## Coalgebraic modal logic: Soundness, completeness and decidability of local consequence (2002)

Venue: | Theoret. Comput. Sci |

Citations: | 50 - 24 self |

### BibTeX

@ARTICLE{Pattinson02coalgebraicmodal,

author = {Dirk Pattinson},

title = {Coalgebraic modal logic: Soundness, completeness and decidability of local consequence},

journal = {Theoret. Comput. Sci},

year = {2002},

volume = {309},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations. 1

### Citations

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(Show Context)
Citation Context ...subsets c,c ′ ⊆ C of C. c ⊆ c ′ =⇒ λ(C)(c) ⊆ λ(C)(c ′ ) Informally, a predicate lifting λ maps predicates over a set C to predicates over the type of observations TC, as defined by T. Naturality (see =-=[9]-=-) is customarily expressed diagrammatically by requiring P(Y ) λ(Y ) �� f −1 �� P(TY ) (Tf) −1 �� P(X) �� P(TX) λ(X) to commute for all functions f : X → Y (where P is the object part of the contravar... |

444 |
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Citation Context ...odal logic and proves soundness, completeness and decidability results. Coalgebraic modal logic is a generalisation of basic propositional modal logic (i.e. of the “basic logic” in the terminology of =-=[3]-=-) and allows us to reason about states of coalgebras for an endofunctor on the category of sets. Coalgebras for an endofunctor on the category of sets provide a uniform framework for modelling state b... |

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Citation Context ...-called terminal sequence (T n 1)n∈ω of the underlying endofunctor T. The terminal sequence (iterated through the class of all ordinal numbers) is frequently 9sused to construct final coalgebras (see =-=[2, 1, 16]-=-). Since we are working with finitary logic, there is no need to iterate the construction further than ω. Using Theorem 4.9, we can determine validity φ |= ψ by just looking at one model: it suffices ... |

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Citation Context ...tems, Kripke models and frames, Moore and Mealy automata and deterministic automata, to name but a few. The use of modal logic as a language for reasoning about coalgebras was first suggested by Moss =-=[10]-=-. The construction of his coalgebraic logic can 1sbe carried out for a large (semantically defined) class of endofunctors. This generality does not come for free: the language of coalgebraic logic is ... |

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Citation Context ...introduces the concept of predicate liftings and shows how predicate liftings give rise to modal languages, interpreted over coalgebras. Predicate liftings were first considered by Jacobs and Hermida =-=[6]-=- in the context of co-induction principles and later by Rößiger [12] and Jacobs [5] in the context of modal logic. There, predicate liftings appear as syntactically defined entities, and naturality is... |

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Citation Context ...ogic is nonstandard in the sense that it does not have modal operators and instead uses functor application to construct formulas. It also lacks a complete axiomatisation. Other approaches, including =-=[5, 8, 12, 13]-=-, restrict attention to a syntactically defined class of endofunctors. This restriction allows the use of a standard language and is amenable to a complete axiomatisation. The present article aims at ... |

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Citation Context ...for all φ ∈ L(Λ). We sometimes write c |=γ φ for c ∈ [φ]γ and take the remaining propositional connectives to be defined as usual. Remark 4.2. Suppose (C,γ) and (D,δ) are T-coalgebras. Following Kurz =-=[7]-=-, we call a pair of states (c,d) ∈ C × D behaviourally equivalent (which we denote by c ∼ d), if there exists a T-coalgebra (E,ǫ) and a pair of coalgebra homomorphisms f : (C,γ) → (E,ǫ) and g : (D,δ) ... |

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Citation Context ...ogic is nonstandard in the sense that it does not have modal operators and instead uses functor application to construct formulas. It also lacks a complete axiomatisation. Other approaches, including =-=[5, 8, 12, 13]-=-, restrict attention to a syntactically defined class of endofunctors. This restriction allows the use of a standard language and is amenable to a complete axiomatisation. The present article aims at ... |

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Citation Context ...ogic is nonstandard in the sense that it does not have modal operators and instead uses functor application to construct formulas. It also lacks a complete axiomatisation. Other approaches, including =-=[5, 8, 12, 13]-=-, restrict attention to a syntactically defined class of endofunctors. This restriction allows the use of a standard language and is amenable to a complete axiomatisation. The present article aims at ... |

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(Show Context)
Citation Context ...al logic and have established soundness, completeness and decidability in the finitary case by induction along the terminal sequence of the underlying endofunctor. A similar argument has been used in =-=[11]-=- in order to establish an expressivity theorem (albeit in a slightly different context). To our knowledge, the use of induction along the terminal sequence is a novel approach to soundness, completene... |

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On the greatest fixed point of a set functor
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Citation Context ...-called terminal sequence (T n 1)n∈ω of the underlying endofunctor T. The terminal sequence (iterated through the class of all ordinal numbers) is frequently 9sused to construct final coalgebras (see =-=[2, 1, 16]-=-). Since we are working with finitary logic, there is no need to iterate the construction further than ω. Using Theorem 4.9, we can determine validity φ |= ψ by just looking at one model: it suffices ... |

14 |
Logics of Time and Computation, volume 7 of CSLI Lecture Notes
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(Show Context)
Citation Context ... atomic propositions and modal operators uniformly via the concept of predicate lifting. This is demonstrated in the next example, which uses the standard definition of the modal operator ✷ (see e.g. =-=[4, 15]-=-). Example 3.5. Suppose A is a set (of atomic propositions) and consider TX = P(X) × P(A) as in Example 2.2. Let γ : C → TC. The first projection π1(X) : TX = P(X) × P(A) → P(X) gives rise to a predic... |