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Calculation Invention and Deduction

Calculation Invention and Deduction. Dr. Simon Colton Imperial College London (Formerly at Edinburgh) YVR in Karlsruhe & Saarbrucken. A Plausible Model for Mathematical Discover. Do some calculations 2. Notice a pattern 3. Prove pattern is true

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Calculation Invention and Deduction

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  1. Calculation Invention and Deduction Dr. Simon Colton Imperial College London (Formerly at Edinburgh) YVR in Karlsruhe & Saarbrucken

  2. A Plausible Model forMathematical Discover • Do some calculations 2. Notice a pattern 3. Prove pattern is true • Computer Algebra Machine Learning Automated Theorem • Proving • Calculation Invention Deduction • MAPLE HR OTTER

  3. A Nice Example • 1926 Hungarian Mathematics Competition: • n(n+1)(n+2)(n+3) is never a square number • Plug and Chug [Calculate] • 1*2*3*4 = 24, 2*3*4*5 = 120, 3*4*5*6 = 360 • Eureka! [Invent] • 20 = 25-1 = 5*5-1, 120=11*11-1, 360 = 19*19-1 • Numbers are all squares – 1 • Therefore [Deduce] • They cannot be square numbers

  4. The HR System • Performs automated theory formation • Start with axioms (or some background) • End with examples, concepts, conjectures, theorems • Cycle of mathematical activity • Invents new concepts based on old ones • Makes conjectures empirically about the concepts • Tries to prove the conjectures using a theorem prover or disprove them using a model generator • Works in number theory, graph theory, various algebras • One of a few programs to add to mathematics

  5. Calculemus Project 1Adding to a Library of Theorems • Work done with Jürgen Zimmer (Saarbrucken) and Geoff Sutcliffe (Miami) • I learned about the Mathweb software bus (task 2.1) • TPTP library of mathematical theorems • To help see who has the best theorem prover • Want to differentiate, not beat provers • HR within MathWeb testing three provers • Found 12,000 theorems in one session • Each prover was beaten by a theorem others could prove • Now added 184 theorems to the library • Only non-human to add to the library

  6. Calculemus Project 2Discriminant Discovery • Work done with Andreas Meier (Saarbrucken) & Volker Sorge (Birmingham) • I learned about proof planning and omega (task 2.3) • What’s the difference between these algebras? • Idempotent element appearing only once on the diagonal • Found by HR as a discriminant in the general case • Part of a larger project to classify residue classes • Using proof planning in Omega, and many other systems • HR found a discriminant for 97% of the 817 pairs given to it

  7. Calculemus Project 3The HOMER System • Work done with Jacques Calmet and Clemens Ballarin at Karlsruhe (task 2.2) • I learned about about Maple (computer algebra system) • Aimed at (recreational) number theorists • HR is given some Maple input (computer algebra) • Makes conjectures about the functions supplied • Uses Otter to prove them • Throws away any which Otter can prove • Because they are unlikely to be interesting to user

  8. Some Results from HOMER • Only non-human to add to • Encyclopedia of integer sequences (80,000+) • Assessed by mathematician (never seen HOMER) • A session in an afternoon produced 59 conjs • 38 proved, 4 false, 17 open • Four were “number theoretically interesting” • Nice results: • Sum of divisors of squares is odd • Phi(n) is square implies Tau(n) is even • Perfect numbers are pernicious

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