1 Introduction
Labeled transition systems, automata, Kripke models, social networks and interpretations in description logics have in common that they are graphbased structures. The relation that specifies whether two states, two actors or two individuals and in such structures and , respectively, behave equivalently or are equivalent from the logical point of view has useful applications in practice. For example, when is the same as , is an equivalence relation and we can use it to minimize or exploit it in classification or clustering. The relation is the largest bisimulation, also called the bisimilarity relation, between and . Bisimulation vanBenthemCorr; HennessyM85 is a natural notion of equivalence that arose in modal logic and labeled transition systems and has been widely studied for all of the mentioned kinds of graphbased structures.
To deal with vagueness and impreciseness, fuzzy graphbased structures are used instead of crisp ones. There are two kinds of bisimulation, namely crisp and fuzzy, between fuzzy graphbased structures. Crisp bisimulations characterize indiscernibility of states/actors/individuals. The HennessyMilner property of crisp bisimulations ai/FanL14; Fan15; FSS2020 states that, if is the largest crisp bisimulation between two imagefinite fuzzy graphbased structures and , then holds iff for all formulas of a certain fuzzy modal/description logic with the Baaz projection operator or involutive negation, where (respectively, ) means the degree in which (respectively, ) has the property (or is an instance of ). On the other hand, fuzzy bisimulations characterize similarity between states/actors/individuals. The HennessyMilner property of fuzzy bisimulations ai/FanL14; Fan15; FSS2020; FBSML states that, if is the greatest fuzzy bisimulation between two imagefinite fuzzy graphbased structures and , then 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 CaoCK11; CaoSWC13; DBLP:journals/fss/WuD16; DBLP:journals/ijar/WuCHC18; DBLP:journals/fss/WuCBD18, weighted automata DamljanovicCI14, fuzzy modal logics EleftheriouKN12; Fan15; aml/MartiM18; fuin/Diaconescu20 and fuzzy description logics FSS2020. Fuzzy bisimulations have been defined and studied for fuzzy automata CiricIDB12; CiricIJD12, weighted/fuzzy social networks ai/FanL14; IgnjatovicCS15, fuzzy modal logics Fan15; FBSML and fuzzy description logics FSS2020; minimizationbyfBS; TFS2020.
This work concerns computing the greatest fuzzy bisimulation between two finite fuzzy graphbased structures. A discussion on related work, based on CompCBarxiv, is presented below.
1.1 Related Work
In CiricIJD12 Ćirić et at. gave an algorithm for computing the greatest fuzzy simulation/bisimulation (of any kind defined in CiricIDB12) between two finite fuzzy automata. They did not provide a detailed complexity analysis. Following CiricIJD12, Ignjatović et at. IgnjatovicCS15 gave an algorithm with the complexity for computing the greatest fuzzy bisimulation between two fuzzy social networks, where is the number of nodes in the networks and is the number of different fuzzy values appearing during the computation. Later Micić et at. MicicJS18 provided algorithms with the complexity for computing the greatest right/left invariant fuzzy quasiorder/equivalence of a finite fuzzy automaton, where is the number of states of the considered automaton and is the number of different fuzzy values appearing during the computation. These relations are closely related to the fuzzy simulations/bisimulations studied in CiricIDB12; CiricIJD12. In TFS2020 Nguyen and Tran provided an algorithm with the complexity for computing the greatest fuzzy bisimulation between two finite fuzzy interpretations in the fuzzy description logic under the Gödel semantics, where is the number of individuals and is the number of nonzero instances of roles in the given fuzzy interpretations. They also adapted that algorithm for computing fuzzy simulations/bisimulations between finite fuzzy automata and obtained algorithms with the same complexity order.
In DBLP:journals/fss/WuCBD18 Wu et al. studied algorithmic and logical characterizations of crisp bisimulations for nondeterministic fuzzy transition systems (NFTSs) CaoSWC13. They gave an algorithm with the complexity for testing crisp bisimulation (i.e., for checking whether two given states are bisimilar), where is the number of states and is the number of transitions in the underlying NFTS. In StanimirovicSC2019 Stanimirović et at. provided algorithms with the complexity for computing the greatest right/left invariant Boolean (crisp) equivalence matrix of a weighted automaton over an additively idempotent semiring. Such matrices are closely related to crisp bisimulations. In CompCBarxiv Nguyen and Tran gave an algorithm with the complexity for computing the (crisp) partition corresponding to the largest crisp bisimulation of a given finite fuzzy labeled graph, where , and are the number of vertices, the number of nonzero edges and the number of different fuzzy degrees of edges of the input graph, respectively. They also studied a similar problem for the setting with counting successors, which corresponds to the case with qualified number restrictions in description logics and graded modalities in modal logics. In particular, they provided an algorithm with the complexity for the considered problem in that setting.
