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Automatic Generation of First Order Theorems

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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

Overview of Talk

- 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

- 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

Theories in Pure Mathematics

- 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

HR: Theory Formation Cycle

- Start with background knowledge
- 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

Inventing New 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))

Making Conjectures

- 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

Proving Theorems

- 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

Disproving Non-Theorems

- 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

Assessing Interestingness

- 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)

Applications of ATF

- Machine Learning
- Learn concept definitions: e.g. seq. ext.
- Theory for prediction tasks
- Theory for puzzle generation

- Constraint Satisfaction Problems
- Conjectures: induced constraints
- Concepts: implied constraints

- Mathematical Discovery
- Exploration of new domains
- Invention of Integer Sequences (NWN)

Application to ATP

- 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)

First Attempt

- 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

Example Theorem

- 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)))
[about pairs of identity elements]

Interface of HR into MathWeb

- MathWeb project in Saarbrücken
- Has access to many first order ATP progs.
- 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

Additional Implementation

- 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

- Calling one prover after another

Experiments

- 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

Future Work: MathWeb #1

- 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”

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

Future Work: MathWeb #2

- 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

Future Work: HR Project

- 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

Web Pages

- 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.

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