Games for the μ-calculus

Niwiński, Damian; Walukiewicz, Igor

doi:10.1016/0304-3975(95)00136-0

Cited by 82 publications

(92 citation statements)

References 8 publications

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“…For the right to left implication assume conversely and show that player II has a winning strategy. See for example [24] or [21] for similar arguments.…”

Section: Theorem 26mentioning

confidence: 98%

Pushdown processes: Games and model checking

Walukiewicz

1996

Lecture Notes in Computer Science

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108

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A pushdown game is a two player perfect information infinite game on a transition graph of a pushdown automaton. A winning condition in such a game is defined in terms of states appearing infinitely often in the play. It is shown that if there is a winning strategy in a pushdown game then there is a winning strategy realized by a pushdown automaton. An EXPTIME procedure for finding a winner in a pushdown game is presented. The procedure is then used to solve the model-checking problem for the pushdown processes and the propositional µ-calculus. The problem is shown to be DEXPTIME-complete.

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“…For the right to left implication assume conversely and show that player II has a winning strategy. See for example [24] or [21] for similar arguments.…”

Section: Theorem 26mentioning

confidence: 98%

Pushdown processes: Games and model checking

Walukiewicz

1996

Lecture Notes in Computer Science

Self Cite

108

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“…The rules {∨ 1 , ∨ 2 , ∧, a , [a] , µ, ν} are called logical rules. The rules ∨ 1 , ∨ 2 , ∧, µ, ν are standard and also the rules a , [a] for reasoning about modalities are natural counterparts to the analogous rules adopted in proof systems for modal (fixed point) logics appeared in the literature (see, e.g., [19], [21] and [15]). The rules {δ, + λ , { }} are called distribution rules and constitute a crucial aspect of the system.…”

Section: Proposition 14mentioning

confidence: 99%

“…[15,21,3]) in which some leaves of the tree are identified with sequents internal to the tree, with the proof looping back to that point. Technically, it is convenient to view such cyclic trees as the infinite trees they unfold to, and to work with general infinite trees, with the finite cyclic ones corresponding exactly to the regular trees (those with only finitely many subtrees).…”

Section: Markov Proofsmentioning

confidence: 99%

“…In the literature on infinitary proof systems for fixed-point logics (see, e.g., [15,21,3]), valid proofs are defined as those preproofs whose infinite branches all contain at least one legitimate sequence (called a valid trace) of fixed-point unfoldings, along which a greatest fixed-point is unfolded infinitely often. This can equivalently be reformulated in terms of a single player game.…”

Section: Markov Proofsmentioning

confidence: 99%

“…This approach is familiar from proof systems for the ordinary (non-probabilistic) modal µ-calculus [15,21] and other fixed-point logics [3]. However, there is a twist in our probabilistic setting.…”

Section: Introductionmentioning

confidence: 99%

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A Proof System for Compositional Verification of Probabilistic Concurrent Processes

Mio

Simpson

2013

Lecture Notes in Computer Science

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Abstract. We present a formal proof system for compositional verification of probabilistic concurrent processes. Processes are specified using an SOS-style process algebra with probabilistic operators. Properties are expressed using a probabilistic modal µ-calculus. And the proof system is formulated as a sequent calculus in which sequents are given a quantitative interpretation. A key feature is that the probabilistic scenario is handled by introducing the notion of Markov proof, according to which proof trees contain probabilistic branches and are required to satisfy a condition formulated by interpreting them as Markov Decision Processes. We present simple but illustrative examples demonstrating the applicability of the approach to the compositional verification of infinite state processes. Our main result is the soundness of the proof system, which is proved by applying the coupling method from probability theory to the game semantics of the probabilistic modal µ-calculus.

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