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A Theory of Theory Formation

A Theory of Theory Formation. Simon Colton Universities of Edinburgh and York. Overview. What is a theory? Four components of the theory of ATF Techniques inside the components Cycles of theory formation Case Studies Applications (briefly) Of both the theories and the process.

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A Theory of Theory Formation

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  1. A Theory of Theory Formation Simon Colton Universities of Edinburgh and York

  2. Overview • What is a theory? • Four components of the theory of ATF • Techniques inside the components • Cycles of theory formation • Case Studies • Applications (briefly) • Of both the theories and the process

  3. What is a Theory? • Theories are (minimally) a collection of: • Objects of interest • Concepts about the objects • Hypotheses relating the concepts • Explanations which prove the hypotheses • Finite Group Theory: • All cyclic groups are Abelian • Inorganic Chemistry: • Acid + Base  Salt + Water

  4. So, We Require: Object Generator Concept Generator Hypothesis Generator Explanation Generator

  5. In Principle, These Could Be: Database, CAS, CSP, Model Generator Machine Learning Program ATP System, Pathway Finder, Visualisation Data Mining Program

  6. In Practice, Current Implementation: Database, Model Generator, (CAS, CSP nearly) The HR Program The HR Program ATP Systems

  7. Object Generation and Explanation Generation • Object Generation: • Machine learning – reading a file, database • In Mathematics • CSP (e.g., FINDER, Solver), CAS (e.g., Maple) • Davis Putnam method (e.g., MACE) • Resolution Theorem Proving (e.g., Otter) • HR must be able to communicate • Read models and concepts from MACE’s output • Read proofs and statistics from Otters output

  8. Concept Generation • Build a new concept from old ones • 10 general production rules (demonstrated later) • Produce both a definition and examples • Throw away concepts using definitions • Tidy definitions up • Repetitions, function conflict, negation conflict • Decide which concepts to use for construction • Plethora of measures of interestingness • Weighted sum of measures

  9. Concept Generation:Lakatos-inspired Techniques • Monster Barring • Remove an object of interest from theory • Counterexample Barring • Except a finite subset of objects from a theorem • E.g., all primes except 2 are odd • Concept Barring • Except a concept from a theorem • All integers other than squares have an even number of divisors • Credit to Alison Pease

  10. Hypothesis Generation:Finding Empirical Relationships • Equivalence conjectures • One concept has the same examples as another • Subsumption conjectures • All examples of one concept are examples of other • Non-existence conjectures • A concept has no examples • Assessment of conjectures • Used to assess the concepts mentioned in them

  11. Hypothesis GenerationExtracting Prime Implicates • Extract implications, then prime implicates • Equivalence conjectures are split: • A & B & C  D & E & F becomes • A & B & C  D, A & B & C  E, etc. • Non-existence conjectures are split: • ¬(A & B & C) becomes: A & B  ¬C, etc. • Extract Prime implicates: • A & B & C  D, try A  D, then B  D, C  D, then A & B  D, etc.

  12. Hypothesis Generation: Imperfect Conjectures • User sets a percentage minimum, say 80% • Near-subsumption conjectures • E.g., primes  odd (99% true) • Also returns the counterexamples: here, 2 • Near-equivalence conjectures • Prime  odd (70% true) • Applicability conjectures • A concept has a (small) finite number of examples • E.g., even prime numbers: 2 is only example

  13. Cycles of Theory Formation • How the individual techniques are employed • Concept driven conjecture making • Finding conjectures to help understand concepts • Exploration techniques • Conjecture driven concept formation • Inventing concepts to fix faulty conjectures • Imperfect conjectures, Lakatos techniques

  14. Concept Driven Cycle (cut-down) Invent Concept Reject Non Existence Equivalence New Concept Subsumptions Implications

  15. Concept Driven Cycle Continued Implications Counterexample Proof Prime Implicates Counterexample Proof

  16. Conjecture Driven Cycle Invent Concept Reject Near Equivalence Applicability Near Subsumption Monster Barring Concept Barring Concept Barring Counterex Barring New Concept Counterex Barring New/Old Concept New/Old Concept Equivalence Implications