As the background, also recall that Hopcroft Hopcroft71 gave an efficient algorithm with the complexity for minimizing states in a (crisp) deterministic finite automaton, and Paige and Tarjan PaigeT87 gave efficient algorithms with the complexity for computing the coarsest partition of a finite (crisp) graph, for both the settings with stability or sizestability. As mentioned in PaigeT87, an algorithm with the same complexity order for the second setting was given earlier by Cardon and Crochemore DBLP:journals/tcs/CardonC82.
1.2 Motivation and Our Contributions
As discussed in the previous subsection, before the current work, the best known algorithm for computing the greatest fuzzy bisimulation between two finite fuzzy graphbased structures under the Gödel semantics was given by Nguyen and Tran TFS2020 and has the complexity . The motivation of the current work is to develop a more efficient algorithm for the same problem.
In this article, by exploiting the ideas and techniques of the works Hopcroft71; PaigeT87; CompCBarxiv; TFS2020, we develop an efficient algorithm with the complexity for computing the fuzzy partition corresponding to the greatest fuzzy autobisimulation of a finite fuzzy labeled graph under the Gödel semantics, where , and are the number of vertices, the number of nonzero edges and the number of different fuzzy degrees of edges of the input graph , respectively. Our notion of fuzzy partition is novel, defined only for finite sets with respect to the Gödel tnorm, with the aim to facilitate the computation of the greatest fuzzy autobisimulation. By using that algorithm, we also provide an algorithm with the complexity for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the Gödel semantics. Taking for the worst case, the latter complexity order can be simplified to . 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 other fuzzy graphbased structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretations in fuzzy description logics.
1.3 The Structure of This Work
The rest of this work is structured as follows. In Section 2, we give preliminaries on fuzzy sets, fuzzy labeled graphs and fuzzy bisimulations. Section 3 is devoted to fuzzy partitions. In Section LABEL:sec:_skeleton_alg, we present the skeleton of our algorithm for computing the fuzzy partition corresponding to the greatest fuzzy autobisimulation of a finite fuzzy labeled graph under the Gödel semantics and prove its correctness. In Section LABEL:sec:_impl, we give details on how to implement that algorithm so that its complexity is of order . In Section LABEL:sec:_comp_FB, we use that improved algorithm to design an algorithm with the complexity for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the Gödel semantics. Section LABEL:sec:_conc contains conclusions.
2 Preliminaries
Recall that a crisp partition of a nonempty set is a set of pairwise disjoint nonempty subsets of whose union is equal to . Given a crisp partition , by a component of we mean an element of the set (we reserve the term “block” for another meaning). Given an equivalence relation on , the crisp partition corresponding to is , where is the equivalence class of w.r.t. (i.e., ).
Given crisp partitions and of , we say that is a refinement of if, for every , there exists such that . In that case we also say that is coarser than . By this definition, every crisp partition is coarser than itself.
2.1 Fuzzy Sets and Operators
We use two fuzzy operators of the Gödel family, which are defined as follows for :
Given a set , a function is called a fuzzy subset of . If is a fuzzy subset of and , then means the fuzzy degree in which belongs to the subset. For and , we write to denote the fuzzy subset of such that for and for .
If and are fuzzy subsets of , then we write to denote that for all . If , then we say that is greater than or equal to . If is a set of fuzzy subsets of , then by we denote the fuzzy subset of specified by . As usual, if and , then is called the greatest element of .
Let , and be nonempty sets. A fuzzy subset of is called a fuzzy relation between and . A fuzzy relation between and itself is called a fuzzy relation on . Given fuzzy relations and , the composition of and , denoted by , is the fuzzy relation between and such that, for every and ,
The converse of is defined by .
A fuzzy relation is

