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## GAMES AND COMPUTER SCIENCE

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**GAMES AND COMPUTER SCIENCE**Theoretical Models 1999 Peter van Emde Boas References available at: http://turing.wins.uva.nl/~peter/teaching/thmod99.html Most Papers will be made available in Library**Games in Computer Science**• Information & Uncertainty (Traub ea. - 80+) • Pebble Game (Register Allocation, Theory) • Tiling Game (Reduction Theory) • Alternating Computation Model and / or trees • Interactive Proofs • Arthur Merlin Games • Zero Knowledge**Game Theory**• Theory of Strategic Interaction • Attributes • Discrete vs. Continuous • Cooperative vs. Non-Cooperative • Full Information vs. Incomplete Information (Knowledge Theory)**Discrete / Continuous**Combinatorial Analysis Backward Induction Number Theory (Conway Guy Berlekamp) Equilibria theory (Nash) Stochasitic Features Optimization Other names of importance: Von Neumann & Morgenstern Aumann Shapley Harsanyi**OTHER ASPECTS**• Single player - no choices • Single player - random moves • Single player - choices : Solitaire • Two players - choices • Two players - choices and random moves • Two players - concurrent moves**Computer Science**• Computation Theory • Complexity Theory • Machine Models • Algorithms • Knowledge Theory • Information Theory**COMPUTATION**• Deterministic • Nondeterministic • Probabilistic • Alternating • Interactive protocols • Concurrency**COMPUTATION**• Notion of Configurations: Nodes • Notion of Transitions: Edges • Non-uniqueness of transition: Out-degree > 1 • Initial Configuration : Root • Terminal Configuration : Leaf • Computation : Branch Tree • Acceptance Condition: Property of trees**Introducing the Opponents**© Games Workshop © Games Workshop URGAT THORGRIM**A Game**Starting with 15 matches players alternatively take 1, 2 or 3 matches away until none remain. The player ending up with an odd number of matches wins the game © Donald Duck 1999 # 35**Questions about this Game**• What if the number of matches is even? • Can any of the two players force a win by clever playing? • How does the winner depend on the number of matches • Is this dependency periodic? If so WHY?**Games as Recognizers**• Construct a map G : S* --> Games (simply computable; Poly-time, Logspace or NC, ….) • Set recognized := {w | G(w) is won (by the first player) } • How does this relate to conventional ways of recognizing languages ?**Games as Recognizers**• Construct a map G : S* --> Games (simply computable; Poly-time, Logspace or NC, ….) • G(w) is guaranteed to be proper • Set recognized := {w | G(w) is won (by the first player) } • Properness conditions frequently involve probabilistic aspects**Game Trees**Thorgrim’s turn Terminal node: Urgat looses Urgat’s turn Terminal node: Thorgrim looses Standard Interpretation: Player unable to move looses the game Root**Game Trees**Thorgrim’s turn T U Terminal node: Urgat’s turn T T U U T Terminal node: Free Interpretation: Winner explicitly designated at terminal node Root**Game Trees**Thorgrim’s turn 2/0 -1/ 4 Terminal node: Urgat’s turn 3/1 1/-1 -3/ 2 1/ 4 5/-7 Terminal node: Non Zero-Sum Game: Payoffs explicitly designated at terminal node Root**Game Trees**SUB-GAME Thorgrim’s turn T U Terminal node: Urgat’s turn T T U U T Terminal node: Free Interpretation: Winner explicitly designated at terminal node Root**Backward Induction**Thorgrim’s turn T U Terminal node: Urgat’s turn T T T U U T T Terminal node: U U Free Interpretation: Winner explicitly designated at terminal node U Root**Backward Induction**Thorgrim’s turn 2/0 -1/ 4 Terminal node: Urgat’s turn 3/1 2/0 1/-1 -3/ 2 1/ 4 5/-7 3/1 Terminal node: -3/ 2 1/ 4 Non Zero-Sum Game: Payoffs explicitly designated at terminal node 1/ 4 Root**Backward Induction**T U 2/0 -1/ 4 T 3/1 T 2/0 U T U T 1/-1 -3/ 2 1/ 4 5/-7 T 3/1 U U -3/ 2 1/ 4 U 1/ 4 At terminal nodes: Pay-off as explicitly given At Thorgrim’s nodes: Pay-off inherited from Thorgrim’s optimal choice At Urgat’s nodes: Pay-off