1. Artificial Intelligence More IJCAI 99 Ian Gent ipg@cs.st-and.ac.uk

2. Artificial Intelligence Three more papers from IJCAI Part I : SAT for Data Encryption Part II: Automated Discovery in Maths Part III: Expert level Bridge player

3. SAT for data encryption • “Using Walk-SAT and Rel-SAT for cryptographic key search” • Fabio Massacci, Univ. di Roma I “La Sapienza” • Proceedings IJCAI 99, pages 290-295 • Challenge papers section • Rel-SAT? A variant of Davis-Putnam with added “CBJ” • Walk-SAT? A successful incomplete SAT algorithm

4. Cryptography background • Plaintext P, Cyphertext C, Key K • (can encode each as sequence of bits) • Cryptographic algorithm is function E • C = EK(P) • If you don’t know K, it is meant to be hard to calculate • P = EK-1(C)

5. Data Encryption Standard • Most widely used encryption standard by banks • Predates more famous “public key” cryptography • DES encodes blocks of 64 bits at a time • Key is length 56 bits • Loop 16 times • break the plaintext in 2 • combine one half with the key using “clever function” f • XOR combination with the other half • swap the two parts • Security depends on the 16 iterations and on f

6. Aim of Paper • Answer question “Can we encode cryptographic key search as a SAT problem so that AI search techniques can solve it?” • Provide benchmarks for SAT research • help to find out which algorithms are best • failures and successes help to design new algorithms • Don’t expect to solve full DES • extensive research by special purpose methods • aim to study use of general purpose methods

7. DES as a SAT problem • Use encoding of DES into SAT • Each bit of C, P, K, is propositional variable • Operation of f is transformed into boolean form • CAD tools used separately to optimise this • Formulae corresponding to each step of DES • This would be huge and unwieldy, so • “clever optimisations” inc. some operations precomputed • Result is a SAT formula (P,K,C) • remember bits are variable, so this encodes the algorithm • not a specific plain text • set some bits (e.g. bits of C) for specific problem

8. Results • We can generate random keys, plaintext • unlimited supply of benchmark problems • problems should be hard, so good for testing algorithms • Results • Walk-SAT can solve 2 rounds of DES • Rel-SAT can solve 3 rounds of DES • compare specialist methods, solving up to 12 rounds • Have not shown SAT can effectively solve DES • Shown an application of SAT,and new challenges

9. Automated Discovery in Maths • “Automatic Concept Formation in Pure Mathematics” • Simon Colton, Alan Bundy • University of Edinburgh • Toby Walsh • University of Strathclyde (now York) • Proceedings of IJCAI-99, pages 786-791 • Machine Learning Section • Introduces the system HR • named for Hardy & Ramunajan, famous mathematicians • Discovered novel mathematical concepts

10. Concept Formation • HR uses a data table for concepts • A concept is a rule satisfied by all entries in the table • Start with some initial concepts • e.g. axioms of group theory • use logical representation of rules, I.e. “predicates” • Now we need to do two things • produce new concepts • identify some of the new ones as interesting • to avoid exponential explosion of dull concepts

11. Production rules • Use 8 production rules to generate new concepts • new table, and definition of new predicate • e.g. “match” production rule • finds rows where columns are equals • e.g. in group theory, general group A*B = C • match rule gives new concept “A*A = A” • Production rules can combine two old concepts • Claim that these 8 can produce interesting concepts • No claim that all interesting concepts covered

12. Heuristic Score of Concepts • Want to identify promising concepts • Parsimony • larger data tables are less parsimonious • Complexity • few production rules necessary means less complex • Novelty • novel concepts don’t already exist • Concepts and production rules can be scored • promising ones used

13. Results • Can use HR to build mathematical theories • This paper uses group theory • HR has introduced novel concepts into the handbook of integer sequences • e.g. Refactorable numbers • the number of factors of a number is itself a factor • e.g. 9 is refactorable • the 3 factors are 1, 3, 9. So 9 is refactorable

14. Expert level bridge play • “GIB: Steps towards an expert level bridge playing program” • Matthew Ginsberg, Oregon University • Proceedings IJCAI 99, pages 584-589 • Computer Game Playing section

15. Expert level bridge play • Aren’t games well attacked by AI? • Deep Blue, beat Kasparov • Chinook, World Man-Machine checkers champion • subject of a later lecture • Connect 4 solved by computer • Little progress on on 19x19 board • because of two types of game • Go, Oriental game huge branching rate • Card games like bridge • because of uncertain information, I.e. other players cards

16. What’s the problem? • If we knew location of all cards, no problem • << 52! Sequences of play, because of suit following • dramatically less than games like chess • one estimate is 10120 • We have imperfect information • estimates of quality of play have to be probabilistic • To date, computer bridge playing very weak • Slightly below average club player • “They would have to improve to be hopeless” • Bob Hamman, six time winner of Bermuda Bowl

17. What’s the solution? • Ginsberg implemented brilliantly simple idea • Pretend we do know the location of cards • by dealing them out at random • Find best play with this known position of cards • score initial move by expected score of hand • Repeat a number of times (e.g. 50, 100) • Pick out move which has best average score • This is called the “Monte Carlo” method • standard name in many areas where random data is generated to simulate real data

18. GIB • Ginsberg implemented (and sells) system called GIB • Best play in given deal found by standard methods • general methods subject of forthcoming lectures • Dealt at random consistent with existing knowledge • cards played to date, bidding history • Separate method for bidding (less successful) • GIB has some good results • won every match in 1998 World Computer Championship • lost to Zia Mahmoud & Michael Rosenberg by 6.4 IMPs • surprisingly close, though only over short match