## 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.

### Citations

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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 ...latter capture Boolean logic 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 transformat... |

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Citation Context ..., respectively. We then define the functor KO b: MSL → MSL as: KO b(A) = ∐ o∈ b O HV A Here we use that the category MSL has arbitrary coproducts—which follows for instance from Linton’s Theorem (see =-=[2]-=-), using that MSL is algebraic over Sets (and thus cocomplete). A map KO b(A) → B in MSL now corresponds to an Ôindexed family of (monotone) functions V A → V B in PoSets. The family of maps □o: P(X) ... |

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Citation Context ...gation), 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 reformulate the expressivity result of [10,8] within our uniform setting of dual adjunctions. Additionally we elaborate on the relation between the discrete and non-discrete Markov chains/pr... |

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Citation Context ...ely supported discrete (sub)distribution functor on Sets; – finite conjunctions for Markov processes, as coalgebras of the Giry functor on the category of measure spaces. The first point goes back to =-=[13]-=-. Here we cast it in the framework of dual adjunctions, with an explicit description of the “modality” endofunctor L on the category BA of Boolean algebras and its relevant properties. An expressivity... |

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Citation Context ...re is a brief “historical” account of how the emergence of dual adjunctions in logical settings can be understood. For reasoning about functors the idea of predicate lifting was used already early in =-=[14,15]-=-. This involves the extension of a predicate (formula) P ⊆ X to a lifted predicate P ⊆ T X, for an endofunctor T : Sets → Sets whose coalgebras X → T X we wish to study. The notion of invariant arises... |

61 |
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Citation Context ...an be interpreted in terms of next or previous states, with respect to some transition system or, more abstractly, coalgebra. The last few years have shown a rapid development in this (combined) area =-=[23,16,27,22,24,5,31,32,29,7,19]-=-. One of the more interesting aspects is the use of dualities or dual adjunctions. Here is a brief “historical” account of how the emergence of dual adjunctions in logical settings can be understood. ... |

55 | Many-sorted coalgebraic modal logic: a model-theoretic study
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Citation Context ...an be interpreted in terms of next or previous states, with respect to some transition system or, more abstractly, coalgebra. The last few years have shown a rapid development in this (combined) area =-=[23,16,27,22,24,5,31,32,29,7,19]-=-. One of the more interesting aspects is the use of dualities or dual adjunctions. Here is a brief “historical” account of how the emergence of dual adjunctions in logical settings can be understood. ... |

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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 ...ϕ is defined, as before, as supp(ϕ) = {x | ϕ(x) ̸= 0}. A function f: X → Y yields a mapping D f(f): D f(X) → D f(Y ) by D f(f)(ϕ) = λy ∈ Y. ∑ x∈f −1 (y) ϕ(x). A coalgebra X → D f(X) is a Markov chain =-=[33,3]-=-. In this context subdistributions (with sum ≤ 1) are more common than distributions (with sum = 1), but the difference does not really matter here. The functor D f preserves injections. What subsets,... |

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(Show Context)
Citation Context |

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Citation Context ...We now present an expressivity result for general, non-discrete, probabilistic systems and logic with the standard 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. [1... |

33 |
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(Show Context)
Citation Context |

28 |
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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 (... |

20 | Fibrations, Logical Predicates and Indeterminates
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(Show Context)
Citation Context ...re is a brief “historical” account of how the emergence of dual adjunctions in logical settings can be understood. For reasoning about functors the idea of predicate lifting was used already early in =-=[14,15]-=-. This involves the extension of a predicate (formula) P ⊆ X to a lifted predicate P ⊆ T X, for an endofunctor T : Sets → Sets whose coalgebras X → T X we wish to study. The notion of invariant arises... |

18 |
Algebraic semantics for coalgebraic logics
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(Show Context)
Citation Context ...are in one-one correspondence, via the adjunction D ⊣ U, with G-coalgebras D(X) → GD(X) in Meas with carriers discrete measure spaces. X � UGD(X) in Sets ======================== D(X) � GD(X) in Meas =-=(20)-=- We use the term discrete Markov process both for a G-coalgebra with carrier discrete measure space in Meas, and for the corresponding UGD-coalgebra in Sets. The whole picture is shown in the followin... |

17 |
Spaces. Number 3
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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))... |

15 |
Bisimulation and cocongruence for probabilistic systems
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(Show Context)
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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Citation Context |

13 | Testing semantics: Connecting processes and process logics
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- 2006
(Show Context)
Citation Context |

12 |
Logical Predicates and Indeterminates
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(Show Context)
Citation Context ...re is a brief “historical” account of how the emergence of dual adjunctions in logical settings can be understood. For reasoning about functors the idea of predicate lifting was used already early in =-=[14,15]-=-. This involves the extension of a predicate (formula) P ⊆ X to a lifted predicate P ⊆ TX , for an endofunctor T :Sets→ Setswhose coalgebras X → TX we wish to study. The notion of invariant arises via... |

11 | Coalgebraic modal logic beyond sets
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- 2007
(Show Context)
Citation Context |

9 |
Modular proof systems for coalgebraic logics
- Ĉırstea, Pattinson
(Show Context)
Citation Context |

7 | Stone coalgebras
- Kupke, Kurz, et al.
- 2003
(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 (... |

5 |
Eilenberg-Moore algebras for stochastic relations
- Doberkat
(Show Context)
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... |

4 |
An Introduction to the Theory of Coalgebras”, course notes from NASSLLI’2003, available from http://www.indiana.edu/∼nasslli/program.html
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(Show Context)
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 |