Like bisimulation, simulation is a well-known notion for comparing observational behaviors of automata and labeled transition systems (LTSs) [1, 2]. Given states and , we say that simulates if the label of is a subset of the label of and, for every -transition from to a state , there exists a -transition from to a state that simulates , where is any action. Such defined simulations preserve positive existential modal formulas, which are modal formulas without implication, negation and universal modal operators. That is, if simulates , then it satisfies all positive existential modal formulas that does. Conversely, given image-finite LTSs and , if a state of satisfies all positive existential modal formulas that a state of does, then the pair belongs to the largest simulation between and (cf. [3, 4]). This is called the Hennessy-Milner property of simulation.
Directed simulation is a stronger notion than simulation. A state directedly simulates a state if:
the label of is a subset of the label of ;
for every -transition from to a state , there exists a -transition from to a state that directedly simulates ;
for every -transition from to a state , there exists a -transition from to a state that is directedly simulated by .
Thus, directed simulation requires both the “forward” and “backward” conditions of bisimulation and is weaker than bisimulation only in that and are not required to have the same label as in the case of bisimulation. Directed simulation was introduced and studied by Kurtonina and de Rijke  for modal logic. They proved that directed simulation characterizes the class of positive modal formulas, like simulation characterizes the class of positive existential modal formulas. Positive modal formulas are modal formulas without implication and negation. Directed simulation has also been formulated and studied for description logics .
For fuzzy structures like fuzzy automata and fuzzy labeled transition systems (FLTSs), researchers have studied both crisp simulations [7, 8, 9, 10, 11] and fuzzy simulations [12, 13, 14, 8, 15]. Crisp/fuzzy bisimulations have also been studied for fuzzy structures by a considerable number of researchers [16, 12, 17, 18, 7, 19, 20, 14, 9, 21, 22, 23, 24, 25, 26]. However, as far as we know, only the work  has concerned crisp directed simulations for fuzzy structures. It only deals with computational aspects.
The current paper concerns logical characterizations of crisp simulations and crisp directed simulations for FLTSs. As related works on logical characterizations of crisp/fuzzy bisimulations or simulations for fuzzy structures, the notable ones are the papers [17, 19, 20, 23, 26] on fuzzy bisimulations, [19, 9, 21, 22, 23] on crisp bisimulations, [13, 8] on fuzzy simulations, and [8, 9, 10] on crisp simulations. We discuss below only the last three works.
In  Pan et at. studied simulations for quantitative transition systems (QTSs), which are a variant of FLTSs without labels for states. The authors provided logical characterizations of cut-based crisp simulations between finite QTSs w.r.t. an existential cut-based crisp Hennessy-Milner logic. A fuzzy threshold, used as the cut for the fuzzy equality relation between actions, is a parameter for both the crisp simulations and the crisp Hennessy-Milner logic under consideration. The main results of  are formulated only for the case when the underlying residuated lattice is a finite Heyting algebra.
In  Wu and Deng provided a logical characterization of crisp simulations for FLTSs w.r.t. a crisp Hennessy-Milner logic, which uses values from the interval as thresholds for modal operators. States of FLTSs considered in  are not labeled. The logical characterization of crisp simulations provided in  is the Hennessy-Milner property formulated w.r.t. a crisp modal logic with a minimal set of constructors, namely with , and , where is an action and .
In  Nguyen introduced and studied cut-based crisp simulations between fuzzy interpretations in fuzzy description logics under the Zadeh semantics. He provided results on preservation of information under such simulations and the Hennessy-Milner property of such simulations w.r.t. fuzzy description logics under the Zadeh semantics.
As seen from the above discussion, logical characterizations of crisp simulations studied in [8, 9] are formulated w.r.t. crisp modal logics, whereas logical characterizations of crisp simulations studied in  are formulated w.r.t. fuzzy description logics under the Zadeh semantics. In addition, crisp simulations studied in [8, 10] are cut-based and, indeed, a form of fuzzy simulations. There was the lack of logical characterizations of crisp simulations between fuzzy structures w.r.t. fuzzy modal/description logics that use a residuated lattice or a t-norm-based semantics. Furthermore, logical characterizations of crisp/fuzzy directed simulations between fuzzy structures have not yet been studied.
