Combining Reasoning Systems for Mathematical Discovery

Transcription

1 Combining Reasoning Systems for Mathematical Discovery Simon Colton Combined Reasoning Group Department of Computing Imperial College, London

2 My one point is... We [AI researchers] should stop thinking (only) about single AI systems for scientific discovery We should start thinking (more) about combining different reasoning systems for scientific discovery

3 To make my point... I ll describe projects where combined systems Have performed tasks which standalone systems cannot Have been more flexible than standalone systems Are better at standard AI tasks than standalone systems All in the context of mathematical discovery

4 Mathematical Discovery Taster Odd refactorable numbers are perfect squares (JIS 2001) Isomorphic/isotopic algebraic classification results (CMUC 2008) Commentationes Mathematicae Universitatis Carolinae

5 1,9,225,... (odd refactorable numbers) The HR System - Example [a,b]: b a [a,b,c] : a = b * c size split match [a,n]: n = {b : b a} [a]: 2 a [a,b] : a = b * b compose negate exists [a,n]: n = {b : b a} & n a exists [a]: (2 a) [a] : b (a = b * b) 1,4,9,16,... (square numbers) [a]: n (n = {b : b a} & n a) 1,2,8,9,12,18,...(refactorable numbers) compose implication conjecture [a]: n (n = {b : b a} & n a) & (2 a)

6 HR System - a few details 22 Production rules (but we normally only use 6 or 7) 3 Main ways to produce conjectures Also has ways to extract prime implicates 22 Measures of interestingness Which drive a heuristic search (new concepts are built from the best old ones)

7 Mathematical Applications HR + Otter + MACE [theory formation about star algebras] HR + Otter + Maple [conjecture making in number theory] Bootstrapping system [deriving classification theorems in loop/quasigroup theory]

8 Star algebras - setup Single axiom algebra : x y z ((x*y)*z = y*(z*x)) HR starts with only this axiom First conjecture: no star algebras exist MACE disproves with a counterexample HR makes conjectures Otter attempts to prove, MACE attempts to disprove We look for interesting true results and open conjectures

9 Star algebras - results a b (b * a = a a * b = a) Mathematical Knowledge Management 2006 with Pedro Torres, Paul Cairns and Volker Sorge Elements with a left/right identity are closed under * Idempotent elements are closed under * Main result: there is no interesting canonical example They are either reducible (repeated row and column) Or are star algebras because commutative & associative

10 The HOMER system - setup The user supplies some number theory concepts from Maple e.g., tau, sigma, phi, isprime, iseven, isodd, issquare Maple is used to calculate background knowledge HR makes conjectures, we prove interesting ones by hand Otter is used to prove conjectures from some given axioms Any proved conjectures are thrown away We also added interesting conjectures to axiom sets

11 The HOMER system - results Experiment 1: tau(n), sigma(n) and isprime(n) Journal of Symbolic Computation 2005 Nice result: isprime(sigma(n)) isprime(tau(n)) Experiment 2: tau, sigma, odd, even, refactorable, issquare Nice result: issquare(a) isodd(sigma(a)) Four teasers about r.f. numbers: even(a) & refactorable(sigma(a)) even(tau(a))

12 The HOMER system - results Conference on Automated Deduction (CADE) 2003 with Sophie Huczynska Experiment 3: With phi function Nice result: even(phi(n)) n > 2 Nice result: issquare(phi(n)) even(tau(n)) Cute because requires the contrapositive to prove

13 Bootstrapping system - overview Fairly complicated system by Sorge and Meier HR is one of a number of reasoning systems Model generators, SAT-solvers, CAS, theorem provers Starts with: (i) axioms of an algebra (ii) size (iii) equivalence relation Ends with: A fully proved classification tree (theorem)

14 Bootstrapping system - example

15 Bootstrapping system - results Isomorphic classification (with HR) 7 full results: Q3,Q4,Q5,IQ6,QG3.6,Loop5,Loop6 2 partial results: QG9.7, Idempotent Loop 7 Isotopism classification (without HR) 2 full results: Loop6, Loop7 Papers: IJCAR 04, ML 06, IJCAR 06, JAR 07, Loops08, CMUC 08 Work with Volker Sorge, Andreas Meier, Roy McCasland

16 Q4 Isomorphism #1

17 Loop6 Isotopism

18 Q4 - Isomorphism #2

19 More Flexibility - TM System Given a non-theorem, theorem provers just say no and model generators give a counterexample, but no explanation TM produces a set of alternative, proved theorems Works using Otter and MACE, but differently: MACE produces supporting and counter examples HR invents concepts true of supporting examples only Otter proves that specialised conjectures are true

20 TM System - Results Electronic Notes on Computer Science 2005 with Alison Pease 83% of 97 non-theorems were modified successfully Theorems taken from the TPTP library (and ruined...) Nice example from ring theory w x ((((w*w)*x)*(w*w) = id) fixed if x (x*x = x+x)

21 Improving AI systems - ICARUS CSP reformulation is an important area We looked at discovering implied constraints Start with a specification of a parameterised CSP CLPFD solver produces solutions to small problems HR suggests some conjectures about the examples Otter is used to prove that they follow from the spec. ICARUS finds those (sets) which make good constraints

22 ICARUS - results European Conference on AI (ECAI) 2006 with John Charnley and Ian Miguel Worked with standard CSP benchmark: QG quasigroups QG3/4: quasigroups are anti-abelian QG3: a*a = b b*b = a

23 Please ask me about... Case-splitting first order theorems Automatic generation of first order theorems for TPTP A rational reconstruction of Graffiti Computational model of Lakatos s philosophy of Maths

24 Some Ongoing Work PhD project of Pedro Torres: Meta-theory formation PhD project of John Charnley: A global workspaces approach to combining reasoners PostDoc project of Alison Pease A computational model of axiom formation using Where mathematics comes from by Lakoff and Nunez Feasibility study: mathematical invention of fitness functions

25 Conclusions The whole is more than a sum of the parts: New tasks, improved efficiency, more flexibility It is time to put AI back together again Workflows of reasoning (like data analysis) Math/scientific discovery is a great application domain Hopefully of benefit to science