Elsevier

Fuzzy Sets and Systems

Volume 301, 15 October 2016, Pages 19-36
Fuzzy Sets and Systems

Logical characterizations of simulation and bisimulation for fuzzy transition systems

https://doi.org/10.1016/j.fss.2015.09.012Get rights and content

Abstract

Simulations and bisimulations are known to be useful for abstracting and comparing formal systems, and they have recently been introduced into fuzzy systems. In this study, we provide sound and complete logical characterizations for simulation and bisimulation, which are defined over fuzzy labeled transition systems via two variants of the Hennessy–Milner Logic. The logic for characterizing fuzzy simulation has neither negation nor disjunction, which is very different from the well-known logical characterizations of probabilistic simulations, although the completeness proofs of our characterization results are inspired by relevant research in probabilistic concurrency theory. The logic for characterizing fuzzy bisimulation also deviates from that for probabilistic bisimulations.

Introduction

The analysis of fuzzy systems has been the subject of active research during the last 60 years and many formalisms have been proposed for modeling them, including fuzzy automata (e.g., see [2], [3], [7], [27], [28], [30], [35], [37], [43]), fuzzy Petri nets [39], fuzzy Markov processes [4], and fuzzy discrete event systems [29], [36], [38].

Recently, a new formal model for fuzzy systems called fuzzy labeled transition systems (FLTSs) was proposed [6], [17], [23]. FLTSs are a natural generalization of the classical labeled transition systems in computer science, where after performing some action, a system evolves from one state into a fuzzy set of successor states instead of a unique state. Many formal description tools for fuzzy systems, such as fuzzy Petri nets and fuzzy discrete event systems [29], [36], are not FLTSs. However, it is possible to translate a system's description in one of these formalisms into an FLTS to represent its behavior.

Bisimulation [33] has been investigated in depth in process algebras because it offers a convenient co-inductive proof technique for establishing behavioral equivalence [31]. Bisimulation has mostly been used for verifying formal systems and it is the foundation of state-aggregation algorithms, which compress models by merging bisimilar states. State aggregation is used routinely as a preprocessing step before model checking [1], [18]. Recently, bisimulation has been introduced into fuzzy systems. For example, Cao et al. [6] considered bisimulations for FLTSs. Bisimulation-based reasoning also appeared for fuzzy automata and fuzzy discrete event systems [10], [11], [17], [32], [41], [44].

Following a seminal study that explored the connection between bisimulation and modal logic [21], many studies have characterized various types of bisimulations using appropriate logics, e.g., [15], [16], [22], [26]. A logic characterizes a bisimulation soundly and completely when two states are bisimilar if and only if they satisfy the same set of logical formulae. The significance of logical characterizations is twofold. Based on a sound and complete logical characterization, the problem of checking whether two states are bisimilar is converted into a logical judgment of whether two states satisfy the same set of logical formulae, which can benefit from traditional logic theories and be assisted by some practical tools. A logical characterization also allows model checking to be performed based on a bisimulation quotient transition system because a logical formula holds for the quotient if and only if it holds for the original transition system.

In the present study, we provide logical characterizations of bisimulation and simulation for FLTSs. Often, a state or system can simulate another but not vice versa. For example, when we check that an implementation matches its specification, we normally do not demand that the implementation performs anything more than that required. It is usually acceptable that the implementation simulates its specification. Hence, it is also interesting to investigate simulations. Unlike other studies of fuzzy systems, we define simulation and bisimulation by virtue of closed subsets of some binary relation (Section 5 provides a detailed discussion). Moreover, some recent studies of FLTSs and fuzzy automata have focused mainly on simulations and bisimulations [6], [10], [13], [25], [41], whereas they did not consider logical characterizations. A logical characterization of fuzzy bisimulation was provided by [17], but the differences from the current study are as follows: (1) the logic used in [17] employs recursive formulae where it interprets a formula as a fuzzy set that gives the measure of satisfaction and unsatisfaction for the formula; and (2) we consider bisimulation and simulation, whereas [17] only considered bisimulation.

