## EXPTIME tableaux for the coalgebraic µ- calculus (2009)

Venue: | Proc. CSL 2009, volume 5771 of Lecture Notes in Computer Science |

Citations: | 5 - 3 self |

### BibTeX

@INPROCEEDINGS{Cîrstea09exptimetableaux,

author = {Corina Cîrstea and Clemens Kupke and Dirk Pattinson},

title = {EXPTIME tableaux for the coalgebraic µ- calculus},

booktitle = {Proc. CSL 2009, volume 5771 of Lecture Notes in Computer Science},

year = {2009},

pages = {179--193},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. The coalgebraic approach to modal logic provides a uniform framework that captures the semantics of a large class of structurally different modal logics, including e.g. graded and probabilistic modal logics and coalition logic. In this paper, we introduce the coalgebraic µ-calculus, an extension of the general (coalgebraic) framework with fixpoint operators. Our main results are completeness of the associated tableau calculus and EXPTIME decidability. Technically, this is achieved by reducing satisfiability to the existence of non-wellfounded tableaux, which is in turn equivalent to the existence of winning strategies in parity games. Our results are parametric in the underlying class of models and yield, as concrete applications, previously unknown complexity bounds for the probabilistic µ-calculus and for an extension of coalition logic with fixpoints. 1

### Citations

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(Show Context)
Citation Context ...duction The extension of a modal logic with operators for least and greatest fixpoints leads to a dramatic increase in expressive power [1]. The paradigmatic example is of course the modal µ-calculus =-=[10]-=-. In the same way that the µ-calculus extends the modal logic K, one can freely add fixpoint operators to any propositional modal logic, as long as modal operators are monotone. Semantically, this pos... |

314 | An automata-theoretic approach to branchingtime model checking - Kupferman, Vardi, et al. |

232 | Tree automata, mu-calculus and determinacy
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(Show Context)
Citation Context ...i ∈ BP , and s is a winning strategy from position b ∈ B if P wins all plays with initial position b that are played according to s. We will use the fact that parity games are history-free determined =-=[5, 13]-=- and that winning regions can be decided in UP ∩ co-UP [9]. Theorem 2. At every position b ∈ B∃ ∪ B∀ in a parity game G = (B∃, B∀, E, Ω) one of the players has a history-free winning ( strategy. Furth... |

156 |
The Complexity of Tree Automata and Logics of Programs
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- 1988
(Show Context)
Citation Context ...ms of the semantics of the underlying modal logic. This apparent simplicity is lost once we move from semantics to syntax: completeness and complexity even of the modal µ-calculus are all but trivial =-=[22, 4]-=-, and µ-calculi arising from other monotone modal logics are largely unstudied, with the notable exception of the graded µ-calculus [12]. Here, we improve on this situation, not by providing a new com... |

155 |
ModaZ logic
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(Show Context)
Citation Context ...fting [[[C]]]X(A) = {(f, (Si)i∈N ) ∈ G(X) | ∃(si)i∈C∀(si)i∈N\C(f((si)i∈N ) ∈ A} that induces the standard semantics of coalition logic. 5. Finally, the similarity type Λ = {} of monotone modal logic =-=[2]-=- has a single unary (we write = ♦) and interpret the ensuing language over monotone neighbourhood frames, that is, coalgebras for the functor / structure M(X) = {Y ⊆ P(P(X)) | Y upwards closed} [[... |

150 | A modal logic for coalitional power in games
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(Show Context)
Citation Context ...EXPTIME bounds, both for the modal and the graded µ-calculus [4, 12], but also previously unknown EXPTIME bounds for the probabilistic and monotone µ-calculus, and for an extension of coalition logic =-=[15]-=- with fixpoint operators. Our main technical results are a syntactical characterisation of satisfiability in terms of (non-)existence of closed tableaux and a game-theoretic characterisation of satisf... |

111 | On the complexity of ω-automata - Safra - 1988 |

61 |
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(Show Context)
Citation Context ...i ∈ BP , and s is a winning strategy from position b ∈ B if P wins all plays with initial position b that are played according to s. We will use the fact that parity games are history-free determined =-=[5, 13]-=- and that winning regions can be decided in UP ∩ co-UP [9]. Theorem 2. At every position b ∈ B∃ ∪ B∀ in a parity game G = (B∃, B∀, E, Ω) one of the players has a history-free winning ( strategy. Furth... |

47 | From nondeterministic Büchi and Streett automata to deterministic parity automata - Piterman |

36 |
progress measures for solving parity games
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(Show Context)
Citation Context ...wins all plays with initial position b that are played according to s. We will use the fact that parity games are history-free determined [5, 13] and that winning regions can be decided in UP ∩ co-UP =-=[9]-=-. Theorem 2. At every position b ∈ B∃ ∪ B∀ in a parity game G = (B∃, B∀, E, Ω) one of the players has a history-free winning ( strategy. Furthermore, for every b ∈ B∃ ∪ B∀, ) ) ⌊d/2⌋ it can be determi... |

