The HR Program forTheorem Generation

Simon Colton

Mathematical Reasoning Group

University of Edinburgh

Overview

Start with the axioms of a domain

Produce 100s of theorems about domain

How do we do this?

Why do we do this?

The HR Program

Machine learning Java program– With special application to mathematics– Performs automated theory formation

Uses five processes to generate theorems– Initialisation from axioms (bootstrapping using MACE)– Production rule based concept formation– Empirical conjecture making (with a little reasoning)– Automated theorem settling (ATP/ModGen)– Theorem post-processing

Concept Formation

10 general production rules Example: Abelian groups

a * b = c

a * b = c & b * a = c

c (a * b = c & b * a = c)

a b c (a * b = c & b * a = c)

compose

exists

forall

Empirical Conjecture Making

Non-existence conjectures– Invents a concept with no examples

Equivalence conjectures– Two concepts have exactly same examples

Implication conjectures– A concept has all the examples of another

A Little Reasoning

HR discards many conjectures:¬( A (p(A) & ¬p(A)) [bad negation]

f(A) = x & f(A) = y & x y [bad instantiation]

a b (p(a,b) & q(a) x (p(a,x) & q(x)))

[unification]

HR also has: – Built-in forward-chaining prover

Settling Conjectures

HR first uses Otter– To try and prove each theorem

If Otter fails– HR uses MACE to try to find a counterex.

Other provers via MathWeb– Bliksem, E, Spass, …– See Jürgen Zimmer’s PaPS talk on Weds

Post-Processing Conjectures

Example: (p(a) & q(a) r(a) & s(a))

Extracts implicates:– p(a) & q(a) r(a), p(a) & q(a) s(a)

Attempts to find prime implicates– Tries: p(a) r(a), then q(a) r(a)– Using Otter each time

Example session

Ring theory axioms RNG-004– 1000 steps in 6481 seconds– 275 prime implicates extracted– 39 with proof length > 10– 30 examples of rings added as counters– 2 of #2 2 of #3 25 of #4 1 of #7

See paper for further details

Applications

Pre-processing AI problems– CSP() ATP(?) ML(??)

Mathematical discovery– Number theory, algebraic domains

Mathematics tutoring– See talk at RADM workshop

Testing ATP programs– HR first non-human to add to TPTP library– Roughly 15 in this year’s CASC comp.

Example TPTP conjecture

Otter and E fail (120 seconds), Spass succeeds:

x y (( z (inv(z)=x & z*y=x) & u (x*u=y & v (v*x=u & inv(v)=x)))( a (inv(a)=x & a*y=x) & b (b*y=x & inv(b)=y)))

[about pairs of identity elements]

Conclusions & Future Work

Automated theory formation– Produces 100s of conjectures– Initialisation, concept formation, empirical conjecture making, ATP & MG, post-processing

Many applications– Pre-proc, TPTP, discovery, tutoring

Applying this to bioinformatics– Deduction and induction combined

http://www.dai.ed.ac.uk/~simonco/research/hr

Please ask me for a demo!