Playing Pushdown Parity Games in a Hurry Joint work with Wladimir - PowerPoint PPT Presentation

Playing Pushdown Parity Games in a Hurry Joint work with Wladimir Fridman (RWTH Aachen University) Martin Zimmermann University of Warsaw September 7th, 2012 GandALF 2012 Naples, Italy Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 1/10

Motivation Playing infinite games in finite time: Ehrenfeucht, Mycielski: positional determinacy of mean-payoff games. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 2/10

Motivation Playing infinite games in finite time: Ehrenfeucht, Mycielski: positional determinacy of mean-payoff games. Jurdziński: small progress measures for parity games. Bernet, Janin, Walukiewicz: permissive strategies for parity games. Björklund, Sandberg, Vorobyov: positional determinacy of parity games. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 2/10

Motivation Playing infinite games in finite time: Ehrenfeucht, Mycielski: positional determinacy of mean-payoff games. Jurdziński: small progress measures for parity games. Bernet, Janin, Walukiewicz: permissive strategies for parity games. Björklund, Sandberg, Vorobyov: positional determinacy of parity games. McNaughton: playing Muller games in finite time using so-called scoring functions . Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 2/10

Motivation Playing infinite games in finite time: Ehrenfeucht, Mycielski: positional determinacy of mean-payoff games. Jurdziński: small progress measures for parity games. Bernet, Janin, Walukiewicz: permissive strategies for parity games. Björklund, Sandberg, Vorobyov: positional determinacy of parity games. McNaughton: playing Muller games in finite time using so-called scoring functions . Fearnley, Neider, Rabinovich, Z.: strong bounds on McNaughton’s scoring functions: yields reduction from Muller to safety games, new memory structure, permissive strategies. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 2/10

Motivation Playing infinite games in finite time: Ehrenfeucht, Mycielski: positional determinacy of mean-payoff games. Jurdziński: small progress measures for parity games. Bernet, Janin, Walukiewicz: permissive strategies for parity games. Björklund, Sandberg, Vorobyov: positional determinacy of parity games. McNaughton: playing Muller games in finite time using so-called scoring functions . Fearnley, Neider, Rabinovich, Z.: strong bounds on McNaughton’s scoring functions: yields reduction from Muller to safety games, new memory structure, permissive strategies. Results hold only for finite arenas. What about infinite ones? Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 2/10

Parity Games Arena A = ( V , V 0 , V 1 , E , v in ) : directed (possibly countable) graph ( V , E ) . positions of the players: partition { V 0 , V 1 } of V . initial vertex v in ∈ V . Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 3/10

Parity Games Arena A = ( V , V 0 , V 1 , E , v in ) : directed (possibly countable) graph ( V , E ) . positions of the players: partition { V 0 , V 1 } of V . initial vertex v in ∈ V . · · · 0 0 0 0 0 0 0 0 · · · 1 1 1 1 1 1 1 1 0 Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 3/10

Parity Games Arena A = ( V , V 0 , V 1 , E , v in ) : directed (possibly countable) graph ( V , E ) . positions of the players: partition { V 0 , V 1 } of V . initial vertex v in ∈ V . · · · 0 0 0 0 0 0 0 0 · · · 1 1 1 1 1 1 1 1 0 Parity game G = ( A , col ) with col : V → { 0 , . . . , d } . Player 0 wins play ⇔ minimal color seen infinitely often even. (Winning / positional) strategies defined as usual. Player i wins G ⇔ she has winning strategy from v in . Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 3/10

Scoring Functions for Parity Games For c ∈ N and w ∈ V ∗ : Sc c ( w ) denotes the number of occurrences of c in the suffix of w after the last occurrence of a smaller color. Formally: Sc c ( ε ) = 0 and  Sc c ( w ) if col ( v ) > c ,   Sc c ( wv ) = Sc c ( w ) + 1 if col ( v ) = c ,  0 if col ( v ) < c .  Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 4/10

Scoring Functions for Parity Games For c ∈ N and w ∈ V ∗ : Sc c ( w ) denotes the number of occurrences of c in the suffix of w after the last occurrence of a smaller color. Remark In a finite arena, a positional winning strategy for Player 0 bounds the scores for all odd c by | V | . Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 4/10

