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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