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

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

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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

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

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

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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

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SLIDE 6

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

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Parity Games

Arena A = (V , V0, V1, E, vin): directed (possibly countable) graph (V , E). positions of the players: partition {V0, V1} of V . initial vertex vin ∈ V .

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 3/10

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Parity Games

Arena A = (V , V0, V1, E, vin): directed (possibly countable) graph (V , E). positions of the players: partition {V0, V1} of V . initial vertex vin ∈ V .

1 1 1 1 1 1 1 1

· · · · · ·

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 3/10

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Parity Games

Arena A = (V , V0, V1, E, vin): directed (possibly countable) graph (V , E). positions of the players: partition {V0, V1} of V . initial vertex vin ∈ V .

1 1 1 1 1 1 1 1

· · · · · · 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 vin.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 3/10

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Scoring Functions for Parity Games

For c ∈ N and w ∈ V ∗: Scc(w) denotes the number of occurrences

  • f c in the suffix of w after the last occurrence of a smaller color.

Formally: Scc(ε) = 0 and Scc(wv) =      Scc(w) if col(v) > c, Scc(w) + 1 if col(v) = c, if col(v) < c.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 4/10

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Scoring Functions for Parity Games

For c ∈ N and w ∈ V ∗: Scc(w) denotes the number of occurrences

  • f 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

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Scoring Functions for Parity Games

For c ∈ N and w ∈ V ∗: Scc(w) denotes the number of occurrences

  • f 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

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SLIDE 13

Scoring Functions for Parity Games

For c ∈ N and w ∈ V ∗: Scc(w) denotes the number of occurrences

  • f 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:

1 1 1 1 1 1 1 1

· · · · · ·

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 4/10

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Pushdown Arenas

Pushdown arena A = (V , V0, V1, E, vin) induced by Pushdown System P = (Q, Γ, ∆, qin): (V , E): configuration graph of P. {V0, V1} induced by partition {Q0, Q1} of Q. vin = (qin, ⊥).

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 5/10

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Pushdown Arenas

Pushdown arena A = (V , V0, V1, E, vin) induced by Pushdown System P = (Q, Γ, ∆, qin): (V , E): configuration graph of P. {V0, V1} induced by partition {Q0, Q1} of Q. vin = (qin, ⊥).

1 1 1 1 1 1 1 1

· · · · · ·

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 5/10

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Pushdown Arenas

Pushdown arena A = (V , V0, V1, E, vin) induced by Pushdown System P = (Q, Γ, ∆, qin): (V , E): configuration graph of P. {V0, V1} induced by partition {Q0, Q1} of Q. vin = (qin, ⊥).

1 1 1 1 1 1 1 1

· · · · · · 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

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Stairs and Stair-Scores

stack height w w finite path starting in vin:

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

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Stairs and Stair-Scores

stack height w w finite path starting in vin: 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

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Stairs and Stair-Scores

reset(w) lstBmp(w) stack height w w finite path starting in vin: 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

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Stairs and Stair-Scores

reset(w) lstBmp(w) stack height w For every color c, define StairScc : V ∗ → N by StairScc(ε) = 0 and StairScc(w) =      StairScc(reset(w)) if minCol(lstBmp(w)) > c, StairScc(reset(w)) + 1 if minCol(lstBmp(w)) = c, if minCol(lstBmp(w)) < c.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

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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

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Stairs and Stair-Scores

reset(w) lstBmp(w) stack height w col: 1 1 1 2 1 StairSc0: StairSc1: StairSc2: 2 2

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

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Stairs and Stair-Scores

reset(w) lstBmp(w) stack height w col: 1 1 1 2 1 StairSc0: StairSc1: StairSc2: 2 2 2 3

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 6/10

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Main Theorem

Finite-time pushdown game: (A, col, k) with pushdown arena A, coloring col, and k ∈ N \ {0}. Rules: Play until StairScc = 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

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Main Theorem

Finite-time pushdown game: (A, col, k) with pushdown arena A, coloring col, and k ∈ N \ {0}. Rules: Play until StairScc = k is reached for the first time for some color c (which is unique). Player 0 wins ⇔ c is even. Let d = |col(V )|.

Theorem

Let G = (A, col) be a pushdown game and k > |Q| · |Γ| · 2|Q|·d · d. Player i wins G if and only if Player i wins (A, col, k). Note: (A, col, k) is a reachability game in finite arena.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 7/10

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Proof Idea

Walukiewicz (96): Reduction from pushdown parity game G to parity game G′ in finite arena A′ (of exponential size): Turn winning strategy σ′ for G′ into winning strategy σ for G.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 8/10

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Proof Idea

Walukiewicz (96): Reduction from pushdown parity game G to parity game G′ in finite arena A′ (of exponential size): Turn winning strategy σ′ for G′ into winning strategy σ for G. One can show more: For every play prefix w in G consistent with σ, there exists play prefix w′ in G′ consistent with σ′ such that StairScc(w) = Scc(w′) for every color c.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 8/10

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Proof Idea

Walukiewicz (96): Reduction from pushdown parity game G to parity game G′ in finite arena A′ (of exponential size): Turn winning strategy σ′ for G′ into winning strategy σ for G. One can show more: For every play prefix w in G consistent with σ, there exists play prefix w′ in G′ consistent with σ′ such that StairScc(w) = Scc(w′) for every color c. If σ′ is positional winning strategy for Player i in G′, then σ bounds the scores of Player 1 − i in G by |A′|. Hence, Player i wins (A, col, k), provided k > |A′|.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 8/10

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Lower Bounds

· · · · · · · · · · · · · · · · · · · · ·

mod2 mod3

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 9/10

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Lower Bounds

· · · · · · · · · · · · · · · · · · · · ·

mod2 mod3

For first n primes p1, . . . , pn: Player 0 has to reach stack height n

j=1 pj > 2n in upper row ⇒ cannot prevent losing player from

reaching exponentially high scores (in the number of states).

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 9/10

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Conclusion

Playing pushdown parity games in finite time: Adapt scores to stair-scores. Exponential threshold stair-score yields equivalent finite-duration game (reachability game in finite tree). (Almost) matching lower bounds on threshold stair-score.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 10/10

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Conclusion

Playing pushdown parity games in finite time: Adapt scores to stair-scores. Exponential threshold stair-score yields equivalent finite-duration game (reachability game in finite tree). (Almost) matching lower bounds on threshold stair-score. Further research: Turn winning strategy for finite-duration game into winning strategy for pushdown game. Permissive strategies for pushdown parity games. Extensions to more general classes of arenas, e.g., higher-order pushdown systems.

Martin Zimmermann University of Warsaw Playing Pushdown Parity Games in a Hurry 10/10