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