In this paper, we formulate and prove logical characterizations of crisp simulations and crisp directed simulations between FLTSs w.r.t. fuzzy modal logics that use a general t-norm-based semantics. The considered logics are fragments of the fuzzy propositional dynamic logic with the Baaz projection operator (). The logical characterizations concern preservation of positive existential (resp. positive) modal formulas under crisp simulations (resp. crisp directed simulations), as well as the Hennessy-Milner property of such simulations.
The rest of this paper is structured as follows. Section II contains definitions about fuzzy sets, fuzzy operators, FLTSs and . In Section III (resp. IV), we define crisp simulations (resp. crisp directed simulations) between FLTSs, formulate and prove their logical characterizations w.r.t. the positive existential (resp. positive) fragments of . Conclusions are given in Section V.
Ii-a Fuzzy Sets and Fuzzy Operators
A fuzzy subset of a set is a function . Given a fuzzy subset of , for means the fuzzy degree of that belongs to the subset. For and , we write , …, to denote the fuzzy subset of such that for and for . Given fuzzy subsets and of a set , we write to denote that for all . A fuzzy subset of is called a fuzzy relation between and .
An operator is called a t-norm if it is commutative and associative, has 1 as the neutral element, and is increasing w.r.t. both the arguments. If , is a left-continuous t-norm and is the operator defined by , then is called the residuum of .
The three well-known t-norms named after Gödel, Łukasiewicz and product are specified below:
The corresponding residua are specified below:
From now on, let be an arbitrary left-continuous t-norm and be its residuum. It is known that is decreasing w.r.t. the first argument and increasing w.r.t. the second argument. Furthermore, iff .
Let be the operator defined by (if then 1 else 0).
Ii-B Fuzzy Labeled Transition Systems
Let be a non-empty set of actions and a non-empty set of state labels. We use to denote an element of and to denote an element of .
An FLTS is a triple , where is a non-empty set of states, is called the transition function, and is called the state labeling function. For , and , means the fuzzy degree of that there is a transition of the action from the state to the state , whereas the fuzzy subset of is the label of and means the fuzzy degree of that belongs to the label of .
An FLTS is image-finite if, for every and , the set is finite. It is finite if , and are finite.
Ii-C Fuzzy PDL with the Baaz Projection Operator
We use as the signature for the logical languages considered in this article. By we denote the fuzzy propositional dynamic logic with the Baaz projection operator. Programs and formulas of over the signature are defined as follows:
actions from are programs of ,
if and are programs of , then , and are also programs of ,
if is a formula of , then is a program of ,
values from the interval and propositions from are formulas of ,
if and are formulas of and is a program of , then , , , , and are formulas of .
Note that we ignore negation, as a formula is usually defined to be .
Given a finite set of formulas with , we define .
Treating an FLTS as a fuzzy Kripke model, a program is interpreted in as a fuzzy relation , whereas a formula is interpreted in as a fuzzy subset . The functions and are specified as follows.
|(if then else 0)|
|(if then 1 else 0)|
Iii Crisp Simulations between FLTSs and Their Logical Characterizations
In this section, we first recall crisp simulations between FLTSs, then define the positive existential fragments and of , and finally formulate and prove logical characterizations of crisp simulations between FLTSs w.r.t. these positive existential fragments of .
Iii-a Crisp Simulations between FLTSs
This subsection is a reformulation of the corresponding one of  (which concerns fuzzy graphs).
Let and be FLTSs. A binary relation is called a (crisp) simulation between and if the following conditions hold for every , and :
Here, and denote the usual crisp logical connectives. Thus, the above conditions mean that:
if holds, then ;
if holds and , then there exists such that and holds.
Let , and let and be the FLTSs specified below and depicted in Fig. 1.
, , , where , , , , , , , and , , , , .
, , , where , , , and , , .
It can be checked that , , is the largest simulation between and .
A (crisp) auto-simulation of is a simulation between and itself.
Let , and be FLTSs and let .
The relation is an auto-simulation of .
If is a simulation between and , and is a simulation between and , then is a simulation between and .
If is a set of simulations between and , then is also a simulation between and .
The proof of this proposition is straightforward.
The largest simulation between two arbitrary FLTSs exists. The largest auto-simulation of a FLTS is a pre-order.
Iii-B The Positive Existential Fragment of
By we denote the largest sublanguage of that disallow the formula constructor and allows implication () only in formulas of the form with . We call the positive existential fragment of .
