The HR Program for Theorem 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

Please ask me for a demo!