Logical Characterizations of Fuzzy Bisimulations in Fuzzy Modal Logics over Residuated Lattices
aa r X i v : . [ c s . L O ] J a n Logical Characterizations of Fuzzy Bisimulations in Fuzzy Modal Logicsover Residuated Lattices
Linh Anh Nguyen
Institute of Informatics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Abstract
There are two kinds of bisimulation, namely crisp and fuzzy , between fuzzy structures such asfuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models and fuzzy interpretationsin description logics. Fuzzy bisimulations between fuzzy automata over a complete residuated lat-tice have been introduced by ´Ciri´c et al . in 2012. Logical characterizations of fuzzy bisimulationsbetween fuzzy Kripke models (respectively, fuzzy interpretations in description logics) over theresiduated lattice [0 ,
1] with the G¨odel t-norm have been provided by Fan in 2015 (respectively,Nguyen et al . in 2020). There was the lack of logical characterizations of fuzzy bisimulations be-tween fuzzy graph-based structures over a general residuated lattice, as well as over the residuatedlattice [0 ,
1] with the Lukasiewicz or product t-norm. In this article, we provide and prove log-ical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices. Theconsidered logics are the fuzzy propositional dynamic logic and its fragments. Our logical char-acterizations concern invariance of formulas under fuzzy bisimulations and the Hennessy-Milnerproperty of fuzzy bisimulations. They can be reformulated for other fuzzy structures such as fuzzylabel transition systems and fuzzy interpretations in description logics.
Keywords: bisimulation, fuzzy bisimulation, fuzzy modal logic, residuated lattice
1. Introduction
Bisimulation is a useful notion for characterizing equivalence of states in transition systems [25,16]. It has been extensively studied in modal logic for characterizing logical indiscernibility of statesand separating the expressive power of modal logics (see, e.g., [1, 2, 13]). Bisimulation can be usedfor minimizing structures. It can also be exploited for studying indiscernibility of individuals andconcept learning in description logics [21].To deal with vagueness, fuzzy structures are used instead of crisp ones. There are two kindsof bisimulation, namely crisp and fuzzy , between fuzzy structures such as fuzzy automata, fuzzytransition systems, fuzzy Kripke models and fuzzy interpretations in description logics. Researchershave studied crisp bisimulations for fuzzy transition systems [4, 5, 28, 26, 27], weighted automata [8],Heyting-valued modal logics [9], G¨odel modal logics [11] and fuzzy description logics [19]. They havealso studied fuzzy bisimulations for fuzzy automata [6, 7], weighted/fuzzy social networks [10, 17],G¨odel modal logics [11] and fuzzy description logics [19, 20, 22].
Email address: [email protected] (Linh Anh Nguyen)
Preprint submitted to arXiv February 1, 2021 his article is devoted to studying logical characterizations of fuzzy bisimulations. In [11] Fanintroduced fuzzy bisimulations between fuzzy Kripke models over the lattice [0 ,
1] using the G¨odelsemantics (i.e., the G¨odel t-norm and its residuum). She provided logical characterizations of suchbisimulations in the basic fuzzy monomodal logic and its extension with converse. The resultsconcern invariance of modal formulas under fuzzy bisimulations and the Hennessy-Milner propertyof fuzzy bisimulations. In [10] Fan and Liau studied fuzzy bisimulations under the name “regularequivalence relations” for weighted social networks. They provided logical characterizations forsuch bisimulations under the G¨odel semantics, including invariance results and the Hennessy-Milner property. In [19] Nguyen et al . defined and studied fuzzy bisimulations for a large class offuzzy description logics under the G¨odel semantics. The work [19] contains results on invariance ofconcepts under such bisimulations and the Hennessy-Milner property of such bisimulations. In [9]Eleftheriou et al . studied bisimulations for Heyting-valued modal logics. Bisimulations definedin [9] are crisp and cut-based (i.e., using fuzzy values as thresholds). As discussed by Fan [11], suchbisimulations give another representation of fuzzy bisimulations. The work [9] contains results onlogical characterizations of the studied bisimulations, including the Hennessy-Milner property.Note that the results on fuzzy bisimulations of all the works [10, 11, 19] are formulated andproved only for fuzzy structures over the lattice [0 ,
1] using the G¨odel semantics. The results of [9]concern only modal logics over Heyting algebras of truth values. Such algebras are residuatedlattices that use (cid:15) = ∧ , and therefore, are closely related to the G¨odel semantics. There was thelack of logical characterizations of fuzzy bisimulations between fuzzy graph-based structures overa general residuated lattice, as well as over the residuated lattice [0 ,
1] with the Lukasiewicz orproduct t-norm.In this article, we provide and prove logical characterizations of fuzzy bisimulations in fuzzymodal logics over general residuated lattices. The considered logics are the fuzzy propositionaldynamic logic and its fragments. Our logical characterizations concern invariance of formulasunder fuzzy bisimulations and the Hennessy-Milner property of fuzzy bisimulations. Our resultsare significant from the theoretical point of view, as they solve an open and interesting problem.They would also have an impact on practical applications, e.g. for studying logical similarityof individuals and concept learning in fuzzy description logics, as they can be reformulated forother fuzzy structures such as fuzzy interpretations in description logics and fuzzy label transitionsystems, and moreover, residuated lattices cover the lattice [0 ,
1] with any t-norm, including theproduct and Lukasiewicz t-norms.The rest of this article is structured as follows. Section 2 contains preliminaries for this work.In Section 3, we define fuzzy bisimulations between fuzzy Kripke models and provide some oftheir basic properties. Sections 4 and 5 contain our results on invariance of formulas under fuzzybisimulations and the Hennessy-Milner property of fuzzy bisimulations, respectively. Section 6contains a discussion on related work. Conclusions are given in Section 7. The work also containstwo appendices: the first one is the proof of a lemma, whereas the second one is a discussion onthe relationship with fuzzy bisimulations between fuzzy automata [6].
