An introduction to combinatorial games · Pure combinatorial games - a de nition Berlekamp, Conway,...
Transcript of An introduction to combinatorial games · Pure combinatorial games - a de nition Berlekamp, Conway,...
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An introduction to combinatorial games
Eric Duchêne - Aline Parreau
LIRIS - Université Lyon 1 - CNRS
Ecole Jeunes Chercheurs en Informatique MathématiqueLyon - 23 janvier 2017
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Introduction
What
What
Why
Why
Who
Who
When
When
Games with:• 2 players playing alternately;• perfect information.
Chess Card games Othello Draughts
Tic Tac Toe Pachisi Go
Main question:
Who is winning and how?
→ Exact and approximate resolutions2/13
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Introduction
What
What
Why
Why
Who
Who
When
When
Maths CS
CombinatoricsNumber theory
Graphs
AILogic
SecurityComplexityWords
1901
Nim gameby Bouton
1939
SpragueGrundy
01
2 01
1976
ConwayTheory
2006
Miserequotients
1998
Deep Blue
2015
AlphaGo
Games and graphs
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Reference books
1976 1981
2007 2013
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Pure combinatorial games - a definition
Berlekamp, Conway, Guy (Winning Ways, 1981)• 2 players: Left and Right, that play alternately and cannot pass theirturn;
• Perfect information, no chance;• Finite number of moves, no draw, always a winner;• Winner determined according to the last move (no scoring)
Chess Card games Othello Draughts
Tic Tac Toe Pachisi Go
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Game tree
N
N R NLPN
RRLLRRRRPLRPLRPP
P P
. . . . . .
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Game tree
N
N R NLPN
RRLLRRRRPLRPLRPP
P P
. . . . . .
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Computing the outcome of Domineering
• Unknown complexity on a n ×m board.• When n and m are fixed:
[Bullcock’s website]
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Decomposing Domineering into a sum of games
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How to play on a big Domineering game ?
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How to play on a big Domineering game ?
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Values of positions of Wythoff
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The PSPACE class
Definition: a decision problem is PSPACE if it can be solve in polynomialspace by a Turing Machine.
The standard PSPACE-complete problem :
Quantified Boolean Formula
• Input : A quantified boolean formula:
Q1x1Q2x2...Qnxn, ϕ(x1, ..., xn)
with Qi ∈ {∃,∀}, xi ∈ {0, 1}• Output : Is the formula true ?
An equivalent problem : QBF-Game
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ED-Geography is PSPACE-complete [Schaeffer 1978]Reduction from QBF-Game :
(x1 ∨ x3) ∧ (x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x3)
x1 x1
x2 x2
x3 x3
C1 C2 C3
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Some nimbers sequencesArcKayles on a path0 0 1 1 2 0 3 1 1 0 3 3 2 2 4 0 5 2 2 3 3 0 1 1 3 0 2 1 1 0 4 5 2 74 0 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 2 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7...
Period 34 with some finite exceptions up to 52
James Bond Game
0 0 0 1 1 1 2 2 0 3 3 1 1 1 0 4...
228 known values, periodicity conjectured
0.106 Game
0 1 0 0 0 1 2 2 2 1 4 4 0 1 0 6...
Period 328226140474, with preperiod 465384263797.
Guy’s conjecture: all finite octal games have periodic nimber sequence.
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Conclusion
Current research questions ?• Graphs and Games: combinatorial games version on graphs• Metatheory: Misère, scoring games, loopy games• Link with other fields:
I Artificial Intelligence for generic gamesI Game versions of parameters of graphsI Logic, automata theory...
Merci !
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