The HOMER System for The HOMER System for Discovery in Number TheoryDiscovery in Number Theory

Simon Colton

Imperial College, London

MotivationMotivation

Newell and Simon:– “Within 10 years (of 1957) a computer will discover

and prove an important mathematical theorem”

Alan Bundy:– “His DReaM = Discovery and Reasoning in Maths”

Bob Kowalski:– “It is more important to discover the most important

theorems than to prove unimportant ones”

An Overview of HOMERAn Overview of HOMER

Aim: – To make conjectures in number theory

User supplies background knowledge:– Number types: primes, squares, etc.– Functions: tau, sigma, phi, etc.

HOMER makes conjectures about them– Conjectures are meant to be interesting– User is involved at certain stages

Otter, Maple, HR and the UserOtter, Maple, HR and the User

OTTER

HR

MAPLEHOMERHOMER

First Order Resolution Prover

Computer Algebra System

Automated TheoryFormation System

USER

Roles of Individual ComponentsRoles of Individual Components

User: Provide Guidance

– Supply background information– Verify/Falsify some conjectures produced

Maple: Generating Empirical Evidence

– Perform calculations Otter: Increase Interestingness via Pruning

– Prove theorems from first principles HR: Make Discoveries

– Invent concepts related to background– Make conjectures relating concepts

Three PhasesThree Phases

Initialisation Phase– User/Maple generate background info• In a format suitable for HR

Discovery Cycle– HR/Maple generate conjectures• Using automated theory formation

Proving & Pruning Cycle– User/Otter prove theorems• HOMER discards any that Otter can prove

Initialisation PhaseInitialisation Phase

User supplies:– Maple code for concepts of interest• Mathematicians don’t want to learn a language

– Integers to work with, e,g,. 1 – 250 & othersHOMER translates to HR’s input– Calls Maple using the code and numbers– Generates pairs of input/output–Writes this in HR’s format• Similar (in this case) to that for ILP systems

ExampleExample

Maple code:

isrefactorable_ := proc(num)

local output := false;

if (member(tau(num),divisors(num))=true) then output := true; fi;

output;

end:

HR input:

h01

isrefactorable_(n)

ascii:isrefactorable_(@n@)

isrefactorable_(1). isrefactorable_(2). isrefactorable_(8).

isrefactorable_(9). isrefactorable_(12). isrefactorable_(18).

Discovery Cycle: Discovery Cycle: 1. Concept Formation1. Concept Formation

HR Forms concepts using production rules – Restricted to 3 out of 10 in HOMER– HR calls Maple again if it needs another calculation

Compose production rule– Composes functions & predicates using conjunction

• Example1: tau(sigma(n)), Example2: odd primes

Split production rule– Performs instantiation of variables to fixed constants

• Example: prime numbers (tau(n) = 2)

Exists production rule– Introduces existential quantification

• Example: exists k s.t. tau(n) = k & sigma(n) = k,

Example InventionExample Invention

Discovery Cycle:Discovery Cycle:2. Conjecture Making2. Conjecture Making Concepts are formed with– Both a definition and a set of examples

Whenever a new concept is invented HR checks whether:

1. It has no examples• Makes a non-existence conjecture

2. It has exactly same examples as another• Makes an equivalence conjecture

3. The examples are subsumed by another• Makes an implication conjecture

Example ConjectureExample Conjecture

HR invents the concept of:– Integers where number of divisors is prime

Later, it in invents the concept of:– Integers where the sum of divisors is prime

Notices that the examples of latter– Are all examples of the former

Makes the conjecture that– isprime(sigma(n)) isprime(tau(n))

Proving and Pruning stageProving and Pruning stage

The user can supply some axioms as background knowledge, or nothing at all

Any conjecture the user proves– Becomes an axiom of the theory

Anything Otter proves using the axioms– Is discarded (hidden from the user)– Because it will be “obviously” true to the user

Proving and Pruning CycleProving and Pruning Cycle1. The User steps in1. The User steps in

For each conjecture, the user is given– The conjecture statement– Up to two alternative conjectures• From which the original follows

The user can choose to:– Say the conjecture (or alternative) is true– Supply a more general theorem– Supply a counterexample to the conjecture– Use HOMER to search for a counterexample

Proving and Pruning CycleProving and Pruning Cycle2. Otter is employed2. Otter is employed

A set of axioms is built up Every new conjecture is passed to Otter

– Along with the axioms – And ground instances of back. concepts

• E.g. isprime(2), tau(4)=3, sigma(5)=6, etc.