2. Preliminaries
In this section, we recall definitions and properties of residuated lattices and fuzzy sets, thenpresent the syntax and semantics of the fuzzy modal logics considered in this article, together withsome related notions. 2 .1. Residuated Lattices and Fuzzy Sets A residuated lattice [14, 3] is an algebra L = h L, ≤ , (cid:15) , ⇒ , , i such that • h L, ≤ , , i is a bounded lattice with the least element 0 and the greatest element 1, • h L, (cid:15) , i is a commutative monoid, • (cid:15) and ⇒ form an adjoint pair, which means that, for every x, y, z ∈ L , x (cid:15) y ≤ z iff x ≤ ( y ⇒ z ) . (1)The expression y ⇒ z is called the residual of z by y . Given a residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i , let ∧ and ∨ denote the join and meet operators associated with the lat-tice. By x ⇔ y we denote ( x ⇒ y ) ∧ ( y ⇒ x ). We use the convention that (cid:15) and ∧ bind strongerthan ∨ , which in turn binds stronger than ⇒ and ⇔ .A residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i is complete (resp. linear ) if the bounded lattice h L, ≤ , , i is complete (resp. linear). It is a Heyting algebra if (cid:15) is the same as ∧ .We will need the following lemma. Although most assertions of this lemma are well-known [3,14] and the proof of this lemma is straightforward, we present the proof in Appendix A to makethis article self-contained. Lemma 2.1 (cf. [3, 14]).
Let L = h L, ≤ , (cid:15) , ⇒ , , i be a residuated lattice. The following prop-erties hold for all x, x ′ , y, y ′ , z ∈ L : x ≤ x ′ and y ≤ y ′ implies x (cid:15) y ≤ x ′ (cid:15) y ′ (2) x ′ ≤ x and y ≤ y ′ implies ( x ⇒ y ) ≤ ( x ′ ⇒ y ′ ) (3) x ≤ y iff ( x ⇒ y ) = 1 (4) x (cid:15) x (cid:15) ( y ∨ z ) = x (cid:15) y ∨ x (cid:15) z (6) x (cid:15) ( x ⇒ y ) ≤ y (7) x (cid:15) ( y ⇒ z ) ≤ ( x ⇒ y ) ⇒ z (8) x (cid:15) ( y ⇔ z ) ≤ ( x ⇒ y ) ⇒ z (9) x (cid:15) ( y ⇔ z ) ≤ y ⇔ x (cid:15) z (10) x ⇒ ( y ⇒ z ) = y ⇒ ( x ⇒ z ) (11) x ⇒ ( y ⇒ z ) ≤ x (cid:15) y ⇒ z (12) x ⇒ ( y ⇔ z ) ≤ x (cid:15) y ⇒ z (13)( x ⇒ y ) (cid:15) ( y ⇒ z ) ≤ x ⇒ z (14)( x ⇔ y ) (cid:15) ( y ⇔ z ) ≤ x ⇔ z (15)( x ⇔ x ′ ) ∧ ( y ⇔ y ′ ) ≤ x ∧ y ⇔ x ′ ∧ y ′ (16)( x ⇔ x ′ ) ∧ ( y ⇔ y ′ ) ≤ x ∨ y ⇔ x ′ ∨ y ′ (17) x ⇔ y ≤ ( z ⇒ x ) ⇔ ( z ⇒ y ) (18) x ⇔ y ≤ ( x ⇒ z ) ⇔ ( y ⇒ z ) . (19)3 n addition, if L is a Heyting algebra, then the following properties hold for all x, x ′ , y, y ′ , z ∈ L : ( x ⇔ x ′ ) ∧ ( y ⇔ y ′ ) ≤ ( x ⇒ y ) ⇔ ( x ′ ⇒ y ′ ) (20) x ≤ ( y ⇔ z ) implies x (cid:15) y = x (cid:15) z. (21) Example 2.2.
Consider the case when L is the unit interval [0 , (cid:15) are the G¨odel, Lukasiewicz and product t-norms. They are specified below together with theircorresponding residua ( ⇒ ). G¨odel Lukasiewicz Product x (cid:15) y min { x, y } max { , x + y − } x · yx ⇒ y (cid:26) x ≤ yy otherwise min { , − x + y } (cid:26) x ≤ yy/x otherwiseNote that, for all of these cases of (cid:15) , the considered residuated lattice is linear and complete, andthe operator (cid:15) is continuous. (cid:4) From now on, let L = h L, ≤ , (cid:15) , ⇒ , , i be an arbitrary residuated lattice.Given a set X , a function f : X → L is called a fuzzy set , as well as a fuzzy subset of X . If f isa fuzzy subset of X and x ∈ X , then f ( x ) means the fuzzy degree of that x belongs to the subset.For { x , . . . , x n } ⊆ X and { a , . . . , a n } ⊆ L , we write { x : a , . . . , x n : a n } to denote the fuzzysubset f of X such that f ( x i ) = a i for 1 ≤ i ≤ n and f ( x ) = 0 for x ∈ X \ { x , . . . , x n } . Givenfuzzy subsets f and g of X , we write f ≤ g to denote that f ( x ) ≤ g ( x ) for all x ∈ X . If f ≤ g ,then we say that g is greater than or equal to f .A fuzzy subset of X × Y is called a fuzzy relation between X and Y . A fuzzy relation between X and itself is called a fuzzy relation on X .Given Z : X × Y → L , the converse Z − : Y × X → L of Z is defined by Z − ( y, x ) = Z ( x, y ).If the underlying residuated lattice is complete, then the composition of fuzzy relations Z : X × Y → L and Z : Y × Z → L , denoted by Z ◦ Z , is defined to be the fuzzy relation