Game, Bisimulation and Model Checking

Game, Bisimulation and Model Checking

Presented by Daniel ChoiProvable Software Lab.

KAIST

Game, Bisimulation and Model Checking, Daniel Choi@PSWLAB, KAIST

• Introduction

• Game Theory– Classification of Games– Notations

• Applications to Model Verification– Bisimulation– Model Checking

• Conclusion2/41


Content


Game theory is the study of the choice of strategies by interacting rational agents.

Roger A. McCain, Game Theory: A Nontechnical Introduction to the Analysis of Strat-egy (South-western, 2004)

3/41

Introduction

Game Theory?


• Nim Game

4/41

Introduction – Game Theory

Simple Example

Take one or more coins at the same rowThe player who takes the last coin wins

Example from : Roger A. McCain, Game Theory: A Nontechnical In-troduction to the Analysis of Strategy (South-western, 2004)


• Players– Shin– Yun-ho

• Assumption– Shin starts first

• Strategies for Shin– Take one coin from the top row– Take one coin from the second row– Take both coins from the second row

5/41

Introduction – Game Theory – Simple Example

Rules for Nim Game

Game, Bisimulation and Model Checking, Daniel Choi@PSWLAB, KAIST 6/41

Introduction – Game Theory – Simple Example

Tree Diagram for Nim

Al wins

Bob wins

Bob wins

Al wins

Al wins

Shin Wins

Yun-ho Wins

Shin Wins

Shin Wins

Yun-ho WinsShin’s turn

Shin’s turn

Yun-ho’s turn


• It can formally analyze the game– E.g. Tree diagram

• Metaphor– Many interactions can be treated and ana-

lyzed as a game

• Game theoretic analysis is to discover which strategy is a person’s best response to the strategies chosen by the others.

7/41


Characteristics of Game Theory


• Worst-Case Analysis– Game between a solver and an adversary

• Network routing– Game between client and environment– Game between router and clients

• Load-sharing in distributed systems– Game between server and client

(or other server)8/41


Computer Science and Game Theory


Game Theory - Motivation

Model Checking

Target ModelRequirement

Properties

(F W)

How about different ap-proaches?

Game Theory


Game Theory - Motivation

Game Theoretical Framework for Model Checking

Target Model

Requirement Properties

(F W)

The model does not satisfy the

requirement properties!!

The model satisfies

the require-ment prop-

erties!!

Player 1

Player 2

Does Player 2 always win?


• Introduction



• Conclusion11/41


Content


• Non-cooperative game– The players act individually, each decision

of a player influences the payoff of the other players

• Cooperative game– The players are allowed to from coalitions

and combine their decision-making prob-lems

12/41

Game Theory

Classification of Games


• Normal form– Decisions of players are simultaneous– The payoffs are represented by a matrix– Ex. Prisoner’s Dilemma

• Extensive form– Decisions of players are sequential– The payoffs are represented by a tree– Ex. Nim game

13/41

Game Theory – Classification of Games

Non-cooperative Game


Game Theory – Classification of Games – Non-cooperative Game

Normal Forms – Prisoner’s Dilemma


• Al Thinks rationally– Case 1: Bob confess

• If Al confess then Al will get 10 years and Bob will get 10 years• If Al does not confess then Al will get 20 years, Bob will get 0 years

– Case 2: Bob does not confess• If Al confess then Al will get 0 years, Bob will get 20 years• If Al does not confess then Al will get 1 years, Bob will get 1 years

15/41


Normal Forms – Prisoner’s Dilemma

AlConfess Don’t

Bob

Con-fess

10 years, 10 years 0, 20 years

Don’t 20, 0 years 1 year, 1 year


• Al Thinks rationally– Case 1: Bob confess

• If Al confess then Al will get 10 years and Bob will get 10 years• If Al does not confess then Al will get 20 years, Bob will get 0 years

– Case 2: Bob does not confess• If Al confess then Al will get 0 years, Bob will get 20 years• If Al does not confess then Al will get 1 years, Bob will get 1 years

16/41


Cooperative Game

AlConfess Don’t

Bob

Con-fess

10 years, 10 years 0, 20 years

Don’t 20, 0 years 1 year, 1 year


• Extensive game– With perfect information

• Any player knows all the moves made before one’s own move

– Without perfect information• A player makes one’s move in the game, one does not

know opponent player’s move

• Computation (run or sequence)– Interaction between “Player” and “Opponent”– Represented by a sequence of move– Opponent always makes the first move

17/41

Game Theory

Notations of Extensive Game


• Definition of Game (MG, λG, PG)– MG : the set of moves of the game– λG : MG → {P, O}

• Labeling function designating each move as by Player or Opponent

– PG : the set of alternating se-quences of moves in MG

• Non-empty, prefix-closed

• Example– MG = {a1, a2, b1, b2, b3}– λG = {(a1, O), (a2, O),

(b1, P), (b2, P), (b3, P)}– PG = {ε , a1, a1b1, a2, a2b2, a2b3}

18/41

Game Theory - Notations of Extensive Game

Definition of Game

a1 a2

b1b2 b3


• A game can be seen as specifying the possi-ble interactions between a System and its Environment

