Peter van Emde Boas: Games and Computer Science 1999 GAMES AND COMPUTER SCIENCE Theoretical Models...

Peter van Emde Boas: Games and Computer Science 1999

GAMES AND COMPUTER SCIENCE

Theoretical Models 1999

Peter van Emde Boas

References available at: http://turing.wins.uva.nl/~peter/teaching/thmod99.htmlMost 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 AnalysisBackward InductionNumber Theory(Conway Guy Berlekamp)

Equilibria theory (Nash)Stochasitic FeaturesOptimization

Other names of importance:Von Neumann & MorgensternAumannShapleyHarsanyi


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


© Games Workshop © Games Workshop

URGAT THORGRIM

Introducing the Opponents


A Game

© Donald Duck 1999 # 35

Starting with 15 matchesplayers alternatively take1, 2 or 3 matches away untilnone remain. The playerending up with an oddnumber of matches winsthe game


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 : * --> Games (simply computable; Poly-time, Logspace or NC, ….)

• Set recognized := {w | (w) is won (by the first player) }

• How does this relate to conventional ways of recognizing languages ?


Games as Recognizers

• Construct a map : * --> Games (simply computable; Poly-time, Logspace or NC, ….)

• (w) is guaranteed to be proper• Set recognized :=

{w | (w) is won (by the first player) }• Properness conditions frequently

involve probabilistic aspects


Game Trees

Root

Terminal node:Thorgrim looses

Thorgrim’s turn

Urgat’s turn

Terminal node:Urgat looses

Standard Interpretation:Player unable to move looses the game


Game Trees

Root

Terminal node:

Thorgrim’s turn

Urgat’s turn

Terminal node:

Free Interpretation:Winner explicitly designated at terminal node

T

TU

U

T

UT


Game Trees

Root

Terminal node:

Thorgrim’s turn

Urgat’s turn

Terminal node:

Non Zero-Sum Game:Payoffs explicitly designated at terminal node

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1


Game Trees

Root

Terminal node:

Thorgrim’s turn

Urgat’s turn

Terminal node:


T

TU

U

T

UT

SUB-GAME


Backward Induction

Terminal node:

Thorgrim’s turn

Urgat’s turn


Root

Terminal node: T

TU

U

T

UT

T

T

UU

U


Backward Induction

Root

Terminal node:

Thorgrim’s turn

Urgat’s turn

Terminal node:

Non Zero-Sum Game:Payoffs explicitly designated at terminal node

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1

2 / 0

3 / 1

1 / 4-3 / 2

1 / 4


Backward Induction2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1

2 / 0

3 / 1

1 / 4-3 / 2

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

T

TU

U

T

UTT

T

UU

U


Analysis of the DD gameExtension used:Thorgrim wins if he hasan odd number when thegame terminates.This allows for even n .

Four types of configurations remain:T/E : Thorgrim has to play and has an even numberT/O : Thorgrim has to play and has an odd numberU/E : Urgat plays, while Thorgrim has an even numberU/O : Urgat plays, while Thorgrim has an odd number

Relevant feature: parity of number of matches collectedso far (not the number itself!)


Backward Induction Tablen U / E U / O T / E T / O

18 U U T / 1 T / 217 U T T / 1 U16 U T U T / 315 U U T / 2 T / 314 U U T / 2 T / 113 T U U T / 112 T U T / 3 U11 U U T / 3 T / 210 U U T / 1 T / 2 9 U T T / 1 U 8 U T U T / 3 7 U U T / 2 T / 3 6 U U T / 2 T / 1 5 T U U T / 1 4 T U T / 3 U 3 U U T / 3 T / 2 2 U U T / 1 T / 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

+ -

+ +- - - ++

-Computation Tree

Configuration Type

Existential

Universal

Negating

Accepting

Rejecting



+ -

+ +- - - ++

-Evaluation Full Computation Tree

This Tree Accepts

Configuration Type

Existential

Universal

Negating

Accepting

Rejecting

+

+

+

+

+

+

-

- -

- -

-

--

++

+

+



Infinite Branches ?

Requires third quality : Indeterminate nodes

Universal node is indeterminate iff it has no rejecting son and at least one indeterminate son

Existential node is indeterminate iff it has no acceptingson and at least one indeterminate one

Negating node is indeterminate iff its son is



Infinite Branches ?

Universal node is accepting iff it has no rejecting son and no indeterminate son (all sons are accepting)

Existential node is accepting iff it has one acceptingSon; indeterminate and rejecting sons don’t matter

Negating node is accepting iff its son is rejecting

Requires Recursive Evaluation of computation tree !


RECURSIVE EVALUATION

…. …. ….Indeterminate :

+ -

+ -

+ -

+ -

+ -

The proper way of Recursive evaluation ???


RECURSIVE EVALUATION

…. …. ….

+ -

+ -

+ -

+ -

+ -

Recursive evaluation ==Solving LEAST FIXED

POINT EQUATION !

+ -Partial order ≤of definedness

Extends to functionsdefined on the tree:

F ≤ G iff x[F(x) ≤ G(x)]

OK NOK


The Knaster Tarski Theorem

DOMAIN :=

SET U with partial order ≤ andleast element

Countable chains have least upper bounds

x0 ≤ x1 ≤ x2 ≤ ….. ≤ xn ≤ xn+1 ≤ …. ---> x =:i xi

i[xi ≤ x] and i[xi ≤ y] ==> x ≤ y

OPERATOR := FUNCTION which is:MONOTONE: x ≤ y ==> (x) ≤ (y)CONTINUOUS: ( i xi ) = i (xi)


The Knaster Tarski Theorem

THEOREM: If is an operator defined over domain Uthen the equation X = ( X ) has a least solution .

This solution is obtained as the limit of the sequence ofiterates: ≤ ≤ ≤ ….

= i i ( )

APPLICATION: U := domain of evaluations of tree := 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

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Peter van Emde Boas: Games and Computer Science 1999 GAMES AND COMPUTER SCIENCE Theoretical Models...

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