be-tween X and Z such that ( Z ◦ Z )( x, z ) = sup { Z ( x, y ) (cid:15) Z ( y, z ) | y ∈ Y } for all h x, z i ∈ X × Z .Let Z be a set of fuzzy relations between X and Y . If Z is finite or the underlying residuatedlattice is complete, then by sup Z we denote the fuzzy relation between X and Y specified by:(sup Z )( x, y ) = sup { Z ( x, y ) | Z ∈ Z} for h x, y i ∈ X × Y . We write Z ∪ Z to denote sup { Z , Z } .A fuzzy relation Z : X × X → L is • reflexive if Z ( x, x ) = 1 for all x ∈ X , • symmetric if Z ( x, y ) = Z ( y, x ) for all x, y ∈ X , • transitive if Z ( x, y ) (cid:15) Z ( y, z ) ≤ Z ( x, z ) for all x, y, z ∈ X .It is a fuzzy equivalence relation if it is reflexive, symmetric and transitive. Let Σ A denote a non-empty set of actions , which are also called atomic programs , and let Σ P denote a non-empty set of propositions , which are also called atomic formulas . The pair h Σ A , Σ P i forms the signature for the fuzzy modal logics considered in this article.4et Φ ⊆ {∪ , → , ? } . By fPDL − Φ we denote the fuzzy propositional dynamic logic without theunion program constructor if ∪ belongs to Φ, without the test operator if ? belongs to Φ, andwithout the full version of implication if → belongs to Φ.In the following, an expression like ∪ / ∈ Φ can be read as “ ∪ is not excluded”. Programs and formulas of fPDL − Φ over a residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i are definedas follows: • if ̺ ∈ Σ A , then ̺ is a program of fPDL − Φ , • if α and β are programs of fPDL − Φ , then – α ◦ β and α ∗ are programs of fPDL − Φ , – if ∪ / ∈ Φ, then α ∪ β is a program of fPDL − Φ , – if ? / ∈ Φ and ϕ is a formula of fPDL − Φ , then ϕ ? is a program of fPDL − Φ , • if a ∈ L , then a is a formula of fPDL − Φ , • if p ∈ Σ P , then p is a formula of fPDL − Φ , • if ϕ and ψ are formulas of fPDL − Φ , α is a program of fPDL − Φ and a ∈ L , then ϕ ∧ ψ , ϕ ∨ ψ , a → ϕ , ϕ → a , ¬ ϕ , [ α ] ϕ and h α i ϕ are formulas of fPDL − Φ , • if ϕ and ψ are formulas of fPDL − Φ and → / ∈ Φ, then ϕ → ψ is a formula of fPDL − Φ .Note that, even when → ∈ Φ, fPDL − Φ allows implications of the form a → ϕ or ϕ → a with a ∈ L . By fPDL we denote fPDL − Φ with Φ = ∅ . By fK we denote the largest sublanguage of fPDL − Φ with Φ = {∪ , → , ? } that disallows the remaining program constructors ( α ◦ β and α ∗ )and the formula constructors ¬ ϕ , ϕ ∨ ψ and [ α ] ϕ . That is, formulas of fK are of the form a , p , ϕ ∧ ψ , a → ϕ , ϕ → a or h ̺ i ϕ , where a ∈ L , p ∈ Σ P , ̺ ∈ Σ A , and ϕ and ψ are formulas of fK .We use letters like • ̺ to denote actions from Σ A , • p and q to denote propositions from Σ P , • a and b to denote values from L , • ϕ and ψ to denote formulas, • α and β to denote programs.Given a finite set Γ = { ϕ , . . . , ϕ n } with n ≤
0, we denote V Γ = ϕ ∧ . . . ∧ ϕ n ∧ , N Γ = ϕ (cid:15) · · · (cid:15) ϕ n (cid:15) . Definition 2.3. A fuzzy Kripke model over a signature h Σ A , Σ P i and a residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i is a pair M = h ∆ M , · M i , where ∆ M is a non-empty set, called the domain , and · M is the interpretation function that maps each p ∈ Σ P to a fuzzy set p M : ∆ M → L and mapseach ̺ ∈ Σ A to a fuzzy relation ̺ M : ∆ M × ∆ M → L . The interpretation function is extended to5omplex programs and formulas as follows, under the condition that the used suprema and infimaexist (e.g., by requiring L to be complete or M to be witnessed as defined shortly).( ϕ ?) M ( x, y ) = (if x = y then ϕ M ( x ) else 0)( α ∪ β ) M ( x, y ) = α M ( x, y ) ∨ β M ( x, y )( α ◦ β ) M ( x, y ) = sup { α M ( x, z ) (cid:15) β M ( z, y ) | z ∈ ∆ M } ( α ∗ ) M ( x, y ) = sup { N { α M ( x i , x i +1 ) | ≤ i < n } | n ≥ , x , . . . , x n ∈ ∆ M , x = x, x n = y } a M ( x ) = a ( ϕ ∧ ψ ) M ( x ) = ϕ M ( x ) ∧ ψ M ( x )( ϕ ∨ ψ ) M ( x ) = ϕ M ( x ) ∨ ψ M ( x )( ϕ → ψ ) M ( x ) = ( ϕ M ( x ) ⇒ ψ M ( x ))( ¬ ϕ ) M ( x ) = ( ϕ → M ( x )([ α ] ϕ ) M ( x ) = inf { α M ( x, y ) ⇒ ϕ M ( y ) | y ∈ S } ( h α i ϕ ) M ( x ) = sup { α M ( x, y ) (cid:15) ϕ M ( y ) | y ∈ S } . (cid:4) Example 2.4.
