Simulation Games

1. Simulation Games Michael Maurer

2. 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

3. Motivation • Capability of mimicking the behavior of another automaton (structural similarities, language containment) • Efficiently reducing the size of finite-state automata (known as quotienting)

4. 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,

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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 c 1 a Quotienting b c a A B 1‘ 3 b b 2 2‘

14. 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

15. Parity Games • Rule for moving the pebble: pebble on vi, Zero (One) moves the pebble to vi+1, such that • If a player can not move, the other one wins • Otherwise the game produces an infinite run • Considering the minimum priority kπthat occurs infinitely often in the run π; Zero wins, if kπis even, One otherwise

16. (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:

17. (Bi)Simulations from Parity Games b a • 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)),..} a 3 1 2 c

18. (Bi)Simulations from Parity Games b a • Example Büchi automaton: • kjhjk pAf ((2,1,a)) = 2 ; pAf ((2,3,c)) = 0 ; pAf ((3,1)) = 1 ; pAf ((1,3)) = 0 ; a 3 1 2 c

19. (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)

20. 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/