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 Machine Learning Program Data Mining Program ATP System, Pathway Finder, Visualisation

In Practice, Current Implementation: Database, Model Generator, (CAS, CSP nearly) 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 EquivalenceNon ExistenceNew Concept Subsumptions Implications Reject

Concept Driven Cycle Continued Implications Counterexample Proof Prime Implicates CounterexampleProof

Conjecture Driven Cycle Invent ConceptReject Near EquivalenceApplicabilityNear Subsumption Concept Barring Counterex Barring New/Old Concept Equivalence Implications Monster Barring New Concept Counterex Barring Concept Barring New/Old Concept

Case Study: Groups Given: Group theory axioms

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

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

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

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

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

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

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

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

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

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

Case Study: Groups Otter fails to prove this conjecture Sorry

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

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

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

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

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

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

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

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

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

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

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

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

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)) = – 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