Let Σ A = { ̺ } , Σ P = { p } and let L be the unit interval [0 , M specified by ∆ M = { u, v, w } , p M = { u : 0 . , v : 0 . , w : 0 . } , ̺ M = {h u, v i : 0 . h u, w i : 0 . } and depicted below: u : 0 . v : 0 . w : 0 . ϕ M ( u ) for some example formulas ϕ using the G¨odel, Lukasiewicz or product t-norm (cid:15) are given below: G¨odel Lukasiewicz Product( h ̺ i p ) M ( u ) 0 . . . ̺ ] p ) M ( u ) 0 . . / h ̺ ∗ i p ) M ( u ) 0 . . . ̺ ∗ ] p ) M ( u ) 0 . . / (cid:4) A fuzzy Kripke model M is witnessed w.r.t. fPDL − Φ if every infinite set under the infimum(resp. supremum) operator in Definition 2.3 has a smallest (resp. biggest) element when consideringonly formulas and programs of fPDL − Φ (cf. [15]). The notion of whether a fuzzy Kripke model M is witnessed w.r.t. fK is defined analogously by restricting to formulas and programs of fK .A fuzzy Kripke model M is image-finite if, for every x ∈ ∆ M and every ̺ ∈ Σ A , the set { y ∈ ∆ M | ̺ M ( x, y ) > } is finite. It is finite if ∆ M , Σ A and Σ P are finite.Observe that every finite fuzzy Kripke model is witnessed w.r.t. fPDL − Φ (and hence also w.r.t. fK ) and every image-finite fuzzy Kripke model is witnessed w.r.t. fK . If the underlying residuatedlattice is finite, then all fuzzy Kripke models are witnessed w.r.t. fPDL − Φ .6 . Fuzzy Bisimulations between Kripke Models In this section, we define fuzzy bisimulations between fuzzy Kripke models, then state andprove some of their basic properties. The relationship with fuzzy bisimulations between fuzzyautomata [6] is presented in Appendix B.
Definition 3.1.
Given fuzzy Kripke models M and M ′ , a fuzzy relation Z : ∆ M × ∆ M ′ → L iscalled a fuzzy bisimulation between M and M ′ if the following conditions hold for all p ∈ Σ P , ̺ ∈ Σ A and all possible values for the free variables: Z ( x, x ′ ) ≤ ( p M ( x ) ⇔ p M ′ ( x )) (22) ∃ y ′ ∈ ∆ M ′ ( Z ( x, x ′ ) (cid:15) ̺ M ( x, y ) ≤ ̺ M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ )) (23) ∃ y ∈ ∆ M ( Z ( x, x ′ ) (cid:15) ̺ M ′ ( x ′ , y ′ ) ≤ ̺ M ( x, y ) (cid:15) Z ( y, y ′ )) . (24) Proposition 3.2.
Suppose that the underlying residuated lattice is complete. If Z is a fuzzy bisim-ulation between fuzzy Kripke models M and M ′ , then it satisfies the following conditions for all x ∈ ∆ M , x ′ ∈ ∆ M ′ and ̺ ∈ Σ A : Z ( x, x ′ ) ≤ inf { p M ( x ) ⇔ p M ′ ( x ) | p ∈ Σ P } (25) Z − ◦ ̺ M ≤ ̺ M ′ ◦ Z − (26) Z ◦ ̺ M ′ ≤ ̺ M ◦ Z. (27) Conversely, if the underlying residuated lattice is also linear, fuzzy Kripke models M and M ′ areimage-finite and Z : ∆ M × ∆ M ′ → L is a fuzzy relation satisfying the conditions (25)–(27), then Z is a fuzzy bisimulation between M and M ′ . The proof of this proposition is straightforward.The above proposition is related to Remark 3.4 of [19]. In [11], Fan studied fuzzy bisimulationsthat are defined for fuzzy Kripke models (over a signature with | Σ A | = 1) using conditions like (25)–(27) and the G¨odel semantics over the unit interval [0 , et al. studied fuzzybisimulations that are defined for interpretations in description logics using conditions like (22)–(24) and the G¨odel semantics over the unit interval [0 , (cid:15) , which is the G¨odel t-norm.The relationship between (22)–(24) and (25)–(27) is characterized by the above proposition. On onehand, the conditions (22)–(24) do not require the underlying residuated lattice to be complete and,as discussed in [19], the style is appropriate for the extension that deals with number restrictions(in description logics) and graded modalities. On the other hand, when restricting to completeresiduated lattices and the case without graded modalities, the conditions (25)–(27) are weakerand make the notion of fuzzy bisimulation stronger for non-image-finite fuzzy Kripke models. Example 3.3.
Let Σ A = { ̺ } , Σ P = { p } and L = [0 , M and M ′ depicted and specified below. M M ′ u : 0 v : 0 . w : 0 . u ′ : 0 v ′ : 0 . w ′ : 0 . ∆ M = { u, v, w } , ∆ M ′ = { u ′ , v ′ , w ′ } , • p M = { u : 0 , v : 0 . , w : 0 . } , p M ′ = { u ′ : 0 , v ′ : 0 . , w ′ : 0 . } , • ̺ M = {h u, v i : 0 . , h u, w i : 1 } , ̺ M ′ = {h u ′ , v ′ i : 1 , h u ′ , w ′ i : 0 . } .In the case when (cid:15) is the G¨odel, Lukasiewicz or product t-norm, the greatest fuzzy bisimulation Z between M and M ′ can be computed as follows: • Z ( v, v ′ ) = (0 . ⇔ .
5) = 1, Z ( w, w ′ ) = (0 . ⇔ .
8) = 1; • Z ( v, w ′ ) = (0 . ⇔ . Z ( w, v ′ ) = (0 . ⇔ . • Z ( v, u ′ ) ≤ (0 . ⇔
0) = 0, Z ( w, u ′ ) ≤ (0 . ⇔
0) = 0; • Z ( u, v ′ ) ≤ (0 ⇔ .
5) = 0, Z ( u, w ′ ) ≤ (0 ⇔ .
