Formats of Winning Strategies for Six Types of Pushdown Games

Fridman, Wladimir

doi:10.4204/eptcs.25.14

Cited by 4 publications

(3 citation statements)

References 13 publications

(10 reference statements)

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“…Recall the one-token game that characterises HDness of a VPA P from the proof of Theorem 6.6. We argued there that the one-token game is a safety game played on the configuration graph of a deterministic VPA whose size is polynomial in the size of P. Therefore, Player 2, when she wins, has a winning strategy that can be implemented by a visibly pushdown transducer, which can be computed in exponential time [Fri10,Wal01]. The proof of Theorem 6.6 shows that such a strategy for Player 2 in the one-token game for VPA induces a resolver.…”

Section: Closure Propertiesmentioning

confidence: 99%

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

Guha,

Jecker,

Lehtinen

et al. 2024

Logical Methods in Computer Science

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We study the expressiveness and succinctness of history-deterministic pushdown automata (HD-PDA) over finite words, that is, pushdown automata whose nondeterminism can be resolved based on the run constructed so far, but independently of the remainder of the input word. These are also known as good-for-games pushdown automata. We prove that HD-PDA recognise more languages than deterministic PDA (DPDA) but not all context-free languages (CFL). This class is orthogonal to unambiguous CFL. We further show that HD-PDA can be exponentially more succinct than DPDA, while PDA can be double-exponentially more succinct than HD-PDA. We also study HDness in visibly pushdown automata (VPA), which enjoy better closure properties than PDA, and for which we show that deciding HDness is ExpTime-complete. HD-VPA can be exponentially more succinct than deterministic VPA, while VPA can be exponentially more succinct than HD-VPA. Both of these lower bounds are tight. We then compare HD-PDA with PDA for which composition with games is well-behaved, i.e. good-for-games automata. We show that these two notions coincide, but only if we consider potentially infinitely branching games. Finally, we study the complexity of resolving nondeterminism in HD-PDA. Every HDPDA has a positional resolver, a function that resolves nondeterminism and that is only dependant on the current configuration. Pushdown transducers are sufficient to implement the resolvers of HD-VPA, but not those of HD-PDA. HD-PDA with finite-state resolvers are determinisable.

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Section: Closure Propertiesmentioning

confidence: 99%

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

Guha,

Jecker,

Lehtinen

et al. 2024

Logical Methods in Computer Science

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“…Such a strategy can be finitely represented by a deterministic pushdown automaton with output (called pushdown transducers, or PDT) reading finite sequences over Σ 1 and outputting a single letter from Σ 2 . These are efficiently computable for Gale-Stewart games with ω-DCFL winning conditions [Fri10,Wal01]. Hence, one can compute a winning strategy for Player 2 in G(L(P d )) and then apply the transformation described in the second part of the proof of Theorem 4.2, which is implementable by deterministic pushdown transducers.…”

Section: Good-for-games Pushdown Automatamentioning

confidence: 99%

Good-for-games $\omega$-Pushdown Automata

Lehtinen¹,

Zimmermann²

2023

Logical Methods in Computer Science

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We introduce good-for-games $\omega$-pushdown automata ($\omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $\omega$-GFG-PDA are more expressive than deterministic $\omega$- pushdown automata and that solving infinite games with winning conditions specified by $\omega$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $\omega$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $\omega$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $\omega$-GFG- PDA and decidability of good-for-gameness of $\omega$-pushdown automata and languages. Finally, we compare $\omega$-GFG-PDA to $\omega$-visibly PDA, study the resources necessary to resolve the nondeterminism in $\omega$-GFG-PDA, and prove that the parity index hierarchy for $\omega$-GFG-PDA is infinite.

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“…This obviously defines a winning strategy because A and B accept the same language. For the other direction one uses the fact that a winning strategy for Classifier can be implemented by a pushdown automaton that reads the moves of Automaton and outputs the moves of Classifier [22,7]. This pushdown automaton for the strategy can easily be converted into P-parity DPDA for L ω (A ).…”

Section: The Parity Index Problemmentioning

confidence: 99%

Decision Problems for Deterministic Pushdown Automata on Infinite Words

Löding¹

2014

Electron. Proc. Theor. Comput. Sci.

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The article surveys some decidability results for DPDAs on infinite words (ω-DPDA). We summarize some recent results on the decidability of the regularity and the equivalence problem for the class of weak ω-DPDAs. Furthermore, we present some new results on the parity index problem for ω-DPDAs. For the specification of a parity condition, the states of the omega-DPDA are assigned priorities (natural numbers), and a run is accepting if the highest priority that appears infinitely often during a run is even. The basic simplification question asks whether one can determine the minimal number of priorities that are needed to accept the language of a given ω-DPDA. We provide some decidability results on variations of this question for some classes of ω-DPDAs.

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Formats of Winning Strategies for Six Types of Pushdown Games

Cited by 4 publications

References 13 publications

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

Good-for-games $\omega$-Pushdown Automata

Decision Problems for Deterministic Pushdown Automata on Infinite Words

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