Automated discovery in pure mathematics
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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

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Automated discovery in pure mathematics

Automated Discovery in Pure Mathematics

Simon Colton

Universities of Edinburgh and York


Overview of talk

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

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

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

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

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

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


Inventing new 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

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

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

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

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

Bootstrapping ATF Cycle


Applications of hr

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

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

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

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

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

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

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

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

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

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

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

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


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