Artificial IntelligenceIntelligence

Ian Gentipg@cs.st-and.ac.uk

More IJCAI 99

Artificial IntelligenceIntelligence

Part I : SAT for Data EncryptionPart II: Automated Discovery in MathsPart III: Expert level Bridge player

Three more papers from IJCAI

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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 290-295Challenge papers section

Rel-SAT? A variant of Davis-Putnam with added “CBJ” Walk-SAT? A successful incomplete SAT algorithm

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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)

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Data Encryption Standard

Most widely used encryption standard by banksPredates more famous “public key” cryptographyDES encodes blocks of 64 bits at a timeKey is length 56 bitsLoop 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

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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

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DES as a SAT problem

Use encoding of DES into SATEach bit of C, P, K, is propositional variableOperation 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

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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 DESShown an application of SAT,and new challenges

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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

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Concept Formation

HR uses a data table for concepts A concept is a rule satisfied by all entries in the tableStart 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

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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 equalse.g. in group theory, general group A*B = Cmatch rule gives new concept “A*A = A”

Production rules can combine two old conceptsClaim that these 8 can produce interesting conceptsNo claim that all interesting concepts covered

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Heuristic Score of Concepts

Want to identify promising conceptsParsimony

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

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Results

Can use HR to build mathematical theoriesThis paper uses group theory HR has introduced novel concepts into the

handbook of integer sequencese.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

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Expert level bridge play

“GIB: Steps towards an expert level bridge playing program”

Matthew Ginsberg, Oregon UniversityProceedings IJCAI 99, pages 584-589Computer Game Playing section

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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

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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

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What’s the solution?

Ginsberg implemented brilliantly simple ideaPretend 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 scoreThis is called the “Monte Carlo” method

standard name in many areas where random data is generated to simulate real data

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GIB

Ginsberg implemented (and sells) system called GIBBest 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