The Homer System

Simon Colton – Imperial College, London

Sophie Huczynska – University of Edinburgh

– Marge! Look at all this great stuff I found at the Marina. It was just sitting in some guy's boat!

HR, Otter and Maple

• HR – Descriptive Machine Learning

• Otter – First Order Theorem Prover

• Maple – Computer Algebra System

• Encapsulated

• Reasoner

• Homer is just a FrontEnd– Specialised to work in number theory (graph theory soon)

Automated Conjecture Making

• Purpose of Homer:

– User supplies some Maple functions in a file

– And optionally some axioms about the functions

– Homer makes conjectures about the functions

• Based on empirical evidence (induction)

• New Spin on ATP:

– If Otter can prove a conjecture from first principles, then it’s unlikely to be of interest (sorry, Bill…)

• Hence Homer discards any theorem Otter proves

Interplay of Systems

User HR

MapleOtter

Trash

Functions

Axioms

Verification

Calculation

requestsCalculation

results

Proved

Not proved

in 5 secs

Conje

ctur

e

The HR Program in Two Slides

• Descriptive induction:– Finds things you didn’t know you were looking for

• Starts with background information– Concepts/examples/axioms

• Forms concepts– Uses 12 production rules to make new from old

• Makes conjectures– By noticing empirical relationships in the concept examples

and generalising the result

The HR Program in Two Slides

• Generic Concept Production Rules:

– Compose, disjunct, exists, forall,

– Match, size, split, equal, negate

• Maths Production Rules:

– Arithmetic (+,*,dirichlet), subalgebra, embed_graph

• Heuristic Search

– Build new concepts from the “best” old ones

– Measure interestingness of concepts

• Using an evaluation function over 20+ measures

Homer Design Decisions

• Make the interface as simple as possible– HR has 300+ GUI objects on-screen

– HOMER has only 10 things to click on

– 5 simple questions at start

– Then, user only responds to conjectures supplied

• Possible responses:– One of a set of alternatives is true

– All false/don’t know/give a generalisation

– Supply a counterexample/search for a counter

– Stop asking now

Five Simple Questions

1

3

4

5

2

Improving Conjecture Quality• Problem with old versions of HR:

– About 90% of conjectures were dull

• Repetition of similar results:– Give Otter each theorem as another axiom

• Drastically reduces the repetition (discard any proved by Otter)

• Easy to prove– Otter (and HR) finds tautologies, and theorems which follow

easily from axioms

• Low applicability– Example: isprime(X) & even(X) isodd(sigma(X))

– Unsolved conjectures are supplied with examples

– Otter is given facts like sigma(2) = 3

An Assessment• Sophie Huczynska

– Number theorist from Glasgow/Edinburgh

– Never used HR/Homer

• Four hour session with Homer using standard functions from number theory– isprime, isodd, iseven, issquare, sigma, tau,

– Also used the phi function (new to Homer/HR)• phi(n) = number of integers less than and coprime to n

– Numbers 1 to 50, no axioms supplied

– HR produced 5000+ conjectures

– Homer only showed Sophie 59

Results from Session

• 38 conjectures proved (4 shown false) by Sophie

– Became more difficult as time progressed

– No results deemed to be dull (tautological)

– Results following from axiom definitions came at start

• 17 conjectures remain open

– 8 out of final 10 are still open (likely to be false)

• Various (now implemented) recommendations

– About Homer and about HR (e.g, Dirichlet convolution)

Illustrative (proved) Conjectures

• 4 Conjectures were said to be “number theoretically interesting” by Sophie

• Examples from Session:

– iseven(phi(n)) n > 2

– issquare(phi(n)) iseven(tau(n))

• “Cute”: requires considering the contrapositive (see paper)

• Old (nice) examples from Homer:

– issquare(n) isodd(sigma(n)) [4th year imperial student]

– isprime(sigma(n)) isprime(tau(n))

– If something is too hard, give it up. The moral, my boy, is to never try anything

Discovery into CAS does go…?