## Exemplaric Expressivity of Modal Logics (2008)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Jacobs08exemplaricexpressivity,

author = {Bart Jacobs and Ana Sokolova},

title = {Exemplaric Expressivity of Modal Logics },

year = {2008}

}

### OpenURL

### Abstract

This paper investigates expressivity of modal logics for transition systems, multitransition systems, Markov chains, and Markov processes, as coalgebras of the powerset, finitely supported multiset, finitely supported distribution, and measure functor, respectively. Expressivity means that logically indistinguishable states, satisfying the same formulas, are behaviourally indistinguishable. The investigation is based on the framework of dual adjunctions between spaces and logics and focuses on a crucial injectivity property. The approach is generic both in the choice of systems and modalities, and in the choice of a “base logic”. Most of these expressivity results are already known, but the applicability of the uniform setting of dual adjunctions to these particular examples is what constitutes the contribution of the paper.

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Citation Context ...n logic) for multitransition systems already exists [31]. 2� � There is also already an expressivity result for Markov chains with the standard modalities and Boolean logic (including negation), cf. =-=[7,25]-=-. Here, we give a proof that finite conjunctions suffice for expressivity for both multitransition systems and Markov chains, just as they do for non-discrete probabilistic systems [10,8]. Then we ref... |

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Citation Context ... an important role for expressivity of the conjunction fragment of suitable modal logics, as we demonstrate below. 9Markov processes On the category Meas we consider the Giry functor (or monad) from =-=[12]-=-. It maps a measure space X = (X, SX) to the space G(X ) = (GX , SG(X )) of subprobability measures ϕ: SX → [0, 1], satisfying ϕ(∅) = 0 and ϕ( ⋃ i Mi) = ∑ i ϕ(Mi) for countable unions of pairwise disj... |

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Citation Context ...ant powerset”, F = “upsets”; or with C = “topological spaces” and A = “frames”. Such situations are studied systematically in [18], and more recently also in the context of coalgebras and modal logic =-=[21,20,5,6,19]-=-. Typically the functor P describes predicates on spaces and the functor F theories of logical models. In this situation it is important to keep track of the direction of arrows. To be explicit, the (... |

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16 |
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Citation Context ...ic and logic with only finite conjunctions, respectively. Section 2 will describe the adjunctions involved. Similar adjunctions have been used in process semantics (see e.g. [1]) or more generally in =-=[18]-=-. Section 3 will enrich these dual adjunctions with endofunctors like T and L in the above diagram (1). It also contains two “folklore” results about the natural transformation involved (the σ in (1))... |

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Citation Context ...modalities and only finite conjunctions. This expressivity result was first shown in [9,10] for Markov processes over analytic spaces, and recently for general Markov processes over any measure space =-=[8]-=-. It is common in the categorial treatment of non-discrete probabilistic systems (cf. [11,9,10]) to make the detour through analytic or Polish spaces. The main reason is that bisimilarity (in terms of... |

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(Show Context)
Citation Context ...ant powerset”, F = “upsets”; or with C = “topological spaces” and A = “frames”. Such situations are studied systematically in [18], and more recently also in the context of coalgebras and modal logic =-=[21,20,5,6,19]-=-. Typically the functor P describes predicates on spaces and the functor F theories of logical models. In this situation it is important to keep track of the direction of arrows. To be explicit, the (... |

4 |
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Citation Context ...0] for Markov processes over analytic spaces, and recently for general Markov processes over any measure space [8]. It is common in the categorial treatment of non-discrete probabilistic systems (cf. =-=[11,9,10]-=-) to make the detour through analytic or Polish spaces. The main reason is that bisimilarity (in terms of spans) can not be described in general measure spaces, due to non-existence of pullbacks. Howe... |

3 |
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Citation Context ...lgebras (of the same functor). One then requires that there exist coalgebra homomorphisms f and g with f(x) = g(y). This formulation is equivalent to the previous one in categories with pushouts, see =-=[28]-=-. Behavioural equivalence coincides with bisimilarity in case the functor involved preserves weak pullbacks. In the context of expressivity of modal logics behavioural equivalence works better, as com... |

1 | LFCS-93-277. Also available as Aarhus Univ - rep |