8) = 0; • ( ̺ M ′ ◦ Z − )( u ′ , v ) = 1, ( ̺ M ′ ◦ Z − )( u ′ , w ) = max { . , (0 . ⇔ . } = 0 . Z ( u, u ′ ) ≤ . • ( ̺ M ◦ Z )( u, w ′ ) = 1, ( ̺ M ◦ Z )( u, v ′ ) = max { . , (0 . ⇔ . } ,thus, the condition (27) only requires Z ( u, u ′ ) ≤ max { . , (0 . ⇔ . } ; • therefore, Z ( u, u ′ ) = max { . , (0 . ⇔ . } .That is, • if (cid:15) is the G¨odel t-norm, then Z = {h u, u ′ i : 0 . , h v, v ′ i : 1 , h w, w ′ i : 1 , h v, w ′ i : 0 . , h w, v ′ i : 0 . } ; • if (cid:15) is the Lukasiewicz t-norm, then Z = {h u, u ′ i : 0 . , h v, v ′ i : 1 , h w, w ′ i : 1 , h v, w ′ i : 0 . , h w, v ′ i : 0 . } ; • if (cid:15) is the product t-norm, then Z = {h u, u ′ i : 0 . , h v, v ′ i : 1 , h w, w ′ i : 1 , h v, w ′ i : 0 . , h w, v ′ i : 0 . } . (cid:4) A fuzzy bisimulation between M and itself is called a fuzzy auto-bisimulation of M . Proposition 3.4.
Let M , M ′ and M ′′ be fuzzy Kripke models. The function ( λ h x, x ′ i ∈ ∆ M × ∆ M . if x = x ′ then 1 else 0) is a fuzzy auto-bisimulationof M . If Z is a fuzzy bisimulation between M and M ′ , then Z − is a fuzzy bisimulation between M ′ and M . If the underlying residuated lattice is complete, Z is a fuzzy bisimulation between M and M ′ , and Z is a fuzzy bisimulation between M ′ and M ′′ , then Z ◦ Z is a fuzzy bisimulationbetween M and M ′′ . If the underlying residuated lattice is linear and Z is a finite set of fuzzy bisimulations between M and M ′ , then sup Z is also a fuzzy bisimulation between M and M ′ . Proof.
The proofs of the first two assertions are straightforward.Consider the third assertion and assume that the premises hold. We have to show that Z ◦ Z satisfies the conditions (22)–(24). • Consider the condition (22). Let x ∈ ∆ M , x ′ ∈ ∆ M ′ , x ′′ ∈ ∆ M ′′ and p ∈ Σ P . We have that Z ( x, x ′ ) ≤ ( p M ( x ) ⇔ p M ′ ( x )) Z ( x ′ , x ′′ ) ≤ ( p M ′ ( x ′ ) ⇔ p M ′′ ( x ′′ )) . Due to (15), ( p M ( x ) ⇔ p M ′ ( x )) (cid:15) ( p M ′ ( x ′ ) ⇔ p M ′′ ( x ′′ )) ≤ ( p M ( x ) ⇔ p M ′′ ( x ′′ )) . By (2), itfollows that Z ( x, x ′ ) (cid:15) Z ( x ′ , x ′′ ) ≤ ( p M ( x ) ⇔ p M ′′ ( x ′′ )) . Therefore, ( Z ◦ Z )( x, x ′′ ) ≤ ( p M ( x ) ⇔ p M ′′ ( x ′′ )), which completes the proof of (22). • Consider the condition (23). Let x, y ∈ ∆ M , x ′′ ∈ ∆ M ′′ and ̺ ∈ Σ A . We need to show thatthere exists y ′′ ∈ ∆ M ′′ such that( Z ◦ Z )( x, x ′′ ) (cid:15) ̺ M ( x, y ) ≤ ̺ M ′′ ( x ′′ , y ′′ ) (cid:15) ( Z ◦ Z )( y, y ′′ ) . (28)Let x ′ be an arbitrary element of ∆ M ′ . Since Z is a fuzzy bisimulation between M and M ′ ,there exists y ′ ∈ M ′ such that Z ( x, x ′ ) (cid:15) ̺ M ( x, y ) ≤ ̺ M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . (29)Since Z is a fuzzy bisimulation between M ′ and M ′′ , there exists y ′′ ∈ M ′′ such that Z ( x ′ , x ′′ ) (cid:15) ̺ M ′ ( x ′ , y ′ ) ≤ ̺ M ′′ ( x ′′ , y ′′ ) (cid:15) Z ( y ′ , y ′′ ) . (30)By (29), (30) and (2), we have that Z ( x ′ , x ′′ ) (cid:15) Z ( x, x ′ ) (cid:15) ̺ M ( x, y ) ≤ Z ( x ′ , x ′′ ) (cid:15) ̺ M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) ≤ ̺ M ′′ ( x ′′ , y ′′ ) (cid:15) Z ( y ′ , y ′′ ) (cid:15) Z ( y, y ′ ) ≤ ̺ M ′′ ( x ′′ , y ′′ ) (cid:15) ( Z ◦ Z )( y, y ′′ ) , which implies (28) since x ′ is an arbitrary element of ∆ M ′ . • The condition (24) can be proved analogously.Consider now the fourth assertion and assume that the premises hold. It is sufficient to considerthe case when Z = { Z , Z } . We need to prove that Z ∪ Z satisfies the conditions (22)–(24). • Consider the condition (22). Let x ∈ ∆ M , x ′ ∈ ∆ M ′ and p ∈ Σ P . Since Z and Z arefuzzy bisimulations between M and M ′ , we have that Z ( x, x ′ ) ≤ ( p M ( x ) ⇔ p M ′ ( x )) and Z ( x, x ′ ) ≤ ( p M ( x ) ⇔ p M ′ ( x )). Hence, ( Z ∪ Z )( x, x ′ ) ≤ ( p M ( x ) ⇔ p M ′ ( x )).9 Consider the condition (23). Let x, y ∈ ∆ M , x ′ ∈ ∆ M ′ and ̺ ∈ Σ A . We need to show thatthere exists y ′ ∈ ∆ M ′ such that( Z ∪ Z )( x, x ′ ) (cid:15) ̺ M ( x, y ) ≤ ̺ M ′ ( x ′ , y ′ ) (cid:15) ( Z ∪ Z )( y, y ′ ) . (31)Without loss of generality, assume that Z ( x, x ′ ) ≤ Z ( x, x ′ ). Since Z is a fuzzy bisimulationbetween M and M ′ , there exists y ′ ∈ ∆ M ′ such that Z ( x, x ′ ) (cid:15) ̺ M ( x, y ) ≤ ̺ M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . Thus, ( Z ∪ Z )( x, x ′ ) (cid:15) ̺ M ( x, y ) = Z ( x, x ′ ) (cid:15) ̺ M ( x, y ) ≤ ̺ M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) ≤ ̺ M ′ ( x ′ , y ′ ) (cid:15) ( Z ∪ Z )( y, y ′ ) . • The condition (24) can be proved analogously. (cid:4)
Corollary 3.5.
