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Characterizing right inverses for spatial constraint systems with applications to modal logic

Michell Guzmán, Salim Perchy, Camilo Rueda, Frank D. Valencia

2018
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Theoretical Computer Science
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Spatial constraint systems are algebraic structures from concurrent constraint programming to specify spatial and epistemic behavior in multi-agent systems. In this paper spatial constraint systems are used to give an abstract characterization of the notion of normality in modal logic and to derive right inverse/reverse operators for modal languages. In particular, a necessary and sufficient condition for the existence of right inverses is identified and the abstract notion of normality is
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... to correspond to the preservation of finite suprema. Furthermore, a taxonomy of normal right inverses is provided, identifying the greatest normal right inverse as well as the complete family of minimal right inverses. These results are applied to existing modal languages such $ as the weakest normal modal logic, Hennessy-Milner logic, and linear-time temporal logic. Some implications of these results are also discussed in the context of modal concepts such as bisimilarity and inconsistency invariance. Keywords: constraint systems, concurrent constraint programming, concurrency theory, modal logic, inverse operators. lief or time, among others. For example, in doxastic modal logic, the logic of belief, the formula i φ (often written as B i φ) specifies that agent i believes φ and the formula i ¬ j ψ specifies that agent i believes that the agent j does not believe ψ. We shall also be interested in inverse modalities. An operator −1 i is a (right) inverse modality for i if the formula i −1 i φ is logically equivalent to φ. Inverse operators arise as, among others, past operators in temporal logic [8], utterances in epistemic logic [9], 5 and backward moves in modal logic for concurrency [10] Kripke Semantics. The most representative semantic models for modal logics are Kripke Structures (KS) [11]. A KS M can be represented as a state graph with n ≥ 1 transition relations 1 −→ M , 2 −→ M , . . . , n −→ M and a function π M that specifies the set of propositions π M (s) that are true at each state (or world) s of M . A pointed KS is a pair (M, s) where s is a state of M . We shall say that (M, s) is model of (or satisfies) a propositional formula p if p ∈ π M (s). Boolean operators are defined as usual; (M, s) is a model of φ ∧ ψ if it is a model of both φ and ψ, and it is a model of ¬φ if it is not model of φ. For the modal case, (M, s) is said to be a model of i φ if (M, t) is a model of φ for every t reachable from s through an i-labelled transition; i.e. s i −→ M t. We shall use [[ φ ]] to denote the set of all models of φ. Different families of KS give rise to different modal logics. For example, the theorems of the S5 epistemic logic are those modal formulae that are satisfied by all pointed KS whose transition relations are equivalences. Normal Modal Operators. In modal logic one is often interested in normal modalities: Roughly, a modal operator i is normal in a given modal logic system if (1) i φ is a theorem whenever φ is a theorem, and (2) Normal modalities are ubiquitous in logic; e.g., the box operator in the the logic system K and its extensions [11] , and the always and next operators as well as the weak past operator from temporal logic [12] , the necessity operator from Hennessy-Milner (HM) logic [13], the knowledge operator from epistemic logic, the belief operator from doxastic logic [14] are all normal. In fact, any operator i whose models are defined with the above-mentioned Kripke semantics, is normal. This paper. Although the notion of spatial constraint system is intended to give an algebraic account of spatial and epistemic assertions, we shall show in this paper that it is sufficiently robust to give an algebraic account of more general modal logic assertions. The main focus of this paper is the study of the above-mentioned extrusion problem 6

doi:10.1016/j.tcs.2018.05.022
fatcat:jv3tihyc4zcepja5u5jkyrqddy