Making Conjectures About Maple Functions

Simon Colton

Universities of Edinburgh & York

Computation, Invention & Deduction

• Perform calculations (CAS)

• Find conjectures (ML)

• Prove theorems (ATP)

• Paul Zeitz, Hungarian maths contest– k = n(n+1)(n+2)(n+3), k is never a square– “Plug and chug” with n=1, n=2, n=3, n=4, ….– Gives: 24, 120, 360, 840, ….– Always square minus 1

General Approach

• Ideal world:– ATP proves a theorem found by ML about

CAS functions

• Problem with this:– CAS functions are too complex for ATP

• Positive spin: ATP proves from 1st principles– Proven conjectures are likely to be less interesting– Use ATP to discard dull results– Find conjectures, not theorems for CAS user

Systems Used

• Maple– Well known CAS system

• HR– Not so well known ATF system (ML)

• Otter– Well known ATP system

• Domain– Number theory

The HR System – Concepts

• Automated theory formation– Invent concepts, make conjecture, prove theorems– Disprove non-theorems and present results

• Concept formation via 10 production rules– Builds new concepts from old ones– Example on next slide

• Heuristic search– Interestingness measures (e.g., complexity)– Limits on the concepts formed (e.g., arity, comp)

Concept Formation

[a] exists b (sigma(a)=b & isprime(b))

[a] : isprime(a) [a,b] : sigma(a)=b

[a,b] : sigma(a)=b & isprime(b)

compose

exists

(Complexity = 4)

The HR System - Conjectures

• HR Makes conjectures empirically– Two concepts are equivalence

– A concept is inconsistent with the axioms

– Examples of one concept all examples of another

• Extracts simpler concepts from these– A & B & C <=> D & E

– A & B & C => D, D & E => A, etc.

• Finds prime implicates: A & B => D

• Also makes applicability conjectures– tau(x) = x only true (empirically) for x = 1 and x = 2

Presentation of Results

• HR has some limited theorem proving– To discard the very dull conjectures:

– f(a) & g(a) g(a) & f(a), f(a)=b & f(a)=c [bc]

– But we didn’t want to re-invent the wheel

• HR can re-write definitions b (tau(a)=b & isprime(b)) becomes isprime(tau(a))

• HR can also sort the conjectures (interestingness)– Applicability = number of entities with non-empty examples

• Conjs about primes score 10/30 for applic (10 primes between 1 & 30)

– Surprisingness (see next slide)

Measuring Surprisingness

Surprisingness = 6

HR Using Maple

• User chooses the Maple functions of interest

• HR calls Maple to supply initial data – User specifies initial input, say, numbers 1 to 10– E.g., tau(10) = 4, sigma(6) = 12, etc.

• HR also calls Maple during theory formation– HR invents concept: tau(sigma(n))– Needs value for: tau(sigma(10)) = tau(18) = 6– Also to find counterexamples to conjectures

• Integration via files (sorry Jürgen, MWeb soon)

HR Using Otter

• Problem with HR – too many conjectures (thesis:6%)• Normally HR supplies Otter with only:

– Axioms of domain and the conjecture statement

• Problematic example– All a (a=1 or a=2 => tau(a) = a)– True but dull– HR supplies tau(1) = 1 and tau(2) = 2 to Otter

• User can identify “axioms” to give to Otter– E.g., isprime(a) tau(a) = 2

• User specifies which conjs to add as axioms– HR re-proves (and discards) many previous results

Experiment• Aims:

– (i) show HR works with Maple– (ii) Show pruning of conjectures works

• Three Maple functions chosen:– tau(n), sigma(n), isprime(n)

• HR given number 1 only, but access to 2-30• Breadth first search ran to completion

– Complexity limit of 6 and some arity constraints

• Then the user steps in– To specify obvious results to add as axioms

• Any proven results ignored

Results• 378 theory formation steps, approx. 2 mins• HR called Maple 120 times

– E.g., for 195 = sigma(72) = sigma(sigma(30))– Numbers 2, 3, 4, 5, 6, 9 and 16 introduced

• Produced 137 implicate conjectures– 43 already proven by Otter (goodbye)– E.g: all a ((sigma(a)=1 & sigma(1)=a) => (a=1))

• We looked through the remaining 94– Added 9 from the first 10 (ordered by complex.)– E.g: [3] all a ((isprime(a)) => (tau(a)=2))– Added another 3 which were “instantiations”

• After re-proving 94 became just 22

Results Continued

• We ordered the 22 by (applic + surp)/2

• Top one was: – isprime(sigma(a)) => isprime(tau(a))– So interesting, we proved it (generalised)– Then we added it as an “axiom”

• Reduced the results down to just 10:tau(tau(a))=a => tau(sigma(sigma(a)))=sigma(a)

tau(tau(a))=a => tau(sigma(a))=a [should’ve gone]

• Are these results interesting ???

Conclusions

• Aims: – HR works with Maple

• No problem with the interface

– HR uses Otter to prune dull conjectures • 137 => 94 => 82 => 22 => 17 => 16 => 10

• Bonus: interesting conjecture(s)

• Question: is this of use to CAS users?

Further Work

• Improved pruning of conjectures– Use Otter’s Knuth Bendix completion (McCasland)– HR now has skolemised repn (discards exists conjs)

• Get ATF embedded in CAS (any offers?)

• Apply HR to more discovery tasks– Roy McCasland and Zariski spaces

• Last line of my thesis:“… if this technology can be embedded into computer algebra systems, we believe

theory formation programs will one day be important tools for mathematicians.”

Acknowledgments

• This work has been supported by– EPSRC grant GR/M98012– EU IHP grant:

• Calculemus HPRN-CT-2000-00102

• This work inspired by collaboration with:– Jacques Calmet and Clemens Ballarin– Calculemus YVR at Karlsruhe