## Automata and fixed point logics: a coalgebraic perspective (2004)

Venue: | Electronic Notes in Theoretical Computer Science |

Citations: | 15 - 7 self |

### BibTeX

@ARTICLE{Venema04automataand,

author = {Yde Venema},

title = {Automata and fixed point logics: a coalgebraic perspective},

journal = {Electronic Notes in Theoretical Computer Science},

year = {2004},

volume = {204},

pages = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of F-automata, devices that operate on pointed F-coalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an F-automaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed F-coalgebras in which p holds.

### Citations

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Citation Context ...tomata are used to classify infinite objects like words, trees or graphs. Interestingly, this research area has provided not only fundamental theoretical results, such as Rabin’s decidability theorem =-=[19]-=-, but also quite concrete applications in computer science, such as tools for the automatic verification of reactive systems, see for instance [6] on model checking. Over the last ten years, the links... |

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Citation Context ...ult concerning Borel games, due to Martin [14]. The fact that winning strategies in parity games can always be taken to be history free, was independently proved in Mostowski [16] and Emerson & Jutla =-=[8]-=-. 3 Automata on infinite objects In this section we intend to supply a gentle introduction to our general definition of an automaton operating on coalgebras, by discussing the shape of some finite aut... |

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Citation Context ...as a history free strategy which is winning from any position in Wini(G). 5sThe determinacy of parity games follows from a far more general game-theoretic result concerning Borel games, due to Martin =-=[14]-=-. The fact that winning strategies in parity games can always be taken to be history free, was independently proved in Mostowski [16] and Emerson & Jutla [8]. 3 Automata on infinite objects In this se... |

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Citation Context ...e paradigmatic examples of coalgebras, as any introduction to the field of coalgebra witnesses. As examples we confine ourselves to mentioning Rutten’s work on automata for finite and infinite words (=-=[21]-=- and [23], respectively). Summarizing the above discussion, we find that the relation between automata theory and (modal) logic has been investigated intensively and successfully, but not uniformly or... |

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Citation Context ...eneral game-theoretic result concerning Borel games, due to Martin [14]. The fact that winning strategies in parity games can always be taken to be history free, was independently proved in Mostowski =-=[16]-=- and Emerson & Jutla [8]. 3 Automata on infinite objects In this section we intend to supply a gentle introduction to our general definition of an automaton operating on coalgebras, by discussing the ... |

53 | Many-sorted coalgebraic modal logic: a model-theoretic study
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Citation Context ... [15]. Over subsequent years, the development and study of modal languages for the specification of properties of coalgebras has been actively pursued and studied by various authors, including Jacobs =-=[9]-=-, Kurz [13], Pattinson [18], and Rößiger [20]. However, given the intended application of coalgebraic modal languages as specification formalisms restricting the behavior of state-based systems, it is... |

50 | Behavioural differential equations: a coinductive calculus of streams, automata, and power series, Theoret
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Citation Context ...matic examples of coalgebras, as any introduction to the field of coalgebra witnesses. As examples we confine ourselves to mentioning Rutten’s work on automata for finite and infinite words ([21] and =-=[23]-=-, respectively). Summarizing the above discussion, we find that the relation between automata theory and (modal) logic has been investigated intensively and successfully, but not uniformly or systemat... |

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Citation Context ...n of automaton is lifted to a category-theoretic level. For instance, there is a series of articles by Arbib and Manes and a theory of functorial automata developed by Adámek, Trnková and others, see =-=[2]-=- (also for further references). Although this work bears some resemblance to ours, there are at least two differences: first, the mentioned research focuses on an algebraic rather than a coalgebraic f... |

43 | A 2-categorical approach to change of base and geometric morphisms i
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Citation Context ... of the functor F by F. It can be proved that an endofunctor on Set can be extended to a relator if and only if it preserves weak pullbacks. This result is usually attributed to Carboni, Kelly & Wood =-=[5]-=-, but it also follows as a special case of an earlier result by Trnková, see [24, Observation 2.10], or [2, section V.2.10] for a proof. In the sequel we will need the following fact; for details, con... |

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Citation Context ...he only work on coalgebraic modal languages in which specimens of fixed point formulas are admitted, or in which the need for coalgebraic modal fixed point logics is discussed, seems to be by Jacobs (=-=[11]-=- and [10], respectively). The coalgebraic perspective on automata may not have been developed so systematically, automata theory contains some of the paradigmatic examples of coalgebras, as any introd... |

