Making Conjectures About Maple Functions Simon Colton Universities of Edinburgh & York

Computation, Invention & Deduction Perform calculations (CAS) Find conjectures (ML) Prove theorems (ATP) Paul Zeitz, Hungarian maths contest –k = n(n+1)(n+2)(n+3), k is never a square –“Plug and chug” with n=1, n=2, n=3, n=4, …. –Gives: 24, 120, 360, 840, …. –Always square minus 1

General Approach Ideal world: –ATP proves a theorem found by ML about CAS functions Problem with this: –CAS functions are too complex for ATP Positive spin: ATP proves from 1 st principles –Proven conjectures are likely to be less interesting –Use ATP to discard dull results –Find conjectures, not theorems for CAS user

Systems Used Maple –Well known CAS system HR –Not so well known ATF system (ML) Otter –Well known ATP system Domain –Number theory

The HR System – Concepts Automated theory formation –Invent concepts, make conjecture, prove theorems –Disprove non-theorems and present results Concept formation via 10 production rules –Builds new concepts from old ones –Example on next slide Heuristic search –Interestingness measures (e.g., complexity) –Limits on the concepts formed (e.g., arity, comp)

Concept Formation [a] exists b (sigma(a)=b & isprime(b)) [a] : isprime(a)[a,b] : sigma(a)=b [a,b] : sigma(a)=b & isprime(b) compose exists (Complexity = 4)

The HR System - Conjectures HR Makes conjectures empirically –Two concepts are equivalence –A concept is inconsistent with the axioms –Examples of one concept all examples of another Extracts simpler concepts from these –A & B & C D & E –A & B & C => D, D & E => A, etc. Finds prime implicates: A & B => D Also makes applicability conjectures –tau(x) = x only true (empirically) for x = 1 and x = 2

Presentation of Results HR has some limited theorem proving –To discard the very dull conjectures: –f(a) & g(a)  g(a) & f(a), f(a)=b & f(a)=c [b  c] –But we didn’t want to re-invent the wheel HR can re-write definitions –  b (tau(a)=b & isprime(b)) becomes isprime(tau(a)) HR can also sort the conjectures (interestingness) –Applicability = number of entities with non-empty examples Conjs about primes score 10/30 for applic (10 primes between 1 & 30) –Surprisingness (see next slide)

Measuring Surprisingness Surprisingness = 6

HR Using Maple User chooses the Maple functions of interest HR calls Maple to supply initial data –User specifies initial input, say, numbers 1 to 10 –E.g., tau(10) = 4, sigma(6) = 12, etc. HR also calls Maple during theory formation –HR invents concept: tau(sigma(n)) –Needs value for: tau(sigma(10)) = tau(18) = 6 –Also to find counterexamples to conjectures Integration via files (sorry Jürgen, MWeb soon)

HR Using Otter Problem with HR – too many conjectures (thesis:6%) Normally HR supplies Otter with only: –Axioms of domain and the conjecture statement Problematic example –All a (a=1 or a=2 => tau(a) = a) –True but dull –HR supplies tau(1) = 1 and tau(2) = 2 to Otter User can identify “axioms” to give to Otter –E.g., isprime(a)  tau(a) = 2 User specifies which conjs to add as axioms –HR re-proves (and discards) many previous results

Experiment Aims: –(i) show HR works with Maple –(ii) Show pruning of conjectures works Three Maple functions chosen: –tau(n), sigma(n), isprime(n) HR given number 1 only, but access to 2-30 Breadth first search ran to completion –Complexity limit of 6 and some arity constraints Then the user steps in –To specify obvious results to add as axioms Any proven results ignored

Results 378 theory formation steps, approx. 2 mins HR called Maple 120 times –E.g., for 195 = sigma(72) = sigma(sigma(30)) –Numbers 2, 3, 4, 5, 6, 9 and 16 introduced Produced 137 implicate conjectures –43 already proven by Otter (goodbye) –E.g: all a ((sigma(a)=1 & sigma(1)=a) => (a=1)) We looked through the remaining 94 –Added 9 from the first 10 (ordered by complex.) –E.g: [3] all a ((isprime(a)) => (tau(a)=2)) –Added another 3 which were “instantiations” After re-proving 94 became just 22

Results Continued We ordered the 22 by (applic + surp)/2 Top one was: –isprime(sigma(a)) => isprime(tau(a)) –So interesting, we proved it (generalised) –Then we added it as an “axiom” Reduced the results down to just 10: tau(tau(a))=a => tau(sigma(sigma(a)))=sigma(a) tau(tau(a))=a => tau(sigma(a))=a [should’ve gone] Are these results interesting ???

Conclusions Aims: –HR works with Maple  No problem with the interface –HR uses Otter to prune dull conjectures  137 => 94 => 82 => 22 => 17 => 16 => 10 Bonus: interesting conjecture(s) Question: is this of use to CAS users?

Further Work Improved pruning of conjectures –Use Otter’s Knuth Bendix completion (McCasland) –HR now has skolemised rep n (discards exists conjs) Get ATF embedded in CAS (any offers?) Apply HR to more discovery tasks –Roy McCasland and Zariski spaces Last line of my thesis: “… if this technology can be embedded into computer algebra systems, we believe theory formation programs will one day be important tools for mathematicians.”

Acknowledgments This work has been supported by –EPSRC grant GR/M98012 –EU IHP grant: Calculemus HPRN-CT This work inspired by collaboration with: –Jacques Calmet and Clemens Ballarin –Calculemus YVR at Karlsruhe