Making Conjectures About Maple Functions

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

2. 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

3. 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 1st principles • Proven conjectures are likely to be less interesting • Use ATP to discard dull results • Find conjectures, not theorems for CAS user

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

5. 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)

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

7. 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

8. 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)

9. Measuring Surprisingness Surprisingness = 6

10. 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)

11. 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

12. 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

13. 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

14. 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 ???

15. 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?

16. Further Work • Improved pruning of conjectures • Use Otter’s Knuth Bendix completion (McCasland) • HR now has skolemised repn (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.”

17. Acknowledgments • This work has been supported by • EPSRC grant GR/M98012 • EU IHP grant: • Calculemus HPRN-CT-2000-00102 • This work inspired by collaboration with: • Jacques Calmet and Clemens Ballarin • Calculemus YVR at Karlsruhe