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The HR Program for Theorem Generation

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

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The HR Program for Theorem Generation

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  1. The HR Program forTheorem Generation Simon Colton Mathematical Reasoning Group University of Edinburgh

  2. Overview • Start with the axioms of a domain • Produce 100s of theorems about domain • How do we do this? • Why do we do this?

  3. 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

  4. Concept Formation • 10 general production rules • Example: Abelian groups a * b = c compose a * b = c & b * a = c exists  c (a * b = c & b * a = c) forall  a b  c (a * b = c & b * a = c)

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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.

  11. 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]

  12. 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

  13. http://www.dai.ed.ac.uk/~simonco/research/hr Please ask me for a demo!

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