33 | Coalgebras and modal logic
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Citation Context ...t and study of modal languages for the specification of properties of coalgebras has been actively pursued and studied by various authors, including Jacobs [9], Kurz [13], Pattinson [18], and Rößiger =-=[20]-=-. However, given the intended application of coalgebraic modal languages as specification formalisms restricting the behavior of state-based systems, it is rather surprising that until now no language... |

30 | Semantical principles in the modal logic of coalgebras
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(Show Context)
Citation Context ...rs, the development and study of modal languages for the specification of properties of coalgebras has been actively pursued and studied by various authors, including Jacobs [9], Kurz [13], Pattinson =-=[18]-=-, and Rößiger [20]. However, given the intended application of coalgebraic modal languages as specification formalisms restricting the behavior of state-based systems, it is rather surprising that unt... |

29 |
A logic for coalgebraic simulation
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Citation Context ...e set of subformulas. Combining ideas from the game semantics for the modal µ-calculus as formulated by Janin & Walukiewicz [12], and the semantic games for coalgebraic languages introduced by Baltag =-=[3]-=-, in Theorem 1 we provide a game-theoretic semantics for this language µL F . Finally, the resemblance between these games and the acceptance games for F-automata leads to the 2smain result of the pap... |

27 |
Automata for the Modal µ-Calculus and Related Results
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Citation Context ...at in many cases, the distinction between automata and formulas has almost disappeared. Of the many interesting results that have been obtained we just mention the connection that Janin & Walukiewicz =-=[12]-=- established between modal fixed point logics, such as the modal µ-calculus, and alternating parity automata operating on labeled transition systems. For an up to date introduction to the world of aut... |

24 | A co-variety-theorem for modal logic
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(Show Context)
Citation Context ...r subsequent years, the development and study of modal languages for the specification of properties of coalgebras has been actively pursued and studied by various authors, including Jacobs [9], Kurz =-=[13]-=-, Pattinson [18], and Rößiger [20]. However, given the intended application of coalgebraic modal languages as specification formalisms restricting the behavior of state-based systems, it is rather sur... |

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Citation Context ...al model of state-based dynamics, and modal logic as a logic for dynamic systems, there are interesting links between the two fields. One of the first to realize this, in the late 1980s, was Abramsky =-=[1]-=-. Barwise & Moss [4], a rich source of material on a great variety of circular systems, contains the outline of Moss’ general approach towards coalgebraic logic, discussed in more detail by Moss [15].... |

18 |
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(Show Context)
Citation Context ...o follows as a special case of an earlier result by Trnková, see [24, Observation 2.10], or [2, section V.2.10] for a proof. In the sequel we will need the following fact; for details, consult Rutten =-=[22]-=-. Fact 2.1 Let F : Set → Set be a functor that preserves weak pullbacks. Then 1. The unique relator F extending F is given, for R ⊆ S × S ′ , by F(R) = F(π) ◦ F(π ′ ) −1 . 2. F is monotone, that is, i... |

16 | Exercises in Coalgebraic Specification
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(Show Context)
Citation Context ...ork on coalgebraic modal languages in which specimens of fixed point formulas are admitted, or in which the need for coalgebraic modal fixed point logics is discussed, seems to be by Jacobs ([11] and =-=[10]-=-, respectively). The coalgebraic perspective on automata may not have been developed so systematically, automata theory contains some of the paradigmatic examples of coalgebras, as any introduction to... |

6 |
Coalgebraic logic. Annals of Pure and
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(Show Context)
Citation Context ...y [1]. Barwise & Moss [4], a rich source of material on a great variety of circular systems, contains the outline of Moss’ general approach towards coalgebraic logic, discussed in more detail by Moss =-=[15]-=-. Over subsequent years, the development and study of modal languages for the specification of properties of coalgebras has been actively pursued and studied by various authors, including Jacobs [9], ... |

5 | Relational automata in a category and theory of languages - Trnková - 1977 |

3 |
Vicious Circles. Volume 60
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Citation Context ...sed dynamics, and modal logic as a logic for dynamic systems, there are interesting links between the two fields. One of the first to realize this, in the late 1980s, was Abramsky [1]. Barwise & Moss =-=[4]-=-, a rich source of material on a great variety of circular systems, contains the outline of Moss’ general approach towards coalgebraic logic, discussed in more detail by Moss [15]. Over subsequent yea... |