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
Functions
Axioms
Verification
User
Trash
HR
Maple
Proved
Otter
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
2
1
3
4
5
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))
Discovery into CAS does go…?
– If something is too hard, give it up. The
moral, my boy, is to never try anything