Simulation Games

Michael Maurer

Overview

Motivation 4 Different (Bi)simulation relations and

their rules to determine the winner Problem with delayed simulation Parity Games Construction of (Bi)simulations as Parity

Games

Motivation

Capability of mimicking the behavior of another automaton (structural similarities, language containment)

Efficiently reducing the size of finite-state automata (known as quotienting)

Simulation Games

4 different Simulation Game Definitions for a given Büchi automaton A :

1) ordinary simulation game,

2) direct (strong) simulation game,

3) delayed simulation game,

4) fair simulation game,

Simulation Games

Played by 2 players: Spoiler and Duplicator At the start: two pebbles (Red and Blue) are

placed on two vertices q0 and q’0 Spoiler chooses a transition

and moves Red to qi+1

Duplicator also chooses a transition

and moves Blue to q‘i+1

If Duplicator can‘t move, the game halts and Spoiler wins

Who will be the winner?

Either the game halts, in which case Spoiler wins

Or the game produces two infinite runs:

and

For each of the 4 simulation games there exist different rules to determine the winner

Rules for the winner

Ordinary simulation: Duplicator wins in any case Fairness conditions are ignored

Duplicator wins as long as the game does not halt

Direct simulation: D wins iff for all i, if then

Rules for the winner

Delayed simulation: D wins iff for all i, if then there exists j ≥ i such

that

Fair simulation: D wins iff there are infinitely many j such that

or only finitely many i such that In other words: if there are infinitely many i such that

, then there are also infinitely many j such that

Simulation Relation

A state q‘ ordinary, direct, delayed, fair simulates a state q, if there is a winning strategy for D

The simulation relation is denoted by , where * stands for one of the 4 simulations

The relations are ordered by containment:

(preorder)

For di, de, f: if then

Bisimulation Games

For all of the mentioned simulations corresponding notions of bisimulation via modification of the game

S can choose in each round which pebble he will move and D has to respond with the other one

Bisimulations define an equivalence relation

Bisimulation winning rules

Fair: an accept state appears infinitely often on one of the 2 runs π and π‘ an accept state must appear infinitely often on the other as well

Delayed: an accept state at position i of either run an accept state at j ≥ i of the other run

Direct: an accept state at position i of either run

an accept state at position i of both runs

Problem with delayed simulation

Quotienting: states that simulate each other are merged

Difficult to find a working definition of a simulation preserving quotient with respect to delayed simulation

Not at all clear how such a quotient should be defined

Problem with delayed simulation

Example for the quotienting problem:

B accepts aω, but A does not Removing transition (1‘,a,1‘) would provide a

simulation-equivalent quotient for A

3

1

22‘

1‘ab

c

b

a

b

cQuotienting

A B

Parity Games

A parity game graph has two disjoint sets of vertices V0 and V1, their union is V

It also has an edge set and a priority function that assigns a priority to each vertex

Played by two players, Zero and One and the game starts by placing a pebble on

Parity Games

Rule for moving the pebble: pebble on v i, Zero (One) moves the pebble to v i+1, such that

If a player can not move, the other one wins Otherwise the game produces an infinite run

Considering the minimum priority kπ that occurs infinitely often in the run π; Zero wins, if kπ is even, One otherwise

(Bi)Simulations from Parity Games

Example: Parity game graph for the fair simulation game

The set of vertices for Zero:

The set of vertices for One: The set of the edges for Zero and One:

The priority function:

(Bi)Simulations from Parity Games Example Büchi automaton:

kjhjk

V0f = {(2,1,a),(2,2,a),(2,3,a),(2,1,b),(2,2,b),(2,3,b),(2,1,c),(2,2,c),(2,3,c),

(3,1,a),(3,2,a),(3,3,a)}

Jhkjh

V1f = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

Hghjg Player 0

Player 1

EAf={((2,1,a),(2,2)),((3,1,a),(3,2)),((2,2,b),(2,2)),((2,2,a),(2,3)),..} U {((1,1),

(2,1,a)),((1,2),(2,2,a)),((2,2),(2,3,b)),..}

1 2

a

c

ab

3

(Bi)Simulations from Parity Games Example Büchi automaton:

kjhjk

pAf ((2,1,a)) = 2 ;

pAf ((2,3,c)) = 0 ;

pAf ((3,1)) = 1 ;

pAf ((1,3)) = 0 ;

1 2

a

c

ab

3

(Bi)Simulations from Paritiy Games

Parity Game constructed: Zero has a winning strategy from (q,q’), iff q is

fairly simulated by q’ Jurdzinkis algorithm as fast algorithm for

computing fair (bi)simulation relations and delayed simulations

Other relations can be constructed from the fair simulation formulas (Handout)

References

Carsten Fritz, Thomas Wilke: Simulation Relations for Alternating Parity Automata and Parity Games. DLT 2006, LNCS 4036, pp. 59-70, Springer-Verlag (2006)

Kousha Etessami, Thomas Wilke, Rebecca A. Schuller: Fair Simulation Relations, Parity Games and State Space Reduction for Büchi Automata. ICALP 2001, LNCS 2076, pp. 694-707, Springer-Verlag (2001)

Carsten Fritz: Simulation-Based Simplification of omega-Automata. PhD thesis, Technische Fakultät der Christian Albrecht Universität zu Kiel (2005) available at http:/e-diss.uni-kiel.de/diss_1644/