Edinburgh and Calculemus

1. Edinburgh and Calculemus Simon Colton Universities of Edinburgh and York

2. Generating Conjectures About Maple Functions (Task 2.2) Simon Colton Universities of Edinburgh and York

3. Some Facts • Mathematicians don’t use ATP or ATF • But they do use CAS • CAS help them make discoveries • The HR program (ATF) makes discoveries • Interesting discoveries are hard to prove • ATP only proves easy things • But easy is the opposite of hard…

4. So, what to suggest? • HR is given maple library functions • HR makes many conjectures • Otter proves some of them • Throw these away – too easy • User chooses conjectures as axioms • Use Otter to prove more easy theorems

5. Calculemus Paid For… • Getting HR into Mathweb • Talk to both Otter and Maple there • Also application to TPTP library • Work done at Saarbrucken • Enabled HR to interface with Maple • Crash course in Maple programming • Work done at Karlsruhe

6. Example – Number Theory • Maple numtheory package, integers 1-100 • tau, sigma, isprime functions • HR exhausted up to complexity 6 • Compose, exists and split production rules • Otter given 10 seconds to prove theorems • Only ground instances supplied, e.g., isprime(3). • Are there any interesting relationships? • If so, can the user find them in HR’s output?

7. Results #1 • Takes two minutes to finish • 180 implicate conjectures produced • 82 of them proved by Otter all a,b (tau(a)=b & sigma(b)=a & isprime(b)  exists c (sigma(b)=c & tau(c)=b)) • Otter removes nearly half the conjs.

8. Results #2 • User goes through • Chooses 10 (true) conjectures • E.g., tau(a)=2  isprime(a) • These are added as axioms • All conjectures proved again • Yield reduces to just 29 conjectures • Possible for user to check through these

9. Results #3 • What’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/surprise • Still some uninteresting ones • But we’re working on that…

10. Future Work • Get all this working inside Mathweb • Make HR more interactive • Ask user on the fly whether a conj. is true • Work with Roy McCasland • EPSRC Visiting Research Fellow • Apply HR to Zariski Spaces

11. Three Possible Challenge Problems (Task 3.5) Simon Colton Universities of Edinburgh and York

12. Overview • Clarke’s Analytica problems • Jurgen Zimmer, Alan Bundy et Al • Lawvere’s Conceptual Mathematics • Alan Smaill • Sutcliffe’s TPTP library • Simon Colton, Jurgen Zimmer, Geoff Sutcliffe

13. Clarke’s Analytica Problems • Analytica • Theorem prover over Mathematica • Written by Ed. Clarke • Challenge problems • Analysis problems (e.g., continuity) • Analytica able to prove the • Jürgen has already talked about this

14. 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.

15. TPTP Problem Generation • TPT Problem library • Roughly 6000 problems • De facto standard for comparing ATPs • Maintained by Geoff Sutcliffe • Problems • Critics complain that people fine-tune • ATPs keep getting better • Vital to keep adding new problems

16. Challenge Problem • Automatically generate new TPTP probs • Use the HR program • Find conjectures empirically • Computation • Use ATPs • To test for theoremhood • To assess rating (number of ATPs which fail) • Deduction

17. Results #1 • Approach #1 – No interaction • 46,000 theorems produced by HR in 10 mins • Sent to Geoff Sutcliffe to assess • Results • All provable by SPASS in 120 seconds • 40 Not provable by E (rating 0.8) • 144 Not provable by vamp/gand/otter (0.6) • Challenge: beat SPASS

18. Results #2 • HR in MathWeb • Uses Otter, Spass, E and Bliksem • 12,000 conjectures tried • All but 70 were proved • For each prover • A theorem which only it could not prove • Beat SPASS 4 times!

19. Next Stage • Add more intelligence to choosing conjs • Add more provers to MathWeb • Offer HR as a mathematical service • User supplies axioms • Requires a certain number of theorems • Which are of a certain difficult • For certain provers

20. Edinburgh’s Proposed Industrial Applications Simon Colton Universities of Edinburgh and York

21. Application 1 on Platform Grant • Configuring the Grid • E-science (large datasets, distributed access) • Edinburgh is the national centre (£££££) • Web Description Service Language • Formally verify that QOS specs are met • E.g., reliability and redundancy • Very preliminary

22. Application 2 on Platform Grant • Bioinformatics • Biochemical 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