By we denote the largest sublanguage of that disallow the program constructors (, , and ), the disjunction operator () and the constructor (with ). That is, only actions from are programs of , and formulas of are of the form , , , or , where , , , and and are formulas of .
An FLTS is said to be witnessed w.r.t. if, for every formula (resp. program ) of and every , if the definition of (resp. ) in Definition 1 uses supremum, then the set under the supremum has the biggest element if it is non-empty.
The notion of whether an FLTS is witnessed w.r.t. is defined analogously.
Observe that if an FLTS is finite, then it is witnessed w.r.t. and . If is image-finite, then it is witnessed w.r.t. .
Iii-C Logical Characterizations of Simulations between FLTSs
A formula is said to be preserved under simulations between FLTSs if, for every FLTSs and that are witnessed w.r.t. , for every simulation between them, and for every and , if holds, then .
All formulas of are preserved under simulations between FLTSs.
This theorem follows immediately from the first assertion of the following lemma.
Let and be FLTSs witnessed w.r.t. and be a simulation between them. Then, the following assertions hold for every , , every formula and every program of , where and are the usual crisp logical connectives:
We prove this lemma by induction on the structure of and .
Consider the assertion (3). Assume that holds. We need to show that . The cases when is a constant or a proposition are trivial. The cases when is of the form , or are also straightforward by using the induction assumptions about and and the definition of and . The remaining cases are considered below.
Case : By the induction assumption, . It is known that the residuum of every continuous t-norm is increasing w.r.t. the second argument. Hence, .
Case : For a contradiction, assume that . Since is witnessed w.r.t. , there exists such that . Since , it follows that and therefore (since for all ). By the induction assumption of (4), there exists such that and holds. Since holds, by the induction assumption, . Since is increasing w.r.t. both the arguments, it follows that
This contradicts the assumption .
Case : Suppose that holds and . Thus,
Since is witnessed w.r.t. , there exists such that
Since for all , we must have that and . Since holds, by the induction assumption, there exists such that holds and . Since holds and , by the induction assumption, there exists such that holds and . Since is increasing, it follows that
Therefore, and the induction hypothesis (4) holds.
Case : Suppose that holds and . Thus, , . W.l.o.g. we assume that . Since holds, by the induction assumption, it follows that there exists such that and holds. Therefore, and the induction hypothesis (4) holds.
Case : Suppose that holds and . If , then by taking , and holds. Assume that . Since is witnessed w.r.t. , there exist and such that , and . Since , we must have that for all (because for all ). Let . For each from to , since holds and , by the induction assumption, there exists such that and holds. Take . Thus, holds. Since is increasing,
Hence, and the induction hypothesis (4) holds.
Let and be FLTSs witnessed w.r.t. and be a simulation between them. Then, for every and , if holds, then for all formulas of .
An FLTS is said to be modally saturated w.r.t. if, for every , , and every infinite set of formulas of , if for every finite subset of there exists such that , then there exists such that and for all . The notion of modal saturatedness is a technical extension of image-finiteness. This is confirmed by the following proposition.
Every image-finite FLTS is modally saturated w.r.t. .
Let be an image-finite FLTS. Let , , and let be an infinite set of formulas of . We prove that is modally saturated w.r.t. by contraposition. Suppose that, for every , there exists such that or . We need to prove that there exists a finite subset of such that, for every , . Let and . Since is image-finite, is finite. For every , since either or and , we must have that . This completes the proof.
The following theorem states the Hennessy-Milner property of crisp simulations between FLTSs.
Let and be FLTSs witnessed and modally saturated w.r.t. . Then, for all formulas of is the largest simulation between and .
Condition (1) holds by the definition of .
Consider Condition (2) and suppose that holds and . Let . We need to show that there exists such that holds. For a contradiction, suppose that, for each , does not hold, which means there exists a formula of such that . For every , let , which is a formula of . Observe that, for every , and . Let . Observe that, for every , either or there exists such that . Since is modally saturated w.r.t. , there exists a finite subset of such that, for every , . Let . It is a formula of . Since is witnessed w.r.t. , it follows that . Since for all , and . Thus, , which contradicts the assumption that holds.
Let and be FLTSs and let and . We write to denote that there exists a simulation between and