Any conjecture that Otter proves– Follows from the axioms (which user knows)– So it will be very easy for the user to prove it– And it won’t say much more than the axioms– So HOMER doesn’t show it to the user

After a while, Otter begins to prune many conjs

An Example (Pruned) TheoremAn Example (Pruned) Theorem

Tau(x) = number of divisors of x Sigma(x) = sum of divisors of x This conjecture looks interesting:– isprime(n) isodd(sigma(tau(n)))

But, this is actually really dull. Why? We know the axiom isprime(n) tau(n)=2– And ground instances: sigma(2)=3, isodd(3).

So, the conjecture is obviously true Otter proves this with no problem– And so it is rightly discarded (user never sees it)

Design Considerations in HOMERDesign Considerations in HOMER Background information given in a format– Which is well known to users

Very few choices for users– Mathematicians are notoriously uncomfortable

with mathematics software• So, there are many defaults in HR and Otter which are set

Very simple user interface– Asked questions they can’t ignore

Given the bare minimum of details– Only the conjectures, nothing else

More Design ConsiderationsMore Design Considerations

Absolute minimal number of dull results– In PhD version of HR• 90% of conjectures were dull• Mostly tautologies (true regardless of semantics)

User can stop interacting at any time– HOMER will continue to find results for them

Use of HTML format – For improved presentation

Three Illustrative ResultsThree Illustrative Results

Refactorable numbers– (number of divisors is itself a divisor)– Odd refactorables are squares– HR’s results contributed to a maths paper

Prime sum of divisors implies prime number of divisors– “Level of an exercise at the end of a chapter”

Square numbers have odd sum of divisors – Not as obvious as it might seem

A Session with A Session with a Number Theorista Number Theorist

Sophie Huczynska (Math PostDoc)– Completely new to HOMER

Session exploring totient function– phi(n) = #integers less than and coprime to n– Completely new to HOMER

Background information– Integers: 1 to 50– Functions: phi, sigma, tau– Number types: odd, even, prime, square

~4 hours (including proving time)– HOMER running for only a fraction of this

Session ResultsSession Results

Presented 59 out of thousands of conjs– 38 have been subsequently proved, – 4 were shown to be false– 17 remain open

Interestingness– None were tautologies– 7 followed trivially from definitions (mostly at start)– Difficulty decreased as session continued

• 2 out of first 30 settled• 8 out of final 10 still open

– Four which are “number theoretically interesting”

Two “Number Theoretically Two “Number Theoretically Interesting” ResultsInteresting” Results(a) phi(n) is odd phi(n) = 1(b) phi(n) = 1 n=1 or n=2Combined to give:– phi(n) is even n > 2 [is this obvious?]

phi(n) is square tau(n) is even– Described as “cute”– Proof required theorem viewed correctly• i.e., by considering the contrapositive

An Evaluation of PotentialAn Evaluation of Potential Sophie’s Evaluation– After more testing with HOMER

Not ready yet for research mathematics– Limited nature of the background concepts

• Nothing likely to surprise the user

– Complication in theorems is not a good thing • Prefer simpler results about complicated functions

Potential Applications– Recreational Mathematics

• User invents something new, HOMER performs exploration

– Setting of tutorial questions• To give students a feel for number theoretic functions

Recommendations for Recommendations for Future Versions Future Versions

Combine with NumbersWithNames– Program which uses Encyclopedia of integer

sequences to make conjectures• Problem: flat file (sequences cannot be extended)

–Would possibly make surprising conjecturesAllow user defined production rules (PRs)–Which are more domain specific• (HR’s PRs work just as well in bioinformatics…)

– E.g., Dirichlet convolution• Implemented yesterday!

A Demonstration…A Demonstration…

Re-invention of Refactorable Results– A number is refactorable if the number of divisors

is itself a divisor– Re-invented by HR in 1998 (original 1990)– 1, 2, 8, 9,…

Also: Anti-tau numbers– Invented by Josh Zelinsky

• High school student inspired by refactorables

– They are coprime with the number of divisors– 1, 3, 4, 5, 7, 11,…