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 q 0 and q’ 0 Spoiler chooses a transition and moves Red to q i+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 a ω, but A does not Removing transition (1‘,a,1‘) would provide a simulation-equivalent quotient for A 3 1 2 2‘ 1‘ a b c b a b c Quotienting AB

14. Parity Games A parity game graph has two disjoint sets of vertices V 0 and V 1, 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 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 k π that occurs infinitely often in the run π; Zero wins, if k π is 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 Example Büchi automaton: kjhjk V 0 f = {(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 V 1 f = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)} Hghjg Player 0 Player 1 E A f ={((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)),..} 12 a c a b 3

18. (Bi)Simulations from Parity Games Example Büchi automaton: kjhjk p A f ((2,1,a)) = 2 ; p A f ((2,3,c)) = 0 ; p A f ((3,1)) = 1 ; p A f ((1,3)) = 0 ; 12 a c a b 3

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/