Automated Discovery in Pure Mathematics

Simon ColtonUniversities 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: 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

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 proversEquivalence 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 onesConcepts 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)*cUsed 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 diagonalHave no identity, or even local identity Commutative pairs are not co-squares

Quasigroup Results

Part of CSP projectQG3 quasigroups: (a*b)*(b*a)=aHR 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 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

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

Demonstration – Zeitz numbers

Hungarian maths competitionMultiply 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: 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