Computation of the greatest right and left invariant fuzzy quasi-orders and fuzzy equivalences

Mičić, Ivana; Jančić, Zorana; Stanimirović, Stefan

doi:10.1016/j.fss.2017.09.004

Cited by 22 publications

(6 citation statements)

References 38 publications

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“…Note that this technique has already given faster algorithms that compute crisp bisimulations for fuzzy automata (cf. [32]). Moreover, since the way to compute approximate solutions that are maximal fuzzy preorders is still an open problem, one direction in future work will be to develop a new methodology to solve this problem.…”

Section: Discussionmentioning

confidence: 99%

“…In the end, if we seek for the greatest fuzzy preorder that is an exact solution to (31) (case when the approximation degree has its greatest value x = 1), then Algorithm 2 does not terminate. Suppose we employ any of the alternative ways to obtain this exact solution (for example, to take a limit of the convergent sequence generated by the Algorithm, see [32] for more details). In that case, we obtain a fuzzy relation in which all rows (i.e., all columns) are mutually different.…”

Section: Applications In Aggregation Of Fuzzy Networkmentioning

confidence: 99%

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On the solvability of weakly linear systems of fuzzy relation equations

Stanimirovic,

Micic

2022

Preprint

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Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on finding solutions and solvability criteria for such systems. The situation is quite different with the so-called weakly linear systems, in which an unknown fuzzy relation is on both sides of the equation or inequality. Precisely, the scholars have only given the characterization of the set of exact solutions to such systems. This paper describes the set of fuzzy relations that solve weakly linear systems to a certain degree and provides ways to compute them. We pay special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. We establish additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. We demonstrate the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks.

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Section: Discussionmentioning

confidence: 99%

Section: Applications In Aggregation Of Fuzzy Networkmentioning

confidence: 99%

On the solvability of weakly linear systems of fuzzy relation equations

Stanimirovic,

Micic

2022

Preprint

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“…Later Micić et at. [18] provided algorithms with the complexity O(ln 5 ) for computing the greatest right/left invariant fuzzy quasi-order/equivalence of a finite fuzzy automaton, where n is the number of states of the considered automaton and l is the number of different fuzzy values appearing during the computation. These relations are closely related to the fuzzy simulations/bisimulations studied in [7,8].…”

Section: Related Workmentioning

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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“…Right and left invariant fuzzy relations were studied in [14,15,36,43]. Their definitions are based on the corresponding crisp counterparts.…”

Section: Introductionmentioning

confidence: 99%

“…above mentioned references and also [31,35,44]). The greatest right and left invariant fuzzy quasi-orders and equivalences were calculated in [14,15,43] by algorithms that approximate the greatest fixpoint by the means of the Kleene fixpoint theorem, and in [36] by partition refinement algorithms. Both variants run in the same, polynomial-time complexity, and rely on the usage of the right and left residuals of fuzzy relations and fuzzy subsets.…”

Section: Introductionmentioning

confidence: 99%

Improved algorithms for computing the greatest right and left invariant Boolean matrices and their application

Stanimirović

Stamenković

Ćirić

2019

Filomat

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We define right and left invariant matrices as Boolean matrices that are solutions to certain systems of matrix equations and inequalities over additively idempotent semirings. We provide improved algorithms for computing the greatest right and left invariant equivalence and quasi-order matrices. The improvements are based on the usage of the well-known partition refinement technique. Afterwards, we present the application of right invariant matrices in the determinization of weighted automata over additively idempotent, commutative and zero-divisor free semirings. In particular, we provide improvements of the well-known determinization method of weighted automata over tropical semirings given by Mohri [Computational Linguistics 23 (2) (1997) 269-311].

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Computation of the greatest right and left invariant fuzzy quasi-orders and fuzzy equivalences

Cited by 22 publications

References 38 publications

On the solvability of weakly linear systems of fuzzy relation equations

On the solvability of weakly linear systems of fuzzy relation equations

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

Improved algorithms for computing the greatest right and left invariant Boolean matrices and their application

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