Let M be a fuzzy Kripke model. If Z is the greatest fuzzy auto-bisimulation of M and the underlying residuated lattice is complete, then Z is a fuzzy equivalence relation. Proof.
Suppose that Z is the greatest fuzzy auto-bisimulation of M and the underlying residuatedlattice is complete. By the assertion 1 of Proposition 3.4, Z is reflexive. By the assertion 2 ofProposition 3.4, Z − is a fuzzy auto-bisimulation of M . Hence, Z − ≤ Z and Z is symmetric. Bythe assertion 3 of Proposition 3.4, Z ◦ Z is a fuzzy auto-bisimulation of M . Hence, Z ◦ Z ≤ Z and Z is transitive. Therefore, Z is a fuzzy equivalence relation. (cid:4)
4. Invariance Results
We say that a formula ϕ is invariant under fuzzy bisimulations w.r.t. fPDL − Φ if, for every fuzzyKripke models M and M ′ that are witnessed w.r.t. fPDL − Φ and for every fuzzy bisimulation Z between M and M ′ , Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )) for all x ∈ ∆ M and x ′ ∈ ∆ M ′ . Theorem 4.1.
All formulas of fPDL − Φ are invariant under fuzzy bisimulations w.r.t. fPDL − Φ ifthe underlying residuated lattice L satisfies the following conditions:if ∪ / ∈ Φ , then L is linear; (32) if → / ∈ Φ or ? / ∈ Φ , then L is a Heyting algebra. (33)This theorem is an immediate consequence of the following lemma. Lemma 4.2.
Let M and M ′ be fuzzy Kripke models that are witnessed w.r.t. fPDL − Φ and Z a fuzzy bisimulation between M and M ′ . Suppose that the underlying residuated lattice satisfiesthe conditions (32) and (33). Then, the following properties hold for every formula ϕ of fPDL − Φ ,every program α of fPDL − Φ and every possible values of the free variables: Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )) (34) ∃ y ′ ∈ ∆ M ′ ( Z ( x, x ′ ) (cid:15) α M ( x, y ) ≤ α M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ )) (35) ∃ y ∈ ∆ M ( Z ( x, x ′ ) (cid:15) α M ′ ( x ′ , y ′ ) ≤ α M ( x, y ) (cid:15) Z ( y, y ′ )) . (36)10 roof. We prove this lemma by induction on the structures of ϕ and α . First, consider theassertion (35). Let x, y ∈ ∆ M and x ′ ∈ ∆ M ′ . It suffices to show that there exists y ′ ∈ ∆ M ′ suchthat Z ( x, x ′ ) (cid:15) α M ( x, y ) ≤ α M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . (37)The base case occurs when α is an atomic program and follows from (23). The induction steps aregiven below. • Case α = ( ψ ?) (and ? / ∈ Φ): If x = y , then α M ( x, y ) = 0 and, by (5), the assertion (37)clearly holds. Suppose x = y and take y ′ = x ′ . By the induction assumption of (34), Z ( x, x ′ ) ≤ ( ψ M ( x ) ⇔ ψ M ′ ( x ′ )). Hence, by (21), Z ( x, x ′ ) (cid:15) ψ M ( x ) = Z ( x, x ′ ) (cid:15) ψ M ′ ( x ′ ) , which implies (37). • Case α = β ∪ γ (and ∪ / ∈ Φ): Without loss of generality, suppose β M ( x, y ) ≥ γ M ( x, y ). Thus, α M ( x, y ) = β M ( x, y ). By the induction assumption of (35), there exists y ′ ∈ ∆ M ′ such that Z ( x, x ′ ) (cid:15) β M ( x, y ) ≤ β M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . Thus, Z ( x, x ′ ) (cid:15) α M ( x, y ) = Z ( x, x ′ ) (cid:15) β M ( x, y ) ≤ β M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) ≤ α M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . • Case α = β ◦ γ : Since M is witnessed w.r.t. fPDL − Φ , there exists z ∈ ∆ M such that α M ( x, y ) = β M ( x, z ) (cid:15) γ M ( z, y ). By the induction assumption of (35), there exist z ′ and y ′ such that: Z ( x, x ′ ) (cid:15) β M ( x, z ) ≤ β M ′ ( x ′ , z ′ ) (cid:15) Z ( z, z ′ ) Z ( z, z ′ ) (cid:15) γ M ( z, y ) ≤ γ M ′ ( z ′ , y ′ ) (cid:15) Z ( y, y ′ ) . Since (cid:15) is associative, it follows that Z ( x, x ′ ) (cid:15) α M ( x, y ) = Z ( x, x ′ ) (cid:15) β M ( x, z ) (cid:15) γ M ( z, y ) ≤ β M ′ ( x ′ , z ′ ) (cid:15) Z ( z, z ′ ) (cid:15) γ M ( z, y ) ≤ β M ′ ( x ′ , z ′ ) (cid:15) γ M ′ ( z ′ , y ′ ) (cid:15) Z ( y, y ′ ) ≤ α M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . • Case α = β ∗ : Since M is witnessed w.r.t. fPDL − Φ , there exist x , . . . , x k ∈ ∆ M such that x = x , x k = y and α M ( x, y ) = β M ( x , x ) (cid:15) · · · (cid:15) β M ( x k − , x k ) . Let x ′ = x ′ . By the induction assumption of (35), there exist x ′ , . . . , x ′ k ∈ ∆ M ′ such that Z ( x i , x ′ i ) (cid:15) β M ( x i , x i +1 ) ≤ β M ′ ( x ′ i , x ′ i +1 ) (cid:15) Z ( x i +1 , x ′ i +1 )11or all 0 ≤ i < k . Since (cid:15) is associative, it