Automated Discovery in Pure Mathematics Simon Colton Universities

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

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

Automated Discoveries #1 Robbins algebras are boolean n Automated theorem proving, Mc. Cune+Wos Quasigroup existence problems (QG 6. 17) n Constraint solvers, John Slaney et al. Inconsistency in Newton’s Principia n Formal methods (NS-analysis), Fleuriot

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

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

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

HR: Theory Formation Cycle Start with background knowledge n user-supplied axioms + concepts 1. 2. 3. 4. Invent a new concept (machine learning) Look for conjectures empirically (d-mining) Prove the conjectures (theorem proving) Disprove the conjectures (model generation) 5. Assess all concepts w. r. t. new concept 1. Invent a new concept n Build it from the most interesting old concepts

Inventing New Concepts Ten General Production Rules (PR) n n Work in all domains (math + non math) Build new concept from one (or two) old ones Example: Abelian groups n n 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))

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

Proving Theorems HR relies on third party theorem provers Equivalence conjectures: n n 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 n n William Mc. Cune et al. Only uses this for finite algebras

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

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

Bootstrapping ATF Cycle

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

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

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 n Quasigroups must have associative triple Have two elements on diagonal Have no identity, or even local identity Commutative pairs are not co-squares

Quasigroup Results Part of CSP project QG 3 quasigroups: (a*b)*(b*a)=a HR conjectured, Otter proved, We interpreted n n n Diagonal elements are all different a*a=b b*b=a a*b=b b*a=a QG 3 quasigroups are anti-Abelian n n a*b = b*a a=b Corollary to one of HR’s results (with our help) 10 x speed up over naïve model

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

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

Refactorable Numbers Number of divisors is itself a divisor n n n 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) n n n 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 2 x refactorable *

Refactorables – Deeper Results Natural density is zero n Kennedy and Cooper 1990 Joshua Zelinsky (hot off the press) n n n 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 w Every integer is the sum of 5 or fewer refactorables Zelinsky uses the results from HR

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 n (1, 2), (8, 9), (1520, 1521), (50624, 50625), … There are no triples of refactorables n n We know there are no quadruples And no triples less than 1053

Demonstration – Zeitz numbers Hungarian maths competition Multiply four consecutive numbers n n n(n+1)(n+2)(n+3) Never a square number Demonstration n Using Numbers. With. Names

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

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