2. Automated Theorem Discovery Simon Colton Universities of Edinburgh and York

3. Automated Theorem Proving Assumption: user supplies theorem Textbook mathematics: theorem & proof Frontline mathematics: exploration & questions Drop the assumption Program supplies the theorems as well Adds value to ATP Talk: generating theorems and applications Examples in finite algebras, number theory

4. Background McCune and Padmanabhan Axiomatisations of finite algebras using Otter et al. Exhaustive search Chou, Bagai, Zytkow, Shanbogue New geometry results using Wu’s method Exhaustive search Fairly predictable results Using theorem prover to extend calculation Relies on power/speed of the theorem prover

5. Approach 1 – Theory Formation The HR program Invents concepts, makes conjectures Proves theorems, finds counterexamples Uses Otter and MACE Proving Restricted to finite algebras Can produce surprising results Also works in number and graph theory

6. Results From Approach 1 Group theory All groups are quasigroups, etc. Important to re-discover well known results Anti-associative algebras (no assoc triple) Cannot be quasigroups or have an identity Must be at least two elements on diagonal Other results in the paper/thesis All new and not obvious to us

7. Approach 2 - Datamining Mathematical information can be mined Encyclopedia of integer sequences 60,000 concepts from many domains (as sequences) Looking for simple relations Subsequences, supersequences, disjoint Surprising results No theorem proving as yet (driving force?) Add proving to mathematical databases

8. Results From Approach 2 Perfect numbers: equal sum of divisors Eg. 6=1+2+3, 28 = 1+2+4+7+14 Written in binary?? 6 = 110, 28 = 11100, 496 = 111110000 Perfect numbers are pernicious (J. Gow) Prove this by ATP? Induction? Probably not……?

9. Approach 3 – Embed in CAS Mathematicians explore with CAS Tools available: calculations, visualisation CAS should also make conjectures/theorems With a theorem prover embedded Using both theory formation and datamining Long term goal of theorem discovery Amazing that there’s little research on this

10. Application 1 – TPTP/CASC CASC – theorem proving competition Uses TPTP: thousand problems for provers Alleged problem Fine tuning possible Possibly use HR to find novel hard theorems In particular in novel domains (eg. anti-assoc) Not entirely succesful yet Difficult theorems for Otter are easy for SPASS

11. Application 2 - CSP Used HR to find additional constraints for CSPs Implied (theorems) and Induced (concepts) Work done with Ian Miguel and Toby Walsh Example: QG3-quasigroups (a*b)*(b*a)=a Results: These are anti-abelian (size 6, 58 sec to 1 sec) All different on the diagonal (a*a) = b -> (b*b) = a Improve model: QG6, size 10, (3.75 hrs to 8 mins)

12. Other Applications Teaching aid For setting exercises Lecturer knows results are true Improving ATP itself Lemma generation Intelligent case-splits

13. Conclusions Conjecturing as important as proving Glue between CAS and ATP Discovery adds value to ATP Tools and data becoming available Theorem provers not good enough yet? More attractive to mathematicians (esp. syntax) More time spent on mathematics problems Build up databases of theorems to use