reflexive if for all ,

symmetric if ,

transitive if .
It is a fuzzy equivalence relation if it is reflexive, symmetric and transitive.
2.2 Fuzzy Bisimulations
A fuzzy labeled graph, hereafter called a fuzzy graph for short, is a structure , where is a nonempty set of vertices, (respectively, ) is a set of vertex labels (respectively, edge labels), is called the fuzzy set of labeled edges, and is called the labeling function of vertices. Given vertices , a vertex label and an edge label , means the degree in which is a member of the label of , and means the degree in which there is an edge from to labeled by . The graph is finite if all the sets , and are finite. It is imagefinite if the set is finite for all and .
Fuzzy graphs are used as fuzzy labeled transition systems (FLTSs), fuzzy automata, fuzzy Kripke models and fuzzy interpretations in fuzzy description logics. For example, in the terminology of FLTSs, vertices, edges, edge labels and vertex labels represent states, transitions, actions and atomic properties of states, respectively. Recall that fuzzy bisimulations have been defined and studied for fuzzy automata CiricIDB12; CiricIJD12, weighted/fuzzy social networks ai/FanL14; IgnjatovicCS15, fuzzy modal logics Fan15; FBSML and fuzzy description logics FSS2020; minimizationbyfBS; TFS2020. We give below their definition, which is based on FSS2020; FBSML and equivalent to the one in Fan15 when and the graphs are imagefinite.
Definition 2.1
Let and be fuzzy graphs over the same signature . A fuzzy relation is called a fuzzy bisimulation between and if the following conditions hold for all , and all possible values for the free variables:
(1)  
(2)  
(3) 
Example 2.2
Let , and let and be the fuzzy graphs depicted and specified as follows:

, , , , ;

, , , , .
It can be checked that , , , , is the greatest bisimulation between and .
Definition 2.3
Given a fuzzy graph , a fuzzy relation is called a fuzzy autobisimulation of , or a fuzzy bisimulation of for short, if it is a fuzzy bisimulation between and itself, i.e., if the following conditions hold for all , and all possible values for the free variables:
(4)  
(5)  
(6) 
It is known that the greatest fuzzy bisimulation of every imagefinite fuzzy graph exists and is a fuzzy equivalence relation (see, e.g., CiricIDB12; minimizationbyfBS; FBSML).
Let and be fuzzy graphs over the same signature . The disjoint union of and , denoted by , is the fuzzy graph such that , and .
Proposition 2.4
Let and be imagefinite fuzzy graphs over the same signature and let . Let be the greatest fuzzy bisimulation of . Then, is the greatest fuzzy bisimulation between and .
This proposition follows from the HennessyMilner property of fuzzy bisimulations (FSS2020, Theorem 3.7).
3 Fuzzy Partitions
Fuzzy partitions have been studied by a considerable number of authors (see, e.g., OVCHINNIKOV1991107; DBLP:conf/ismvl/Schmechel95; DBLP:journals/isci/BaetsCK98; DBLP:journals/fss/CiricIB07 and references therein). In these cited papers, a fuzzy partition is defined to be the set of all fuzzy equivalence classes of some fuzzy equivalence relation. The notion defined in this way has many interesting properties. In particular, it can be defined for an infinite set over a complete residuated lattice. In this section, we introduce and study a novel notion of fuzzy partition, which is defined only for finite sets with respect to the Gödel tnorm, with the aim to facilitate the computation of the greatest fuzzy autobisimulation of a finite fuzzy labeled graph under the Gödel semantics.
In this section, let be a finite set. To represent a collection of elements from we use an abstract class (type) called block with two subclasses, which are named fuzzy block and crisp block. The intuition behind these kinds of blocks is as follows.

