Automated Discovery in Pure Mathematics

1. Automated Discovery in Pure Mathematics Simon Colton Universities of Edinburgh and York

2. Overview of Talk • Some example discoveries • ATP, CSP, CAS, ad-hoc methods • The HR system • Automated theory formation • Overview of applications • Application to mathematical discovery • Finite algebras, number theory, refactorables • Demonstration • NumbersWithNames program

3. Automated Discoveries #1 • Robbins algebras are boolean • Automated theorem proving, McCune+Wos • Quasigroup existence problems (QG6.17) • Constraint solvers, John Slaney et al. • Inconsistency in Newton’s Principia • Formal methods (NS-analysis), Fleuriot

4. Automated Discoveries #2 • Mersenne prime: 26972593 – 1 • Distributed (internet) search, CAS • New geometry results • Chou using Wu’s method • Simple axiomatisations of algebras • Group: x(y(((zz-1)(uy)-1)x))-1=u • McCune and Kunen, ATP

5. Automated Discoveries #3 • Fajtlowicz’s Graffiti graph theory program • All G, Chrom+Rad < MaxDeg+FreqMaxDeg • 60+ papers about it’s conjectures • Bailey’s PSQL algorithm • New formula for : i (1/16i)(4/(8i+1)-2/(8i+4)-1/(8i+5)-1/(8i+6)) • Easier to calculate nth hex digit of 

6. Theories in Pure Mathematics • Concepts • Examples and definitions • Statements • Conjectures and theorems • Explanations • Proofs, counterexamples • e.g., pure maths:group theory • Concepts: cyclic groups, Abelian groups • Conjecture: cyclic groups are Abelian • Examples provide empirical evidence • Simple proof for explanation

7. HR: Theory Formation Cycle • Start with background knowledge • user-supplied axioms + concepts • Invent a new concept (machine learning) • Look for conjectures empirically (d-mining) • Prove the conjectures (theorem proving) • Disprove the conjectures (model generation) • Assess all concepts w.r.t. new concept • Invent a new concept • Build it from the most interesting old concepts

8. Inventing New Concepts • Ten General Production Rules (PR) • Work in all domains (math + non math) • Build new concept from one (or two) old ones • Example: Abelian groups • Given: [G,a,b,c] : a*b=c • Compose PR: [G,a,b,c] : a*b=c & b*a=c • Exists PR: [G,a,b] :  c (a*b=c & b*a=c) • Forall PR: [G] :  a b ( c (a*b=c & b*a=c))

9. Making Conjectures • Theory formation step • Attempt to invent a new concept • Concept has same examples as previous one • HR makes an equivalence conjecture • Concept has no examples • HR makes a non-existence conjecture • Examples of one concept are all examples of another concept • HR makes an implication conjecture

10. Proving Theorems • HR relies on third party theorem provers • Equivalence conjectures: • Sets of implication conjectures • From which prime implicates are extracted • E.g.  a (a*a=a a=id) • a*a=a  a=id, a=id  a*a=a • HR uses the Otter theorem prover • William McCune et al. • Only uses this for finite algebras

11. Disproving Non-Theorems • Any conjectures which Otter can’t prove • HR looks for a counterexample • Using the MACE model generator • Also written by William McCune • Other possibilities: • Computer algebra, constraint satisfaction • Counterexamples are added to the theory • Fewer similar non-theorems are made later

12. Assessing Interestingness • New concepts from interesting old ones • Concepts measured in terms of: • Intrinsic values, e.g. complexity of definition • Relational values, e.g. novelty of categorisation • Concepts also assessed by conjectures • Quality, quantity of conjectures involving conc. • Conjectures also assessed • Difficulty of proof (proof length from Otter) • Surprisingness (of LHS and RHS definitions)

13. Bootstrapping ATF Cycle

14. Applications of HR • Puzzle generation • Next in sequence, odd one out • Automated theorem proving • Discovering useful lemmas • Constraint satisfaction problems • Discovering additional constraints • Machine learning tasks • Puzzle solving, prediction tasks • Studying machine creativity • Multi-agent, cross-domain, meta-level