Scoring Functions for Parity Games For c ∈ N and w ∈ V ∗ : Sc c ( w ) denotes the number of occurrences of c in the suffix of w after the last occurrence of a smaller color. Remark In a finite arena, a positional winning strategy for Player 0 bounds the scores for all odd c by | V | . Corollary In a finite arena, Player 0 wins ⇔ she can prevent a score of | V | + 1 for all odd c (safety condition). Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 4/10

Scoring Functions for Parity Games For c ∈ N and w ∈ V ∗ : Sc c ( w ) denotes the number of occurrences of c in the suffix of w after the last occurrence of a smaller color. Remark In a finite arena, a positional winning strategy for Player 0 bounds the scores for all odd c by | V | . Corollary In a finite arena, Player 0 wins ⇔ she can prevent a score of | V | + 1 for all odd c (safety condition). The remark does not hold in infinite arenas: · · · 0 0 0 0 0 0 0 0 · · · 1 1 1 1 1 1 1 1 0 Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 4/10

Pushdown Arenas Pushdown arena A = ( V , V 0 , V 1 , E , v in ) induced by Pushdown System P = ( Q , Γ , ∆ , q in ) : ( V , E ) : configuration graph of P . { V 0 , V 1 } induced by partition { Q 0 , Q 1 } of Q . v in = ( q in , ⊥ ) . Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 5/10

Pushdown Arenas Pushdown arena A = ( V , V 0 , V 1 , E , v in ) induced by Pushdown System P = ( Q , Γ , ∆ , q in ) : ( V , E ) : configuration graph of P . { V 0 , V 1 } induced by partition { Q 0 , Q 1 } of Q . v in = ( q in , ⊥ ) . · · · 0 0 0 0 0 0 0 0 · · · 1 1 1 1 1 1 1 1 0 Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 5/10

Pushdown Arenas Pushdown arena A = ( V , V 0 , V 1 , E , v in ) induced by Pushdown System P = ( Q , Γ , ∆ , q in ) : ( V , E ) : configuration graph of P . { V 0 , V 1 } induced by partition { Q 0 , Q 1 } of Q . v in = ( q in , ⊥ ) . · · · 0 0 0 0 0 0 0 0 · · · 1 1 1 1 1 1 1 1 0 Pushdown parity game G = ( A , col ) where col is lifting of col : Q → { 0 , . . . , d } to configurations. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 5/10

Stairs and Stair-Scores stack height w w finite path starting in v in : Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Stairs and Stair-Scores stack height w w finite path starting in v in : Stair in w : position s. t. no subsequent position has smaller stack height (first and last position are always a stair). reset ( w ) : prefix of w up to second-to-last stair. lstBmp ( w ) : suffix after second-to-last stair. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Stairs and Stair-Scores reset ( w ) lstBmp ( w ) stack height w w finite path starting in v in : Stair in w : position s. t. no subsequent position has smaller stack height (first and last position are always a stair). reset ( w ) : prefix of w up to second-to-last stair. lstBmp ( w ) : suffix after second-to-last stair. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Stairs and Stair-Scores reset ( w ) lstBmp ( w ) stack height w For every color c , define StairSc c : V ∗ → N by StairSc c ( ε ) = 0 and  StairSc c ( reset ( w )) if minCol ( lstBmp ( w )) > c ,   StairSc c ( w ) = StairSc c ( reset ( w )) + 1 if minCol ( lstBmp ( w )) = c ,  0 if minCol ( lstBmp ( w )) < c .  Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Stairs and Stair-Scores reset ( w ) lstBmp ( w ) stack height w col : 1 1 1 2 1 Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Stairs and Stair-Scores reset ( w ) lstBmp ( w ) stack height w col : 1 1 1 2 1 StairSc 0 : 2 StairSc 1 : 2 StairSc 2 : 0 Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Stairs and Stair-Scores reset ( w ) lstBmp ( w ) stack height w col : 1 1 1 2 1 StairSc 0 : 2 2 StairSc 1 : 2 3 StairSc 2 : 0 0 Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

Main Theorem Finite-time pushdown game: ( A , col , k ) with pushdown arena A , coloring col , and k ∈ N \ { 0 } . Rules: Play until StairSc c = k is reached for the first time for some color c (which is unique). Player 0 wins ⇔ c is even. Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 7/10