A crisp block is a collection of elements from , whereas a fuzzy block is a collection of blocks called subblocks. If is a crisp block, then we define to be the set of elements of . If is a fuzzy block, then we define to be the set of subblocks of , and inductively define to be . In other words, is the set of all elements belonging the collection represented by .

A fuzzy block has the attribute with the meaning that, treating as a fuzzy equivalence class of a fuzzy equivalence relation , for every , , and furthermore, if and are different subblocks of , and , then .

A crisp block has the attribute .
We assume the following restrictions:

if is a crisp block, then ;

if is a fuzzy block, then ;

if a fuzzy block is a subblock of a fuzzy block , then ;

if and are different subblocks of a block , then the sets and are disjoint.
We also assume that blocks and are equal iff:

either both and are crisp and ,

or both and are fuzzy, and (in the sense that each block from is equal to some block from and vice versa).
A crisp block with is denoted by . A fuzzy block with and is denoted by , where is the denotation of , for .
A block is called a fuzzy partition of if .
Definition 3.1
Given a fuzzy equivalence relation , the fuzzy partition corresponding to is the block defined inductively as follows:

if for all , then is the crisp block such that ;

else:

let ;

let be the (crisp) equivalence relation on such that iff ;

let be all the (crisp) equivalence classes of ;

for , let be the restriction of to and let be the fuzzy partition corresponding to ;

is the fuzzy block such that and .

In the above definition, since is a fuzzy equivalence relation, the relation defined as is really a (crisp) equivalence relation. This guarantees that the fuzzy partition corresponding to a fuzzy equivalence relation is well defined. Here, we implicitly use the assumption that is finite and is the Gödel tnorm.
Definition 3.2
Let be a fuzzy partition of . The fuzzy equivalence relation corresponding to is the fuzzy relation defined inductively as follows:

if is a crisp block, then for all ;

else:

let and ;

for , let and let be the fuzzy equivalence relation corresponding to ;

for , (if for some then else ).

It is easy to check that the fuzzy relation defined above is really a fuzzy equivalence relation. It is also straightforward to prove the following proposition.
Proposition 3.3
Let be a fuzzy partition of and a fuzzy equivalence relation. Then, is the fuzzy equivalence relation corresponding to iff is the fuzzy partition corresponding to .

Example 3.4
Let . Let be the fuzzy partition of depicted at the left hand side of Figure 1 and specified as follows:

is a fuzzy block, with and ;

is a fuzzy block, with and ;

is a crisp block, with ;

is a crisp block, with ;


is a fuzzy block, with and ;

is a crisp block, with ;

is a crisp block, with ;


is a crisp block, with .
The fuzzy block is denoted by .
Let be the fuzzy relation specified at the right hand side of Figure 1. It is the fuzzy equivalence relation corresponding to the fuzzy partition .
Let , , , and be the fuzzy subsets of specified as follows:

,

,

,

,

.
They are the fuzzy equivalence classes of the fuzzy equivalence relation . Using the notion of fuzzy partition defined in OVCHINNIKOV1991107; DBLP:conf/ismvl/Schmechel95; DBLP:journals/isci/BaetsCK98; DBLP:journals/fss/CiricIB07, the fuzzy partition corresponding to the fuzzy equivalence relation is the set . The fuzzy block considered in this example gives a more compact representation for this fuzzy partition.
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