follows that Z ( x , x ′ ) (cid:15) α M ( x , x k )= Z ( x , x ′ ) (cid:15) β M ( x , x ) (cid:15) · · · (cid:15) β M ( x k − , x k ) ≤ β M ′ ( x ′ , x ′ ) (cid:15) Z ( x , x ′ ) (cid:15) β M ( x , x ) (cid:15) · · · (cid:15) β M ( x k − , x k ) ≤ β M ′ ( x ′ , x ′ ) (cid:15) β M ′ ( x ′ , x ′ ) (cid:15) Z ( x , x ′ ) (cid:15) β M ( x , x ) (cid:15) · · · (cid:15) β M ( x k − , x k ) ≤ . . . ≤ β M ′ ( x ′ , x ′ ) (cid:15) · · · (cid:15) β M ′ ( x ′ k − , x ′ k ) (cid:15) Z ( x k , x ′ k ) ≤ α M ′ ( x ′ , x ′ k ) (cid:15) Z ( x k , x ′ k ) . Taking y ′ = x ′ k , we obtain (37).The assertion (36) can be proved analogously as for (35).Consider the assertion (34). The case when ϕ = a is trivial. The case when ϕ = p follows fromthe condition (22). The case when ϕ = ¬ ψ is reduced to the case when ϕ = ( ψ → • Case ϕ = ψ ∧ ξ : We have ϕ M ( x ) = ψ M ( x ) ∧ ξ M ( x ) and ϕ M ′ ( x ′ ) = ψ M ′ ( x ′ ) ∧ ξ M ′ ( x ′ ). Bythe induction assumption of (34), Z ( x, x ′ ) ≤ ( ψ M ( x ) ⇔ ψ M ′ ( x ′ )) (38) Z ( x, x ′ ) ≤ ( ξ M ( x ) ⇔ ξ M ′ ( x ′ )) . (39)By (16), ( ψ M ( x ) ⇔ ψ M ′ ( x ′ )) ∧ ( ξ M ( x ) ⇔ ξ M ′ ( x ′ )) ≤ ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )) . (40)The assertion (34) follows from (38), (39) and (40). • The case ϕ = ( ψ ∨ ξ ) is similar to the previous case, using (17) instead of (16). • Case ϕ = ( ψ → ξ ) (and → / ∈ Φ): The proof is similar to the proof of the case when ϕ = ψ ∧ ξ ,using (20) instead of (16). • Case ϕ = ( a → ψ ): We have ϕ M ( x ) = ( a ⇒ ψ M ( x )) and ϕ M ′ ( x ′ ) = ( a ⇒ ψ M ′ ( x ′ )). Bythe induction assumption of (34), Z ( x, x ′ ) ≤ ( ψ M ( x ) ⇔ ψ M ′ ( x ′ )) . The assertion (34) followsfrom this and (18). • The case ϕ = ( ψ → a ) is similar to the previous case, using (19) instead of (18). • Case ϕ = h α i ψ : Since M is witnessed w.r.t. fPDL − Φ , there exists y ∈ ∆ M such that ϕ M ( x ) = α M ( x, y ) (cid:15) ψ M ( y ) . (41)By the induction assumption of (35), there exists y ′ ∈ ∆ M ′ such that Z ( x, x ′ ) (cid:15) α M ( x, y ) ≤ α M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ) . (42)By definition, α M ′ ( x ′ , y ′ ) (cid:15) ψ M ′ ( y ′ ) ≤ ϕ M ′ ( x ′ ) . (43)12y the induction assumption of (34), Z ( y, y ′ ) ≤ ( ψ M ( y ) ⇔ ψ M ′ ( y ′ )) . (44)By (42) and (1), Z ( x, x ′ ) ≤ ( α M ( x, y ) ⇒ Z ( y, y ′ ) (cid:15) α M ′ ( x ′ , y ′ )) . By (44), (2) and (3), it follows that Z ( x, x ′ ) ≤ α M ( x, y ) ⇒ ( ψ M ( y ) ⇔ ψ M ′ ( y ′ )) (cid:15) α M ′ ( x ′ , y ′ ) . Since (cid:15) is commutative, by (10) and (3), it follows that Z ( x, x ′ ) ≤ ( α M ( x, y ) ⇒ ( ψ M ( y ) ⇔ α M ′ ( x ′ , y ′ ) (cid:15) ψ M ′ ( y ′ ))) . By (13), it follows that Z ( x, x ′ ) ≤ ( α M ( x, y ) (cid:15) ψ M ( y ) ⇒ α M ′ ( x ′ , y ′ ) (cid:15) ψ M ′ ( y ′ )) . By (41), (43) and (3), it follows that Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇒ ϕ M ′ ( x ′ )) . Analogously, it can be shown that Z ( x, x ′ ) ≤ ( ϕ M ′ ( x ′ ) ⇒ ϕ M ( x )) . Therefore, Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )) . • Case ϕ = [ α ] ψ : Since M ′ is witnessed w.r.t. fPDL − Φ , there exists y ′ ∈ ∆ M ′ such that ϕ M ′ ( x ′ ) = ( α M ′ ( x ′ , y ′ ) ⇒ ψ M ′ ( y ′ )) . (45)By the induction assumption of (36), there exists y ∈ ∆ M such that Z ( x, x ′ ) (cid:15) α M ′ ( x ′ , y ′ ) ≤ α M ( x, y ) (cid:15) Z ( y, y ′ ) . (46)By definition, ϕ M ( x ) ≤ ( α M ( x, y ) ⇒ ψ M ( y )) . (47)By the induction assumption of (34), Z ( y, y ′ ) ≤ ( ψ M ( y ) ⇔ ψ M ′ ( y ′ )) . (48)By (46) and (1), Z ( x, x ′ ) ≤ ( α M ′ ( x ′ , y ′ ) ⇒ Z ( y, y ′ ) (cid:15) α M ( x, y )) . By (48), (2) and (3), it follows that Z ( x, x ′ ) ≤ ( α M ′ ( x ′ , y ′ ) ⇒ ( ψ M ( y ) ⇔ ψ M ′ ( y ′ )) (cid:15) α M ( x, y )) . (cid:15) is commutative, by (9) and (3), it follows that Z ( x, x ′ ) ≤ ( α M ′ ( x ′ , y ′ ) ⇒ (( α M ( x, y ) ⇒ ψ M ( y )) ⇒ ψ M ′ ( y ′ ))) . By (11), it follows that Z ( x, x ′ ) ≤ (( α M ( x, y ) ⇒ ψ M ( y )) ⇒ ( α M ′ ( x ′ , y ′ ) ⇒ ψ M ′ ( y ′ ))) . By (47), (45) and (3), it follows that Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇒ ϕ M ′ ( x ′ )) . Analogously, it can be shown that Z ( x, x ′ ) ≤ ( ϕ M ′ ( x ′ ) ⇒ ϕ M ( x )) . Therefore, Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )) . This completes the proof. (cid:4)
The following lemma is a counterpart of Lemma 4.2 for fK . Lemma 4.3.