inherited from Urgat’s optimal choice For strictly competetive games this is the Max-Min rule**Analysis of the DD game**Extension used: Thorgrim wins if he has an odd number when the game terminates. This allows for even n . Relevant feature: parity of number of matches collected so far (not the number itself!) Four types of configurations remain: T/E : Thorgrim has to play and has an even number T/O : Thorgrim has to play and has an odd number U/E : Urgat plays, while Thorgrim has an even number U/O : Urgat plays, while Thorgrim has an odd number**Backward Induction Table**nU / EU / OT / ET / O 18UUT / 1T / 2 17 U T T / 1 U 16 U T U T / 3 15 U U T / 2T / 3 14 U U T / 2T / 1 13 TU U T / 1 12 TU T / 3 U 11 U U T / 3 T / 2 10UUT / 1T / 2 9 U T T / 1 U 8 U T U T / 3 7 U U T / 2T / 3 6 U U T / 2T / 1 5 TU U T / 1 4 TU T / 3 U 3 U U T / 3 T / 2 2UUT / 1T / 2 1 U T T / 1 U 0 U T U T**What is the Strategy?**• Play to number 0 or 1 (mod4) • Switch your parity on every turn • Start right: to even if n mod 8 {5,6,7,0} and to odd if n mod 8 {1,2,3,4} • Question: explain the correctness of this strategy, otherwise than by inspecting the table.....**Alternating Computation**Configuration Type Existential Universal Negating + - + Accepting - - + + - + - Rejecting Computation Tree**Alternating Computation**+ Configuration Type + - - + + Existential + - Universal + + + - - - Negating - + - + + - + Accepting - - + + - + - Rejecting Evaluation Full Computation Tree This Tree Accepts**Alternating Computation**Infinite Branches ? Requires third quality :Indeterminatenodes Universalnode isindeterminateiff it has no rejecting son and at least oneindeterminateson Existentialnode isindeterminateiff it has noaccepting son and at least oneindeterminateone Negatingnode isindeterminateiff its son is**Alternating Computation**Infinite Branches ? Universalnode isacceptingiff it has no rejecting son and noindeterminateson (all sons are accepting) Existentialnode isacceptingiff it has oneaccepting Son; indeterminateand rejecting sons don’t matter Negatingnode isacceptingiff its son is rejecting Requires RecursiveEvaluation of computation tree !**RECURSIVE EVALUATION**+ - ^ The proper way of Recursive evaluation ??? + - ^ ^ + - + - ^ + - ^ …. …. …. Indeterminate : ^**RECURSIVE EVALUATION**+ - ^ Recursive evaluation == Solving LEAST FIXEDPOINT EQUATION ! + - ^ + - Partial order ≤ of definedness Extends to functions defined on the tree: F ≤ G iff "x[F(x) ≤ G(x)] ^ + - ^ + - ^ + - ^ …. …. …. OK NOK**The Knaster Tarski Theorem**SET U with partial order ≤ and least element ^ Countable chains have least upper bounds DOMAIN := x0≤ x1 ≤ x2 ≤ ….. ≤ xn ≤ xn+1 ≤ …. ---> xw=:i xi "i[xi ≤ xw] and "i[xi ≤ y] ==> xw ≤ y FUNCTION F which is: MONOTONE: x ≤ y ==> F(x) ≤ F(y) CONTINUOUS: F( i xi ) = iF(xi) OPERATOR :=**The Knaster Tarski Theorem**THEOREM: If F is an operator defined over domain U then the equation X = F( X ) has a least solution W . This solution is obtained as the limit of the sequence of iterates: ^ ≤ F( ^ ) ≤ F(F( ^ )) ≤ …. W = iFi ( ^ ) APPLICATION: U := domain of evaluations of tree F := single application of recursive rule**Back to Alternation**• For an accepting tree there exists a witness subtree for acceptance (and similar for rejection) • Witness subtree contains a single accepting son for every accepting node, and a single rejecting son for every rejecting node • A witness subtree is finite, even when the tree itself is infinite! • Infinite branches are irrelevant!**Negating Nodes ?**• Create for every node its dual node which yields the “same” transitions • Dual of accepting node is rejecting • Dual of rejecting node is accepting • Dual of universal node is existential • Dual of existential node is universal • Dual of Dual is identity • Replace every negating node by an existential one, dualizing the entire subtree below it (think de Morgan!)**Eliminating Negating Nodes**+ + + - - + - - + + + - + + - - + + + - - - + + + - - - - + - + + - - + - + + + - - + + - + - - + - + - + - - Dualized nodes -