  17. Case Study: Groups Given: Group theory axioms

  18. Case Study: Groups Davis Putnam Method MACE model generator finds a model of size 1

  19. Case Study: Groups HR Reads MACE’s Output Extracts concepts: Element, Multiplication, Identity, Inverse

  20. Case Study: Groups Match Production Rule Invents the concept idempotent elements (a*a=a)

  21. Case Study: Groups Equivalence Finding Makes Conjecture: a*a=a  a is the identity element

  22. Case Study: Groups Resolution Theorem Proving Otter proves this in less than a second

  23. Case Study: Groups Extracts Prime Implicates a*a = a  a=identity, a=identity  a*a=a End of cycle

  24. Case Study: Groups Compose Production Rule Later: Invents the concept of triples of elements (a,b,c) for which a*b=c & b*a=c

  25. Case Study: Groups Exists Production Rule Invents concept of pairs (a,b) for which there exists an element c such that: a*b=c & b*a=c

  26. Case Study: Groups Forall Production Rule Invents the concept of groups for which all pairs of elements have such a c: Abelian groups

  27. Case Study: Groups Equivalence Finding Makes the Conjecture: G is a group if and only if it is Abelian

  28. Case Study: Groups Sorry Otter fails to prove this conjecture

  29. Case Study: Groups Davis Putnam Method MACE finds a counterexample: Dihedral Group of size 6 (non-Abelian)

  30. Case Study: Groups Assessment of Concepts Concept of Abelian groups allowed into theory Theory recalculated in light of new object of interest

  31. Case Study: Goldbach Given: Integers 1 to 100, Concepts: Divisors, Addition

  32. Case Study: Goldbach Split Production Rule Invents: Even Numbers (divisible by 2)

  33. Case Study: Goldbach Size Production Rule Invents: Number of Divisors (tau function)

  34. Case Study: Goldbach Split Production Rule Invents: Prime numbers (2 divisors)

  35. Case Study: Goldbach Compose Production Rule Half an hour later: Invents: Goldbach numbers (sum of 2 primes)

  36. Case Study: Goldbach Near Equivalence Finding Conjectures: Even numbers are Goldbach numbers (with one exception, the number 2)

  37. Case Study: Goldbach Counterexample Barring (Split) Forces: Concept of being the number 2

  38. Case Study: Goldbach Counterexample Barring (Negate) Forces concept: Even numbers except 2

  39. Case Study: Goldbach Subsumption Finding Conjectures: Even numbers except 2 are Goldbach Numbers (Goldbach’s Conjecture)

  40. Case Study: Goldbach Absolutely No Chance Passes the conjecture to an inductive theorem prover?

  41. Applications of Theories • Puzzle generation • Which is the odd one out: 4, 9, 16, 24 • Which is the odd one out: 2, 9, 8, 3 • Problem generation • TPTP library: find theorem to differentiate Spass & E • See AI and Maths paper • Prediction tests: (e.g., Progol animals file) • P(mammal | has_milk) = 1.0 • P(mammal | habitat(water)) = 0.125 • Take average over all Bayesian probabilities

  42. Applications of Theory Formation • Identifying concepts (e.g., Michalski trains) • Forward look ahead mechanism (see ICML-00 paper) • Simplifying problems • Lemma generation for ATP • Constraint generation for CSP (see CP-01 paper) • Identifying outliers • How unique an object of interest is • Inventing concepts • Integer sequences (and conjectures), • See AAAI-00 paper, Journal of Integer Sequences

  43. Conclusions • Presented a snapshot of the theory of ATF • Autonomous • Four components, numerous techniques • Uses third party software • Concept driven and conjecture driven cycles • Applies to many machine learning tasks • Concept identification, puzzle generation, • Predictions, problem simplification

  44. Welcome to the Next Level • For any of the four components • Substitute a human for interactive ATF • Roy McCasland (hopefully), mathematician • Work on Zariski spaces with HR • For any of the four components • Substitute another agent for multi-agent ATF • Alison Pease’s PhD, cognitive modelling • Lakatos style reasoning and machine creativity

  45. Theory Formation in Bioinformatics? • Can work with non-maths data • Can form near-conjectures • Needs to relax notion of equality • Multi-agent approach definitely needed

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