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

A Theory of Theory FormationA Theory of Theory Formation

Simon Colton

Universities of Edinburgh and York

OverviewOverview

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?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:So, We Require:

Object

GeneratorConcept

Generator

Hypothesis Generator

Explanation Generator

In Principle, These Could Be:In Principle, These Could Be:

Database, CAS, CSP, Model Generator

Machine Learning Program

Data Mining Program

ATP System, Pathway Finder, Visualisation

In Practice, In Practice, Current Implementation:Current Implementation:

Database, Model Generator, (CAS,

CSP nearly)

The HR Program

The HR Program

ATP Systems

Object Generation and Object Generation and Explanation GenerationExplanation 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 GenerationConcept 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:Concept Generation:Lakatos-inspired TechniquesLakatos-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:Hypothesis Generation:Finding Empirical RelationshipsFinding Empirical RelationshipsEquivalence conjectures

– One concept has the same examples as anotherSubsumption conjectures

– All examples of one concept are examples of otherNon-existence conjectures

– A concept has no examples Assessment of conjectures

– Used to assess the concepts mentioned in them

Hypothesis GenerationHypothesis GenerationExtracting Prime ImplicatesExtracting Prime Implicates

Extract implications, then prime implicatesEquivalence 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: Hypothesis Generation: Imperfect ConjecturesImperfect 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 FormationCycles of Theory Formation

How the individual techniques are employedConcept 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)Concept Driven Cycle (cut-down)

Invent Concept

EquivalenceNon Existence New Concept

Subsumptions

Implications

Reject

Concept Driven Cycle ContinuedConcept Driven Cycle Continued

Implications

Counterexample Proof

Prime Implicates

Counterexample Proof

Conjecture Driven CycleConjecture Driven CycleInvent Concept Reject

Near Equivalence Applicability Near Subsumption

Concept Barring

Counterex Barring

New/Old Concept

Equivalence

Implications

Monster Barring

New Concept

Counterex Barring

Concept Barring

New/Old Concept

Case Study: GroupsCase Study: Groups

Given: Group theory axioms


MACE model generator finds a model of size 1

Davis Putnam Method


Extracts concepts:Element, Multiplication, Identity, Inverse

HR Reads MACE’s Output


Invents the concept idempotent elements (a*a=a)

Match Production Rule


Makes Conjecture: a*a=a a is the identity element

Equivalence Finding


Otter proves this in less than a second

Resolution Theorem Proving


a*a = a a=identity, a=identity a*a=aEnd of cycle

Extracts Prime Implicates


Later: Invents the concept of triples of elements(a,b,c) for which a*b=c & b*a=c

Compose Production Rule


Invents concept of pairs (a,b) for which thereexists an element c such that: a*b=c & b*a=c

Exists Production Rule


Invents the concept of groups for which all pairsof elements have such a c: Abelian groups

Forall Production Rule


Makes the Conjecture: G is a group if and only if it is Abelian

Equivalence Finding


Otter fails to prove this conjecture

Sorry


MACE finds a counterexample:Dihedral Group of size 6 (non-Abelian)

Davis Putnam Method


Concept of Abelian groups allowed into theoryTheory recalculated in light of new object of interest

Assessment of Concepts

Case Study: GoldbachCase Study: Goldbach

Given: Integers 1 to 100, Concepts: Divisors, Addition


Invents: Even Numbers (divisible by 2)

Split Production Rule


Invents: Number of Divisors (tau function)

Size Production Rule


Invents: Prime numbers (2 divisors)

Split Production Rule


Half an hour later:Invents: Goldbach numbers (sum of 2 primes)

Compose Production Rule


Conjectures: Even numbers are Goldbach numbers(with one exception, the number 2)

Near Equivalence Finding


Forces: Concept of being the number 2

Counterexample Barring (Split)


Forces concept: Even numbers except 2

Counterexample Barring (Negate)


Conjectures: Even numbers except 2 are GoldbachNumbers (Goldbach’s Conjecture)

Subsumption Finding


Passes the conjecture to an inductive theorem prover?

Absolutely No Chance

Applications of TheoriesApplications 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 FormationApplications 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

ConclusionsConclusions

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 LevelWelcome 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 Theory Formation in Bioinformatics?Bioinformatics?

Can work with non-maths dataCan form near-conjecturesNeeds to relax notion of equalityMulti-agent approach definitely needed

Date post:	01-Jan-2016
Category:	Documents
Upload:	jane-obrien
View:	215 times
Download:	0 times

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

Documents