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

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

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

Automated Discoveries #2 Mersenne prime: – 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

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 

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

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

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))

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

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

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

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)

Bootstrapping ATF Cycle

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

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)

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

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

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)

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

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 *

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 ) 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

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 10 53

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

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

Web Pages HR: NumbersWithNames program: Encyclopedia of Integer Sequences: