## Coalgebraic modal logic beyond Sets (2007)

Venue: | In MFPS XXIII |

Citations: | 11 - 3 self |

### BibTeX

@INPROCEEDINGS{Klin07coalgebraicmodal,

author = {Bartek Klin},

title = {Coalgebraic modal logic beyond Sets},

booktitle = {In MFPS XXIII},

year = {2007},

pages = {177--201},

publisher = {Elsevier}

}

### OpenURL

### Abstract

Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be

### Citations

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Citation Context ... for useful discussions, and to Alexander Kurz and Sam Staton for comments on a previous draft of this paper. 2s2 Preliminaries Klin The reader is assumed to be acquainted with basic category theory; =-=[2,22]-=- are good references. An epimorphism e : X → Y is strong if for every commutative square (i) with m mono there exists a unique diagonal d : Y → U such that (ii) commutes. X e �� �� Y f �� U � � g �� m... |

511 | Algebraic laws for nondeterminism and concurrency
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Citation Context ...systems are normally specified with a modal logic. Various logics have been developed to describe properties of different kinds of systems, e.g., Hennessy-Milner logic for labelled transition systems =-=[14]-=-, probabilistic modal logic [17] for probabilistic systems, or logics for systems with name binding [24,8]. Importantly, such logics are expressive, i.e., they characterize their respective notions of... |

319 | Universal coalgebra, a theory of systems
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Citation Context ...ominal sets. Keywords: coalgebra, modal logic, locally presentable category 1 Introduction In recent years, coalgebra has received much attention as a unifying abstract approach to transition systems =-=[29,16]-=-. Many kinds of systems considered in theoretical computer science, including labelled, probabilistic and timed ones, are modeled as coalgebras for certain functors (called behaviour functors in this ... |

306 | The linear time–branching time spectrum II: the semantics of sequential systems with silent moves
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Citation Context ...ive notions of bisimilarity. However, non-expressive fragments of these logics are also often used to characterize other notions of process equivalence, e.g., trace equivalence or testing equivalence =-=[13]-=-. A successful abstract theory of transition systems must provide a general perspective on modal logics and their properties. The first abstract approach to logics for coalgebras was coalgebraic logic... |

242 | A tutorial on (co)algebras and (co)induction
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Citation Context ...ominal sets. Keywords: coalgebra, modal logic, locally presentable category 1 Introduction In recent years, coalgebra has received much attention as a unifying abstract approach to transition systems =-=[29,16]-=-. Many kinds of systems considered in theoretical computer science, including labelled, probabilistic and timed ones, are modeled as coalgebras for certain functors (called behaviour functors in this ... |

223 | A new approach to abstract syntax with variable binding
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Citation Context ...φ) = ff y∈δ therefore ρ⋆ Φ (β) �= ρ⋆ Φ (γ) and ρ⋆ Φ is pointwise monic. 6.2 Nominal Sets and Systems with Name Binding We begin by recalling the basics of nominal sets. For more information, see e.g. =-=[12]-=-. 15 ψysKlin Throughout this section, fix a countably infinite set N = {a, b, c, . . .} of names. An action of the symmetric group Sym(N ) (i.e., the group of permutations of N ) on a set X is a funct... |

219 |
Locally presentable and accessible categories
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Citation Context ...ny regular cardinal κ. All definitions, results and proofs given in this paper work for the more general case with no change. For more information and intuition on locally presentable categories, see =-=[3,23]-=-. For an endofunctor L on a category C, an algebra is an object X (the carrier), with a map g : LX → X (the structure). An algebra morphism from g : LX → X to h : LY → Y is a map f : X → Y in C such t... |

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Terminal coalgebras in well-founded set theory, Theoret
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Citation Context ...rtesian closed and strongly locally finitely presentable, and any set with at least two elements is an internal cogenerator. Moreover, all functors on Set preserve monos with nonempty domains, and in =-=[4]-=- it was shown how to modify any functor on Set so that it preserve all monos, without a substantial change in its category of coalgebras. For a specific application, consider B = Pω(A×−) for a fixed s... |

110 | Modal logic for mobile processes
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Citation Context ...ties of different kinds of systems, e.g., Hennessy-Milner logic for labelled transition systems [14], probabilistic modal logic [17] for probabilistic systems, or logics for systems with name binding =-=[24,8]-=-. Importantly, such logics are expressive, i.e., they characterize their respective notions of bisimilarity. However, non-expressive fragments of these logics are also often used to characterize other... |

92 | Coalgebraic logic
- Moss
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Citation Context ...ssful abstract theory of transition systems must provide a general perspective on modal logics and their properties. The first abstract approach to logics for coalgebras was coalgebraic logic of Moss =-=[25]-=-, providing expressive logics for essentially all functors on Set. However 1 Supported by EPSRC grant EP/D039045/1. 2 Email: bklin@inf.ed.ac.uk c○2007 Published by Elsevier Science B. V.sKlin general,... |

70 | Probabilistic extensions of process algebras
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Citation Context ...ith a modal logic. Various logics have been developed to describe properties of different kinds of systems, e.g., Hennessy-Milner logic for labelled transition systems [14], probabilistic modal logic =-=[17]-=- for probabilistic systems, or logics for systems with name binding [24,8]. Importantly, such logics are expressive, i.e., they characterize their respective notions of bisimilarity. However, non-expr... |

68 |
Accessible Categories: the Foundations of Categorical Model Theory
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Citation Context ...ny regular cardinal κ. All definitions, results and proofs given in this paper work for the more general case with no change. For more information and intuition on locally presentable categories, see =-=[3,23]-=-. For an endofunctor L on a category C, an algebra is an object X (the carrier), with a map g : LX → X (the structure). An algebra morphism from g : LX → X to h : LY → Y is a map f : X → Y in C such t... |

61 |
Specifying coalgebras with modal logic. Theoret
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Citation Context ...pplications of the behaviour functor to formulas, and it does not provide simple and natural modalities like those known from Hennessy-Milner or similar logics. On the other hand, logics developed in =-=[15,19,26,28]-=- are close to their usual concrete presentations, but their expressivity depends on some conditions imposed on the behaviour functor. For example, modalities in [26] are predicate liftings, which map ... |

48 | Semantics of name and value passing
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Citation Context ...d as coalgebras for certain functors (called behaviour functors in this context) on the category Set of sets and functions. Other categories have also been considered, for example presheaf categories =-=[11]-=- or the category Nom of nominal sets [10] to model process algebras with name binding. The coalgebraic approach provides an abstract view on notions of coinduction and bisimulation. Properties of tran... |

39 | Expressivity of coalgebraic modal logic: the limits and beyond
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Citation Context ... predicates on BX, where B is the behaviour functor, and the resulting modal logic is expressive provided enough predicate liftings exist for B. This approach was analyzed and generalized by Schröder =-=[30]-=-, who noted that predicate liftings are equivalent to functions B2 → 2 and considered polyadic modal logic, where modalities of any arity, such as functions B(2 n ) → 2, are allowed. He then proved po... |

33 |
Duality for logics of transition systems
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Citation Context ...gic, and the proof of its expressivity, to accessible functors on locally presentable categories that satisfy some additional conditions. Our approach is inspired by recent work by Kurz and Bonsangue =-=[6,7,20,21]-=-, who use Stone dualities to obtain logics for coalgebras on arbitrary categories, and by that by Pavlovic, Mislove and Worrell [27], who exploit logical connections between data and tests to develop ... |

33 | Coalgebras and modal logic
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Citation Context ...pplications of the behaviour functor to formulas, and it does not provide simple and natural modalities like those known from Hennessy-Milner or similar logics. On the other hand, logics developed in =-=[15,19,26,28]-=- are close to their usual concrete presentations, but their expressivity depends on some conditions imposed on the behaviour functor. For example, modalities in [26] are predicate liftings, which map ... |

31 | Semantical Principles in the Modal Logic of Coalgebras
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Citation Context ...pplications of the behaviour functor to formulas, and it does not provide simple and natural modalities like those known from Hennessy-Milner or similar logics. On the other hand, logics developed in =-=[15,19,26,28]-=- are close to their usual concrete presentations, but their expressivity depends on some conditions imposed on the behaviour functor. For example, modalities in [26] are predicate liftings, which map ... |

