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# Automatic Generation - PowerPoint PPT Presentation

Automatic Generation of First Order Theorems. Simon Colton Universities of Edinburgh and York Funded by EPSRC grant GR/M98012 and the Calculemus Network. Overview of Talk. Automated Theory Formation Principles Implementation in the HR system Applications Application to Theorem Generation

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### Automatic Generation of First Order Theorems

Simon Colton

Universities of Edinburgh and York

Funded by EPSRC grant GR/M98012 and the Calculemus Network

• Automated Theory Formation

• Principles

• Implementation in the HR system

• Applications

• Application to Theorem Generation

• HR adds to the TPTP library

• HR becomes a MathWeb service

• Future Directions

• Scientific theories about a domain contain:

• Concepts, examples, definitions,

• hypotheses, explanations, etc.

• e.g. chemistry:acids

• Concepts: Acid, Base, Salt

• Hypothesis: Acid + Base  Salt + Water

• Experiments for plausibility/evidence

• Reaction pathways for explanation

• Concepts have examples and definitions

• Hypotheses are “conjectures”

• Explanations are proofs

• Conjectures become “theorems”

• e.g pure maths:group theory

• Concepts: cyclic groups, Abelian groups

• Conjecture: cyclic groups are Abelian

• Examples provide empirical evidence

• Proof for explanation

• user-supplied axioms + concepts

• Invent a new concept (machine learning)

• Look for conjectures empirically (d-mining)

• Prove the conjectures (theorem proving)

• Disprove the conjectures (model generation)

• Assess all concepts w.r.t. new concept

• Invent a new concept

• Build it from the most interesting old concepts

• Ten General Production Rules (PR)

• Work in all domains (math + non math)

• Build new concept from one (or two) old ones

• Example: Abelian groups

• Given: [G,a,b,c] : a*b=c

• Compose PR: [G,a,b,c] : a*b=c & b*a=c

• Exists PR: [G,a,b] :  c (a*b=c & b*a=c)

• Forall PR: [G] :  a b ( c (a*b=c & b*a=c))

• Theory formation step

• Attempt to invent a new concept

• Concept has same examples as previous one

• HR makes an equivalence conjecture

• Concept has no examples

• HR makes a non-existence conjecture

• HR can also make implication conjectures

• Examples of one concept are all examples of another concept

• HR relies on third party theorem provers

• Equivalence conjectures:

• Sets of implication conjectures

• From which prime implicates are extracted

• E.g.  a (a*a=a a=id)

• a*a=a  a=id, a=id  a*a=a

• HR uses the Otter theorem prover

• William McCune

• Only uses this for finite algebras

• Any conjectures which Otter can’t prove

• HR looks for a counterexample

• Using the MACE model generator

• Also written by William McCune

• Other possibilities: CAS, CSP

• Counterexamples are added to the theory

• Fewer similar non-theorems are made later

• New concepts from interesting old ones

• Concepts measured in terms of:

• Intrinsic values, e.g. complexity of definition

• Relational values, e.g. novelty of categorisation

• Concepts also assessed by conjectures

• Quality, quantity of conjectures involving conc.

• Conjectures also assessed

• Difficulty of proof (proof length from Otter)

• Surprisingness (of lhs and rhs definitions)

• Machine Learning

• Learn concept definitions: e.g. seq. ext.

• Theory for puzzle generation

• Constraint Satisfaction Problems

• Conjectures: induced constraints

• Concepts: implied constraints

• Mathematical Discovery

• Exploration of new domains

• Invention of Integer Sequences (NWN)

• Big project: using ATF to improve ATP

• Sub-project:

• Using AFT to assess ATP programs

• Compare first order ATP programs

• Using a large set of HR’s conjectures

• Facilitate comparison:

• Using MathWeb (Zimmer,Franke,…)

• Using SystemOnTPTP (Sutcliffe)

• Aim: add to the TPTP library

• 5882 test problems for first order provers

• Otter, SPASS, E, Vampire, etc.

• New provers are tested using TPTP

• HR produced 46,000 group conjectures

• In ten minutes.

• Around 200 of these were worthy of TPTP

• All provable by SPASS in 120 seconds

• 153 provable by only SPASS and E only

• 42 provable by only SPASS

• Otter and E could not prove this:

• x y

(( z (inv(z)=x & z*y=x) &

 u (x*u=y &  v (v*x=u & inv(v)=x)))

( a (inv(a)=x & a*y=x) &

• b (b*y=x & inv(b)=y)))

• MathWeb project in Saarbrücken

• E, Otter, SPASS, Vampire, Bliksem, …

• Idea: HR passes conjectures to MathWeb

• MathWeb translates conjectures using tptp2x

• MathWeb calls the provers

• Interface

• Via sockets at the moment

• Later by XMLRPC for better standardization

• By Zimmer, Colton and Franke

• Changes to HR

• Improvements in quantity of theorems

• Ability to write conjectures in TPTP format

• Changes to MathWeb

• Calling one prover after another

(1000s of times in a row)

• Quicker interaction with tptp2x

• Integration of the E system

• Possible experiments:

• Which one proves most of HR’s theorems 1st

• Compare the average times

• How many timeouts for each prover

• Watch this space for results…..

• Saturday: 9000 group theory theorems proved by SPASS, E & Otter, before a crash!

• Preliminary (unsurprising) result

• Average times: SPASS < E < Otter

• Try HR on more provers in MathWeb

• Vampire, Bliksem

• Offer HR as a new MathWeb service

• User says: “Give me 1,000 theorems which

SPASS and E take over 10 secs. to prove”

• Interface HR and model generators in MW

• Use MACE, etc. to disprove theorems

• Interface HR and CSP, CAS in MW

• Infinite Group theory with Bundy and Sorge

• Aim: Beat SPASS……

• SPASS is too good for HR in group theory

• 46,000 theorems and SPASS proved them all!

• Part two of my Calculemus project:

• With Jacques Calmet & Clemens Ballarin in Karlsruhe

• HR invents new domains

• Adds and constrains new operators for finite algebras

• “Grow” difficult theorems from prime implicates

• Colton: Express HR as a ML program

• Try domains other than maths

• Walsh: Integrate HR

• With every maths program ever written

• Bundy:

• Build an automated mathematician

• Mathweb:

• www.mathweb.org

• HR:

• www.dai.ed.ac.uk/~simonco/research/hr

• NumbersWithNames program:

• www.machine-creativity.com/programs/nwn

• Demonstration: Tomorrow @ 2pm? Room 208.