15. Application to Mathematical Discovery • Exploration of algebras using HR • Anti-associative algebras • Quasigroups • Number theory results • Encyclopedia of Integer Sequences • Using HR and NumbersWithNames • Refactorable numbers • Results and open conjectures • Problem solving (Zeitz numbers)

16. Anti-associative Algebras(Novel domain to me) • all a,b,c a*(b*c)  (a*b)*c • Used HR with Otter and MACE (2 hours) • 34 examples, sizes 2 to 6 (exists each size) • AAAs are not: abelian or quasigroups • Quasigroups must have associative triple • Have two elements on diagonal • Have no identity, or even local identity • Commutative pairs are not co-squares

17. Quasigroup Results • Part of CSP project • QG3 quasigroups: (a*b)*(b*a)=a • HR conjectured, Otter proved, We interpreted • Diagonal elements are all different • a*a=b  b*b=a • a*b=b  b*a=a • QG3 quasigroups are anti-Abelian • a*b = b*a  a=b • Corollary to one of HR’s results (with our help) • 10x speed up over naïve model

18. Neil Sloane’s Encyclopedia of Integer Sequences • Large database of sequences • E.g., Primes: 2, 3, 5, 7, 11, 13,… • Contains 67,000+ sequences (36 years) • A new sequence must be novel, infinite, interesting • HR has invented 20 new sequences • All supplied with interesting theorems (our proof) • Datamining the Encyclopedia itself • NumbersWithNames program (details ommitted)

19. Some Nice Results • Number of divisors, (n), is a prime • 2, 3, 4, 5, 7, 9, 11, 13, … • m(n) is prime  (n) is prime • g(n) = #squares dividing n • 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, … • numbers setting the record for g(n) • 1, 4, 16, 36, 144, 576, … • Squares of the highly composite numbers • Perfect numbers are pernicious

20. Refactorable Numbers • Number of divisors is itself a divisor • 1, 2, 8, 9, 12, 18, 24, 36, 40, … • HR’s first success [not in Encyclopedia] • Turned out to be a re-invention (1990) • Preliminary results (* - made by HR) • Infinitely many refactorables • Odd refactorables are perfect squares * • Congruent to 0, 1, 2 or 4 mod 8 * • Perfect numbers are not refactorable * • m,n relprim and refactorable  mn refactorable • x refactorable  2x refactorable *

21. Refactorables – Deeper Results • Natural density is zero • Kennedy and Cooper 1990 • Joshua Zelinsky (hot off the press) • T(n) < 0.5 B(n) with finitely many counterexamples (max 1013) • T(n) = #refacs < n, B(n) = #primes < n • Assuming Goldbach’s strong conjecture • Every integer is the sum of 5 or fewer refactorables • Zelinsky uses the results from HR

22. Refactorables – Questions….. • Numbers n!/3 are refactorable* • Numbers for which ((n))=n are refactorable* (x) = #integers less than or equal to and coprime to x • There are infinitely many pairs of refactorables • (1,2), (8,9), (1520,1521), (50624,50625), … • There are no triples of refactorables • We know there are no quadruples • And no triples less than 1053

23. Demonstration – Zeitz numbers • Hungarian maths competition • Multiply four consecutive numbers • n(n+1)(n+2)(n+3) • Never a square number • Demonstration • Using NumbersWithNames

24. Future Work: HR Project • McCasland? • Use HR to explore Zariski spaces • Colton: Express HR as a ML program • Try domains other than maths (bioinformatics) • Walsh: Integrate HR • With every maths program ever written • In particular Maple computer algebra • Bundy: • Build an automated mathematician

25. Web Pages • HR: • www.dai.ed.ac.uk/~simonco/research/hr • NumbersWithNames program: • www.machine-creativity.com/programs/nwn • Encyclopedia of Integer Sequences: • www.research.att.com/~njas/sequences