can easily be formed bymanipulation of old ones, but the search space is combinatorially explosive.One way to guide the search through this space is to form concepts thatfacilitate the solving of problems. We call this problem-driven concept-formation.Poof analysis is one technique for problem-driven concept-formation inmathematics. We have investigated its application in a classic example-theinvention of uniform convergence. This investigation has suggested asystematic technique for poof correction and concept formation, and led tothe discovery of alternative corrections and concepts. The technique hasalso been applied to the problem of disambiguation in natural languageunderstanding. The technique has been partially implemented.ACKNOWLEDGMENTSI am grateful for feedback on earlier versions of this paper to members ofthe mathematical reasoning group in the Edinburgh AI Department, and toAlan Sinclair of the Mathematics Department at Edinburgh.This research was supported by SERC grant GR/B/20826.REFERENCESBurton, R. R. (1982). Diagnosing bugs in a simple procedural skill, Academic Press,London; pp 157-183.Ge1ernter, H. (1963). Realization of a Geometry theorem-proving machine. InFeigenbaum E and Feldman J (eds) Computers and Thought, pp 134-52.McGraw-Hili, New York.Lakatos, I. (1976). Proofs and refutations; The logic of Mathematical discovery,Cambridge University Press, Cambridge.Lenat, D. B. (1982). AM: An Artificial Intelligence approach to discovery inMathematics as Heuristic Search. In Knowledge-based systems in artificialintelligence. McGraw-Hili. Also available from Stanford as TechReport AIM286.Mitchell, T. M. (1983). Learning and Problem Solving. In Bundy A (ed) Proc EighthIJCAI. pp 1139-1151, Morgan Kaufmann, Los Altos.Silver, B. (1986). Precondition Analysis: Learning Control Information, inMichalski R S, Carbonell J G and Mitchell T M (eds) Machine Learning, 2,Morgan Kaufmann, Los Altos. Earlier version available from Univ of Edinburghas DAI Research Paper 220.Simmons, G. F. (1963). Introduction to Topology and Modern Analysis. McGraw-Hill, New York.

Poof Analysis: A technique for Concept Formation

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