27 | PSPACE bounds for rank 1 modal logics
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- 2006
(Show Context)
Citation Context ...ableau game, and the announced EXPTIME upper bound follows once we can ensure that legality of moves in the tableau game can be decided in exponential time. Related Work. Our treatment is inspired by =-=[14, 21, 19]-=-, but we note some important differences. In contrast to [14], we use parity games that directly correspond to tableaux, together with parity automata to detect bad traces. Moreover, owing to the gene... |

26 | A finite model construction for coalgebraic modal logic - Schröder - 2006 |

24 |
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(Show Context)
Citation Context ...ghbourhood frames, that is, coalgebras for the functor / structure M(X) = {Y ⊆ P(P(X)) | Y upwards closed} [□]X(A) = {Y ∈ M(X) | A ∈ Y }which recovers the standard semantics in a coalgebraic setting =-=[8]-=-. It is readily verified that all structures above are indeed monotone. 3 The Model-Checking Game We start by describing a characterisation of model checking in terms of parity games that generalises ... |

24 |
Games for the mu-calculus
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- 1996
(Show Context)
Citation Context ...ableau game, and the announced EXPTIME upper bound follows once we can ensure that legality of moves in the tableau game can be decided in exponential time. Related Work. Our treatment is inspired by =-=[14, 21, 19]-=-, but we note some important differences. In contrast to [14], we use parity games that directly correspond to tableaux, together with parity automata to detect bad traces. Moreover, owing to the gene... |

24 | Completeness of Kozen’s axiomatisation of the propositional µ-calculus
- Walukiewicz
- 1995
(Show Context)
Citation Context ...ms of the semantics of the underlying modal logic. This apparent simplicity is lost once we move from semantics to syntax: completeness and complexity even of the modal µ-calculus are all but trivial =-=[22, 4]-=-, and µ-calculi arising from other monotone modal logics are largely unstudied, with the notable exception of the graded µ-calculus [12]. Here, we improve on this situation, not by providing a new com... |

23 |
Modal and Temporal
- Stirling
- 2001
(Show Context)
Citation Context ...stablish a small model theorem. We start by describing a parity game that characterizes model checking for the coalgebraic µ-calculus. As in the model-checking game for the modal µ-calculus (see e.g. =-=[20]-=-), we allow greatest and least fixpoints to be unfolded ad libitum. Truth of a formula in a particular state of a model then follows, if only greatest fixpoints are unfolded infinitely often on the to... |

16 |
so many possible worlds
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- 1972
(Show Context)
Citation Context ... | supp(f) finite} [〈n〉]X(A) = {f ∈ B(X) | ∑ f(x) > n}. where supp(f) = {x ∈ X | f(x) ̸= 0} is the support of f. Note that this semantics differs from the Kripke semantics for both graded modal logic =-=[7]-=- and the graded µcalculus, but both types of semantics induce the same satisfiability problem: Kripke frames are multigraphs where each edge has multiplicity one, and the unravelling of a multigraph c... |

16 | Automata and fixed point logics: a coalgebraic perspective
- Venema
- 2006
(Show Context)
Citation Context ...ableau game, and the announced EXPTIME upper bound follows once we can ensure that legality of moves in the tableau game can be decided in exponential time. Related Work. Our treatment is inspired by =-=[14, 21, 19]-=-, but we note some important differences. In contrast to [14], we use parity games that directly correspond to tableaux, together with parity automata to detect bad traces. Moreover, owing to the gene... |

10 |
On the expressivity of the modal mu-calculus
- Bradfield
(Show Context)
Citation Context ... and for an extension of coalition logic with fixpoints. 1 Introduction The extension of a modal logic with operators for least and greatest fixpoints leads to a dramatic increase in expressive power =-=[1]-=-. The paradigmatic example is of course the modal µ-calculus [10]. In the same way that the µ-calculus extends the modal logic K, one can freely add fixpoint operators to any propositional modal logic... |

7 |
The complexity of the graded mu-calculus
- Kupferman, Vardi
- 2002
(Show Context)
Citation Context ... complexity even of the modal µ-calculus are all but trivial [22, 4], and µ-calculi arising from other monotone modal logics are largely unstudied, with the notable exception of the graded µ-calculus =-=[12]-=-. Here, we improve on this situation, not by providing a new complexity result for a specific fixpoint logic, but by providing a generic and uniform treatment of modal fixpoint logics on the basis of ... |

3 |
Modular Games for Coalgebraic Fixed Point Logics
- Cı̂rstea, Sadrzadeh
- 2008
(Show Context)
Citation Context ...by describing a characterisation of model checking in terms of parity games that generalises [20, Theorem 1, Chapter 6] to the coalgebraic setting. The modelchecking game is a variant of the one from =-=[3]-=-. A parity game played by ∃ (Éloise) and ∀ (Abelard) is a tuple G = (B∃, B∀, E, Ω) where B = B∃ ∪ B∀ is the disjoint union of positions owned by ∃ and ∀, respectively, E ⊆ B × B indicates the allowed ... |