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Automated Discovery in Pure Mathematics

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Automated Discovery in Pure Mathematics

Simon Colton

Universities of Edinburgh and York

- 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

- 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

- 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

- 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

- New formula for :

- 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

- 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

- 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))

- 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

- 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

- 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

- 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)

- 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

- 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)

- 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

- 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

- 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)

- 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

- 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 *

- 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

- 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

- Hungarian maths competition
- Multiply four consecutive numbers
- n(n+1)(n+2)(n+3)
- Never a square number

- Demonstration
- Using NumbersWithNames

- 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

- 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