Modal Equivalence and Bisimilarity in Many-valued Modal Logics with Many-valued Accessibility Relations

Diaconescu, Denisa

doi:10.3233/fi-2020-1920

Cited by 13 publications

(5 citation statements)

References 23 publications

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“…Given two fuzzy systems of the same kind such as fuzzy automata, fuzzy LTSs, fuzzy/weighted social networks, fuzzy Kripke models or fuzzy interpretations in a description logic, one can define a simulation or bisimulation between them either as a crisp relation or as a fuzzy relation between the sets of states (respectively, actors or individuals) of the systems. Bisimulations as crisp relations have been studied for fuzzy LTSs [4,5,[29][30][31], many-valued/fuzzy modal logics [9][10][11], weighted/fuzzy automata [8,32] and fuzzy description logics [19]. Simulations as crisp relations have been studied for fuzzy LTSs [20,24,31], weighted/fuzzy automata [8,32] and fuzzy description logics [16].…”

Section: Related Workmentioning

confidence: 99%

Fuzzy Simulations and Bisimulations between Fuzzy Automata

Nguyen¹

2022

Preprint

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Simulations and bisimulations between two fuzzy automata over a complete residuated lattice were defined by Ćirić et al. (2012) as fuzzy relations between the sets of states of the automata. However, they act as a crisp relationship between the automata. In particular, if there exists a (forward) bisimulation between two fuzzy automata, then the fuzzy languages recognized by them are crisply equal. Approximate simulations and bisimulations introduced by Stanimirović et al. (2020) aim at fuzzifying this phenomenon. However, they are defined only for fuzzy automata over a complete Heyting algebra and do not give the exact relationship between states of the automata. In this article, we introduce and study fuzzy simulations and bisimulations between fuzzy automata over a complete residuated lattice. These notions are novel and have good properties. They are defined for fuzzy automata over any complete residuated lattice. We prove that the fuzzy language recognized by a fuzzy automaton is fuzzily preserved by fuzzy simulations and fuzzily invariant under fuzzy bisimulations. We also prove that the notions of fuzzy simulation and bisimulation have the Hennessy-Milner properties, which are a logical characterization of the greatest fuzzy simulation or bisimulation between two fuzzy automata. In addition, we provide results showing that our notions of fuzzy simulation and bisimulation are more general and refined than the notions of simulation and bisimulation introduced by Ćirić et al. and the notions of approximate simulation and bisimulation introduced by Stanimirović et al.

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Section: Related Workmentioning

confidence: 99%

Fuzzy Simulations and Bisimulations between Fuzzy Automata

Nguyen¹

2022

Preprint

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“…When moving to residuated lattices, the use of the Baaz projection operator is more suitable [6]. Crisp bisimulations have been studied for fuzzy transition systems [10][11][12][13], weighted automata [14], Heyting-valued modal logics [15], fuzzy/many-valued modal logics [8,16,17] and FDLs [7,9]. Fuzzy bisimulations have been studied for fuzzy automata [18][19][20], weighted/fuzzy social networks [21,22], fuzzy modal logics [6,8] and FDLs [9].…”

Section: Related Workmentioning

confidence: 99%

“…-Consider the case where U ∈ Φ. Since I = ∅, by (18) and Condition (17) with I, I ′ and Z replaced by I ′ , I/ ∼ Φ and Z ′′ , respectively, for every 1 ≤ i ≤ n, there exists…”

Section: Minimizing Finite Fuzzy Interpretations: Logical Propertiesmentioning

confidence: 99%

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Minimizing interpretations in fuzzy description logics under the Gödel semantics by using fuzzy bisimulations

Nguyen

Nguyên

2019

IFS

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The problem of minimizing finite fuzzy interpretations in fuzzy description logics (FDLs) is worth studying. For example, the structure of a fuzzy/weighted social network can be treated as a fuzzy interpretation in FDLs, where actors are individuals and actions are roles. Minimizing the structure of a fuzzy/weighted social network makes it more compact, thus making network analysis tasks more efficient. In this work, we study the problem of minimizing a finite fuzzy interpretation in a FDL by using the largest crisp auto-bisimulation. The considered FDLs use the Baaz projection operator and their semantics is specified using an abstract algebra of fuzzy truth values, which can be any linear and complete residuated lattice. We provide an efficient algorithm with a complexity of O((m log l + n) log n) for minimizing a given finite fuzzy interpretation I, where n is the size of the domain of I, m is number of nonzero instances of atomic roles of I and l is the number of different fuzzy values used for instances of atomic roles of I. We prove that the fuzzy interpretation returned by the algorithm is minimal among the ones that preserve fuzzy TBoxes and ABoxes under certain conditions.

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“…The Hennessy-Milner property of fuzzy bisimulations [12,13,21,20] states that, if Z is the greatest fuzzy bisimulation between two image-finite fuzzy graph-based structures G and G ′ , then Z(x, x ′ ) = inf{ϕ G (x) ⇔ ϕ G ′ (x ′ ) | ϕ is a formula of a certain fuzzy modal/description logic}, where ⇔ is the fuzzy equivalence in the considered logic. Crisp bisimulations have been defined and studied for fuzzy transition systems [3,4,31,29,30], weighted automata [9], fuzzy modal logics [11,13,17,10] and fuzzy description logics [21]. Fuzzy bisimulations have been defined and studied for fuzzy automata [7,8], weighted/fuzzy social networks [12,16], fuzzy modal logics [13,20] and fuzzy description logics [21,22,24].…”

Section: Introductionmentioning

confidence: 99%

Computing the Fuzzy Partition Corresponding to the Greatest Fuzzy Auto-Bisimulation of a Fuzzy Graph-Based Structure

Nguyen¹

2021

Preprint

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Fuzzy graph-based structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretations in fuzzy description logics are useful in various applications. Given two states, two actors or two individuals x and x ′ in such structures G and G ′ , respectively, the similarity degree between them can be defined to be Z(x, x ′ ), where Z is the greatest fuzzy bisimulation between G and G ′ with respect to some t-norm-based fuzzy logic. Such a similarity measure has the Hennessy-Milner property of fuzzy bisimulations as a strong logical foundation. A fuzzy bisimulation between a fuzzy structure G and itself is called a fuzzy auto-bisimulation of G. The greatest fuzzy auto-bisimulation of an image-finite fuzzy graph-based structure is a fuzzy equivalence relation. It is useful for classification and clustering.In this paper, we design an efficient algorithm with the complexity O((m log l + n) log n) for computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a finite fuzzy labeled graph G under the Gödel semantics, where n, m and l are the number of vertices, the number of non-zero edges and the number of different fuzzy degrees of edges of G, respectively. Our notion of fuzzy partition is novel, defined only for finite sets with respect to the Gödel t-norm, with the aim to facilitate the computation of the greatest fuzzy auto-bisimulation. By using that algorithm, we also provide an algorithm with the complexity O(m • log l • log n + n 2 ) for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the Gödel semantics. This latter algorithm is better (has a lower complexity order) than the previously known algorithms for the considered problem. Our algorithms can be restated for the other mentioned fuzzy graphbased structures.

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Modal Equivalence and Bisimilarity in Many-valued Modal Logics with Many-valued Accessibility Relations

Cited by 13 publications

References 23 publications

Fuzzy Simulations and Bisimulations between Fuzzy Automata

Fuzzy Simulations and Bisimulations between Fuzzy Automata

Minimizing interpretations in fuzzy description logics under the Gödel semantics by using fuzzy bisimulations

Computing the Fuzzy Partition Corresponding to the Greatest Fuzzy Auto-Bisimulation of a Fuzzy Graph-Based Structure

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