The logic used to characterize fuzzy bisimulation is very similar to Larsen and Skou's probabilistic extension of the Hennessy–Milner Logic.1 The completeness proof for our logical characterization of fuzzy bisimulation is also inspired by [22], who characterized probabilistic bisimulation. Indeed, there is only a slight difference between the FLTS model and probabilistic labeled transition systems (PLTSs). This may create the impression that the current study is a straightforward generalization of the study of PLTSs, but this is not the case. For PLTSs, disjunction is necessary to characterize simulation, whereas it is not for FLTSs. For PLTSs, negation is not necessary for characterizing bisimulation and binary conjunction is already sufficient, whereas for FLTSs, both negation and infinite conjunction are necessary to characterize bisimulation for general FLTSs, which may be infinitely branching. Moreover, different techniques are needed to prove characterization theorems for FLTSs and PLTSs. For example, in the case of PLTSs, the well-known πλ theorem holds, which greatly simplifies the completeness proof for the logical characterization of bisimulation. However, the πλ theorem is invalid for FLTSs, so we adopt a different approach to prove completeness, where we try to construct a characteristic formula for each equivalence class, i.e., the formula is satisfied only by the states in that equivalence class. Sections 4.2 and 4.3 provide more details.

The remainder of this paper is organized as follows. We briefly review some of the basic concepts used in this study in Section 2. In Section 3, we describe some properties of simulations and bisimulations for FLTSs. In particular, similarity and bisimilarity are shown to be closed under the parallel composition of FLTSs. Section 4 presents the logical characterization theorems. In this section, we also analyze why the logics characterizing bisimulations for FLTSs and PLTSs are different. We introduce some related research in Section 5. Finally, we give our conclusions in Section 6 by providing a summary of the differences between the logical characterizations of FLTSs and PLTSs, as well as discussing future research.

Section snippets

Preliminaries

In this section, we briefly recall some basic concepts and terminologies from set theory and fuzzy set theory, before introducing FLTSs.

Let S be an ordinary set. A fuzzy set μ of S is a function that assigns each element s of S with a value μ(s) in the real unit interval [0,1]. The support of μ is the set supp(μ)={sS|μ(s)>0}. We denote F(S) as the set of all fuzzy sets in S and Ff(S) as the set of all fuzzy sets with finite-support, i.e., Ff(S)={μF(S)|supp(μ)is finite}. Whenever supp(μ) is a

Simulation and bisimulation

In this section, we introduce our notions of simulation and bisimulation for FLTSs, and we discuss their properties.

Based on the idea of defining bisimulations for PLTSs [16], we require that if (s,t) is a pair of states in a simulation relation, then t can mimic all the stepwise behaviors of s with respect to R. Thus, if s can perform an action on a possibility distribution μ, then t can perform the same action on another possibility distribution ν such that μ and ν are related via a relation

Logical characterizations of bisimulation and simulation

Specification, i.e., the description of the required properties of an implementation, is a major issue for transition systems [42]. These properties are best expressed as formulae in a logic language. In this section, we introduce a variant of the Hennessy–Milner Logic [21] to characterize bisimulation. We also show that a negation-free sub-logic is sufficient to characterize simulation.

Related work

Fuzzy simulations and bisimulations have attracted much attention from researchers in the field. Next, we briefly summarize some of the recent research in this area.

D'Errico and Loreti [17] proposed a notion of fuzzy bisimulation and applied it to fuzzy reasoning. Kupferman and Lustig [25] defined a latticed simulation between two lattice-valued Kripke structures, which they applied to latticed games. Pan et al. [32] studied simulation for lattice-valued doubly labeled transition systems. Cao

Conclusion and future work

In this study, we investigated two fuzzy variants of the Hennessy–Milner Logic and characterized bisimulations and simulations for FLTSs soundly and completely. Compared with the logical characterizations for PLTSs, the following are the main differences.

  • Characterizing simulations. For PLTSs, disjunction is necessary and binary conjunction is already sufficient [16], whereas for FLTSs, infinite conjunction is generally necessary but disjunction is not, i.e., the logic Lsi. The logic with binary

Acknowledgements

The first author was supported by the Zhejiang Provincial Natural Science Foundation of China (LY13F020046) and Zhejiang Provincial Education Department Fund of China (Y201223001). The second author was supported by the National Natural Science Foundation of China (61173033, 61261130589) and ANR 12IS02001 PACE. The first author would also like to thank Dr. Haiyu Pan for helpful discussions and suggestions. The authors would like to thank the anonymous reviewers and the language editor for their

References (45)

Cited by (54)

  • Computing crisp bisimulations for fuzzy structures

    2024, International Journal of Approximate Reasoning
View all citing articles on Scopus
View full text