28 |
Presenting Functors by Operations and Equations
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Citation Context ...gic, and the proof of its expressivity, to accessible functors on locally presentable categories that satisfy some additional conditions. Our approach is inspired by recent work by Kurz and Bonsangue =-=[6,7,20,21]-=-, who use Stone dualities to obtain logics for coalgebras on arbitrary categories, and by that by Pavlovic, Mislove and Worrell [27], who exploit logical connections between data and tests to develop ... |

25 |
Comparing operational models of name-passing process calculi
- Fiore, Staton
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Citation Context ...a generator by (iii), therefore h = k. Consider the following functor on Nom: BX = Pω(N × X N + N × N × X + N × [N ]X + X). This is the functor for late bisimulation on systems with name binding (see =-=[11,10,9,5]-=- for a comparison), i.e., observational equivalence coincides with late bisimilarity. B is finitary on Nom. To apply the framework of polyadic modal logic, choose C = D = Nom and F = G = 2 − . As we h... |

20 | A congruence rule format for name-passing process calculi
- Fiore, Staton
- 2009
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Citation Context ...led behaviour functors in this context) on the category Set of sets and functions. Other categories have also been considered, for example presheaf categories [11] or the category Nom of nominal sets =-=[10]-=- to model process algebras with name binding. The coalgebraic approach provides an abstract view on notions of coinduction and bisimulation. Properties of transition systems are normally specified wit... |

20 | Towards a duality result in the modal logic of coalgebras
- Jacobs
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Citation Context |

16 |
Abstract and Concrete Categories, WileyInterscience
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Citation Context ... for useful discussions, and to Alexander Kurz and Sam Staton for comments on a previous draft of this paper. 2s2 Preliminaries Klin The reader is assumed to be acquainted with basic category theory; =-=[2,22]-=- are good references. An epimorphism e : X → Y is strong if for every commutative square (i) with m mono there exists a unique diagonal d : Y → U such that (ii) commutes. X e �� �� Y f �� U � � g �� m... |

14 |
Coalgebras and their logics
- Kurz
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Citation Context ...gic, and the proof of its expressivity, to accessible functors on locally presentable categories that satisfy some additional conditions. Our approach is inspired by recent work by Kurz and Bonsangue =-=[6,7,20,21]-=-, who use Stone dualities to obtain logics for coalgebras on arbitrary categories, and by that by Pavlovic, Mislove and Worrell [27], who exploit logical connections between data and tests to develop ... |

13 | Testing semantics: Connecting processes and process logics
- Pavlović, Mislove, et al.
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Citation Context ...ur approach is inspired by recent work by Kurz and Bonsangue [6,7,20,21], who use Stone dualities to obtain logics for coalgebras on arbitrary categories, and by that by Pavlovic, Mislove and Worrell =-=[27]-=-, who exploit logical connections between data and tests to develop an abstract theory of testing. In those works, as in the present paper, contravariant adjunctions provide the infrastructure for lin... |

11 | Proof Systems for Pi-Calculus Logics
- Dam
(Show Context)
Citation Context ...ties of different kinds of systems, e.g., Hennessy-Milner logic for labelled transition systems [14], probabilistic modal logic [17] for probabilistic systems, or logics for systems with name binding =-=[24,8]-=-. Importantly, such logics are expressive, i.e., they characterize their respective notions of bisimilarity. However, non-expressive fragments of these logics are also often used to characterize other... |

10 |
On final coalgebras of continuous functors
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Citation Context ...e equated by [ ] ♭ . This functional presentation is easily generalized to cover logics where another set is used for “truth values”; for example, in some probabilistic logics the continuous interval =-=[0, 1]-=- is used instead of 2. Abstracting from the category of sets, consider any symmetric monoidal closed category (C, ⊗, ⊸) with a chosen object Ω. The contravariant internal hom-functor 4sKlin − ⊸ Ω on C... |

10 |
Strongly complete logics for coalgebras
- Kurz, Rosick´y
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Citation Context |

8 | The least fibred lifting and the expressivity of coalgebraic modal logic
- Klin
- 2005
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Citation Context ...d expressive logics (and to present their non-expressive fragments) for transition systems. In §6.1, the familiar setting of sets and functions is studied. Schröder’s polyadic coalgebraic modal logic =-=[30,18]-=- is shown to be a special case of the present approach, hence all examples covered there are examples here a well. However, for completeness we describe the classical example of finitary Hennessy-Miln... |