Edinburgh and Calculemus

Simon ColtonUniversities of Edinburgh and

York

Generating Conjectures About Maple Functions

(Task 2.2)

Simon ColtonUniversities of Edinburgh and

York

Some FactsMathematicians don’t use ATP or ATF

But they do use CASCAS help them make discoveriesThe HR program (ATF) makes discoveries

Interesting discoveries are hard to prove

ATP only proves easy thingsBut easy is the opposite of hard…

So, what to suggest?HR is given maple library functions

HR makes many conjectures

Otter proves some of themThrow these away – too easy

User chooses conjectures as axioms

Use Otter to prove more easy theorems

Calculemus Paid For…Getting HR into Mathweb

Talk to both Otter and Maple thereAlso application to TPTP libraryWork done at Saarbrucken

Enabled HR to interface with MapleCrash course in Maple programmingWork done at Karlsruhe

Example – Number TheoryMaple numtheory package, integers 1-100

tau, sigma, isprime functions

HR exhausted up to complexity 6Compose, exists and split production rules

Otter given 10 seconds to prove theoremsOnly ground instances supplied, e.g., isprime(3).

Are there any interesting relationships?If so, can the user find them in HR’s output?

Results #1Takes two minutes to finish180 implicate conjectures produced

82 of them proved by Otterall a,b (tau(a)=b & sigma(b)=a &

isprime(b) exists c (sigma(b)=c & tau(c)=b))

Otter removes nearly half the conjs.

Results #2User goes through

Chooses 10 (true) conjecturesE.g., tau(a)=2 isprime(a)These are added as axiomsAll conjectures proved again

Yield reduces to just 29 conjecturesPossible for user to check through these

Results #3What’s left?

isprime(tau(a)) isprime(tau(tau(a))isprime(sigma(a)) isprime(tau(a)isprime(tau(a)) tau(sigma(tau(a)))=tau(a)

User can order by applic/surpriseStill some uninteresting ones

But we’re working on that…

Future WorkGet all this working inside Mathweb

Make HR more interactiveAsk user on the fly whether a conj. is true

Work with Roy McCaslandEPSRC Visiting Research FellowApply HR to Zariski Spaces

Three Possible Challenge Problems (Task 3.5)

Simon ColtonUniversities of Edinburgh and

York

OverviewClarke’s Analytica problems

Jurgen Zimmer, Alan Bundy et Al

Lawvere’s Conceptual MathematicsAlan Smaill

Sutcliffe’s TPTP librarySimon Colton, Jurgen Zimmer, Geoff Sutcliffe

Clarke’s Analytica Problems

AnalyticaTheorem prover over MathematicaWritten by Ed. Clarke

Challenge problemsAnalysis problems (e.g., continuity)Analytica able to prove the

Jürgen has already talked about this

Lawvere’s Conceptual Maths

This book, while being an introductory text, gives insight into the search for solutions and abstraction of ideas that goes on in mathematical practice, in a way unusual in mathematical texts.  It also gives many interesting abstract characterisations of mathematical concepts. Some examples: Brouwer's fixed point theorems for a real interval, and for the closed disc, are derivable from a few intuitive properties of continuity; similarly for Banach's fixed-point theorem for contraction maps.  For another example, the generality of this approach allows a characterisation of the notion of "diagonal“ argument, which is then applicable in many versions (e.g., Cantor's diagonal argument, or Goedel's diagonalisation in his theorem about incompleteness of formal systems.  The challenge is to formalise these ideas in such a way as to allow exploration of the search spaces involved,while retaining the intuitive feel for the concepts involved.

TPTP Problem GenerationTPT Problem library

Roughly 6000 problemsDe facto standard for comparing ATPsMaintained by Geoff Sutcliffe

ProblemsCritics complain that people fine-tuneATPs keep getting better

Vital to keep adding new problems

Challenge ProblemAutomatically generate new TPTP probsUse the HR program

Find conjectures empiricallyComputation

Use ATPsTo test for theoremhoodTo assess rating (number of ATPs which fail)Deduction

Results #1Approach #1 – No interaction

46,000 theorems produced by HR in 10 minsSent to Geoff Sutcliffe to assess

ResultsAll provable by SPASS in 120 seconds40 Not provable by E (rating 0.8)144 Not provable by vamp/gand/otter (0.6)Challenge: beat SPASS

Results #2HR in MathWeb

Uses Otter, Spass, E and Bliksem12,000 conjectures triedAll but 70 were proved

For each proverA theorem which only it could not prove

Beat SPASS 4 times!

Next StageAdd more intelligence to choosing conjsAdd more provers to MathWebOffer HR as a mathematical service

User supplies axiomsRequires a certain number of theoremsWhich are of a certain difficultFor certain provers

Edinburgh’s Proposed Industrial Applications

Simon ColtonUniversities of Edinburgh and

York

Application 1 on Platform Grant

Configuring the GridE-science (large datasets, distributed access)Edinburgh is the national centre (£££££)

Web Description Service LanguageFormally verify that QOS specs are metE.g., reliability and redundancy

Very preliminary

Application 2 on Platform Grant

BioinformaticsBiochemical structures as logical expr.Composition as inference processes

Infer compound properties from comp.Proof planning to synthesise structures

Critics to analyse failed attempts

Again, very preliminary