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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 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 So, We Require: Object Generator Concept Generator Hypothesis Generator Explanation Generator In Principle, These Could Be: Database, CAS, CSP, Model Generator Data Mining Program Machine Learning Program ATP System, Pathway Finder, Visualisation In Practice, Current Implementation: Database, Model Generator, (CAS, CSP nearly) The HR Program The HR Program ATP Systems 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 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 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 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 Hypothesis Generation Extracting 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. 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 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 Concept Driven Cycle (cut-down) Invent Concept Non Existence Equivalence Reject New Concept Subsumptions Implications Concept Driven Cycle Continued Implications Counterexample Proof Prime Implicates Counterexample Proof Conjecture Driven Cycle Invent Concept Near Equivalence Concept Barring Applicability Reject Near Subsumption Monster Barring Counterex Barring Concept Barring New Concept Counterex Barring New/Old Concept New/Old Concept Equivalence Implications Case Study: Groups Given: Group theory axioms Case Study: Groups Davis Putnam Method MACE model generator finds a model of size 1 Case Study: Groups HR Reads MACE’s Output Extracts concepts: Element, Multiplication, Identity, Inverse Case Study: Groups Match Production Rule Invents the concept idempotent elements (a*a=a) Case Study: Groups Equivalence Finding Makes Conjecture: a*a=a a is the identity element Case Study: Groups Resolution Theorem Proving Otter proves this in less than a second Case Study: Groups Extracts Prime Implicates a*a = a a=identity, a=identity a*a=a End of cycle 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 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 Case Study: Groups Forall Production Rule Invents the concept of groups for which all pairs of elements have such a c: Abelian groups Case Study: Groups Equivalence Finding Makes the Conjecture: G is a group if and only if it is Abelian Case Study: Groups Sorry Otter fails to prove this conjecture Case Study: Groups Davis Putnam Method MACE finds a counterexample: Dihedral Group of size 6 (non-Abelian) Case Study: Groups Assessment of Concepts Concept of Abelian groups allowed into theory Theory recalculated in light of new object of interest Case Study: Goldbach Given: Integers 1 to 100, Concepts: Divisors, Addition Case Study: Goldbach Split Production Rule Invents: Even Numbers (divisible by 2) Case Study: Goldbach Size Production Rule Invents: Number of Divisors (tau function) Case Study: Goldbach Split Production Rule Invents: Prime numbers (2 divisors) Case Study: Goldbach Compose Production Rule Half an hour later: Invents: Goldbach numbers (sum of 2 primes) Case Study: Goldbach Near Equivalence Finding Conjectures: Even numbers are Goldbach numbers (with one exception, the number 2) Case Study: Goldbach Counterexample Barring (Split) Forces: Concept of being the number 2 Case Study: Goldbach Counterexample Barring (Negate) Forces concept: Even numbers except 2 Case Study: Goldbach Subsumption Finding Conjectures: Even numbers except 2 are Goldbach Numbers (Goldbach’s Conjecture) Case Study: Goldbach Absolutely No Chance Passes the conjecture to an inductive theorem prover? 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 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 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 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 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