• Games classify behaviors– Programs will be modeled by strategies

• Strategies are rules specifying how the System should actually play

– Deterministic strategy σ on a game G (σ ⊆ PGeven)

• ε ∈ σ• sab ∈ σ ⇒ s ∈ σ• sab, sac ∈ σ ⇒ b = c

19/41

Game Theory - Notations of Extensive Game - Definition of Game

Strategies


• Introduction



• Conclusion20/41


Content


Application to the Model Verification

CCS Overview



CCS Overview



CCS Overview



CCS Overview



CCS Overview



CCS Overview



CCS Overview



CCS Overview


Application to the Model Verification – CCS Overview

Bisimulation


Application to the Model Verification – CCS Overview

Bisimulation


• Bisimulation relation can be modeled as a Bisimu-lation game (Equivalence Game)– Observer can repeatedly interact with a process by

choosing an available transition from it– Observer match their selections so that they can pro-

ceed with further corresponding choices

• Equivalence game G(E0, F0)– Player I and II : Observers who make choices of transi-

tions– Player I attempts to show initial processes are different– Player II attempts to show two processes are equivalent

31/41


Game-theoretical View


• Equivalence game G(E0, F0)– Player I chooses a transition Ej –a-> Ej+1 and then player II

chooses a transition with the same label Fj –a-> Fj+1

– Player I chooses a transition Fj –a-> Fj+1 and then player II chooses a transition with the same label Ej –a-> Ej+1

• Player I win– Player I can choose a transition and player II will be unable to

match it

• Player II win– If the play is infinite – If the play reaches the position (En, Fn) and both processes

have no available transitions 32/41


Game-theoretical View


Clock and Clock2 is equiva-

lent!

Clock and

Clock2 is different!

33/41

Application to the Model Verification - Game-theoretical View

Example

tick

Clock

tick

Clock2

tick

Player I Player II

G(Clock, Clock2)

(Clock, Clock2), (Clock, tick.Clock2), (Clock, Clock2) …. Player II wins


Clock and Clock’ is equiva-

lent!

Clock and

Clock’ is different!

34/41


Example

0

Clock

tick

Clock’

tick

Player I Player II

G(Clock, Clock2)

(Clock, 0)Player I wins

tick


• Winning Strategy π– If the player wins every play in which the player uses π

• PropositionFor any game G(E,F) either player I or player II has a

history-free winning strategy

• History-free : Rules do not depend on what happened previously in the play

• If player II has a winning strategy for G(E, F) then E is game equivalent to process F

35/41


Winning Strategy


• E is game equivalent to F iff E is bisimular to F– Only if

• Showing that the relation R = {(E, F) | E and F are game equivalent} is a bisimulation.

– Player I’s move : E –a-> E’ (this is possible move by player I)Player II can respond with F –a-> F’ (by game equivalent relation)

– Player I’s move : F –a-> F’ (this is possible move by player I) Player II can respond with E –a-> E’ (by game equivalent relation)

– If• There is a bisimulation relation R s.t. (E, F) ∈ R, Construct

a winning strategy for player II for the game G(E, F)• In any play, whatever move player I makes player II re-

sponds by making sure that the resulting pair of pro-cesses remain in the relation R

36/41


Game Equivalent


tea

U ≡ 1p.(1p.tea.U + 1p.cof -fee.U)

cof -fee

tea1p

1p 1p 1p

1p 1p

1p

V ≡ 1p.1p.tea.V + 1p.1pcoffee.V

37/41

Applications - Bisimulation

Examples : Vending Machine

U and V is bisimula-tion rela-

tion!

U and V is not bisim-ulation re-

lation!

Player I Player II

G(U, V)


Applications - Bisimulation

Examples : Vending Machine

Player I Player II

V -1p->1p.tea.V

(U, V) -> (1p.tea.U + 1p.coffee.U,

1p.tea.V)

U -1p-> 1p.tea.U + 1p.coffee.U

1p.tea.U + 1p.coffee.U, 1p.tea.V-1p-> coffee.U

(1p.tea.U + 1p.coffee.U, 1p.tea.V)

-> (coffee.U, tea.V)

1p.tea.V -1p-> tea.V)

It is not bisimulation rela-tion


• Introduction



• Conclusion39/41


Content


• Semantics of Interaction

• Computability Logic: A Formal Theory of Interaction (In Interactive Comput-ing)

• Model mu-calculus– Property Checking Game– Model Checking Game

40/41

Conclusion

Further Study


• Overview of Game Theory and Using to Model the Knowl-edge of Multi-Agent Systemby Thuy Lien PHAM, Marc BUI, Michel LAMURE In Actes de la Pre-mière Conférence Internationale RIVF'03 Rencontres en Informa-tique Vietnam-France, RIVF'03

• Bisimulation, Model Checking and Other Gamesby Colin Stirling, In Notes for Mathfit instructional meeting on games and computation, Edinburgh, June 1997

• Games and Model Mu-Calculusby Colin Stirling, In TACAS 1996 Lecture Notes in Computer Science 1055, 298-312, 1996

• Semantics and Logics of ComputationEdited by A. Pitts and P. Dybjer, Cambrige Press, 1996

41/41

Reference

List of Papers

Documents

Game, Bisimulation and Model Checking