Let M and M ′ be fuzzy Kripke models that are witnessed w.r.t. fK and Z a fuzzybisimulation between M and M ′ . Then, the following properties hold for every x ∈ ∆ M , x ′ ∈ ∆ M ′ and every formula ϕ of fK : Z ( x, x ′ ) ≤ ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )) . This lemma can be proved analogously as done for the assertion (34) of Lemma 4.2, by using (23)and (24) instead of (35) and (36), respectively. Roughly speaking, the proof is a simplification ofthe proof of Lemma 4.2.
Remark 4.4.
Analyzing the proof of Lemma 4.2, it can be seen that the condition (33) ( L is aHeyting algebra if → / ∈ Φ or ? / ∈ Φ) can be replaced by the conditions (20) and (21). This alsoapplies to Theorem 4.1. (cid:4)
Remark 4.5.
To justify that a condition like (33) (or (20) and (21) together) is essential forLemma 4.2 and Theorem 4.1, we show that, if → / ∈ Φ or ? / ∈ Φ, L = [0 ,
1] and (cid:15) is the Lukasiewiczor product t-norm, then there exist finite fuzzy Kripke models M and M ′ , a fuzzy bisimulation Z between M and M ′ , x ∈ ∆ M , x ′ ∈ ∆ M ′ and a formula ϕ of fPDL − Φ such that Z ( x, x ′ ) ( ϕ M ( x ) ⇔ ϕ M ′ ( x ′ )). Let • Σ A = ∅ , Σ P = { p, q } , ϕ = ( p → q ), ψ = [ p ?] q , • ∆ M = { v } , p M = { v : 0 . } , q M = { v : 0 . } , • ∆ M ′ = { v ′ } , p M ′ = { v ′ : 0 . } , q M ′ = { v ′ : 0 . } , • (cid:15) be the Lukasiewicz or product t-norm, • Z be the greatest fuzzy bisimulation between M and M ′ .14f (cid:15) is the Lukasiewicz t-norm, then • Z ( v, v ′ ) = min { ( p M ( v ) ⇔ p M ′ ( v ′ )) , ( q M ( v ) ⇔ q M ′ ( v ′ )) } = 0 . • ϕ M ( v ) = ψ M ( v ) = 1, ϕ M ′ ( v ′ ) = ψ M ′ ( v ′ ) = 0 . • ( ϕ M ( v ) ⇔ ϕ M ′ ( v ′ )) = ( ψ M ( v ) ⇔ ψ M ′ ( v ′ )) = 0 . (cid:15) is the product t-norm, then • Z ( v, v ′ ) = min { ( p M ( v ) ⇔ p M ′ ( v ′ )) , ( q M ( v ) ⇔ q M ′ ( v ′ )) } = 0 . • ϕ M ( v ) = ψ M ( v ) = 1, ϕ M ′ ( v ′ ) = ψ M ′ ( v ′ ) = 1 / • ( ϕ M ( v ) ⇔ ϕ M ′ ( v ′ )) = ( ψ M ( v ) ⇔ ψ M ′ ( v ′ )) = 1 / Z ( v, v ′ ) ( ϕ M ( v ) ⇔ ϕ M ′ ( v ′ )) and Z ( v, v ′ ) ( ψ M ( v ) ⇔ ψ M ′ ( v ′ )). (cid:4)
5. The Hennessy-Milner Property
In this section, we present and prove the Hennessy-Milner property of fuzzy bisimulations. It isformulated for the class of modally saturated models, which is larger than the class of image-finitemodels. Our notion of modal saturatedness is a counterpart of the ones given in [12, 2, 19].A fuzzy Kripke model M is said to be modally saturated (w.r.t. fK and the underlying resid-uated lattice L ) if, for every a ∈ L \ { } , every x ∈ ∆ M , every ̺ ∈ Σ A and every infinite set Γ offormulas in fK , if for every finite subset Λ of Γ there exists y ∈ ∆ M such that ̺ M ( x, y ) (cid:15) ϕ M ( y ) ≥ a for all ϕ ∈ Λ, then there exists y ∈ ∆ M such that ̺ M ( x, y ) (cid:15) ϕ M ( y ) ≥ a for all ϕ ∈ Γ. Proposition 5.1.
All image-finite fuzzy Kripke models are modally saturated.
Proof.
Let M be an image-finite fuzzy Kripke model, let a ∈ L \ { } , x ∈ ∆ M , ̺ ∈ Σ A and letΓ be an infinite set of formulas in fK . Assume that, for every finite subset Λ of Γ, there exists y ∈ ∆ M such that ̺ M ( x, y ) (cid:15) ϕ M ( y ) ≥ a for all ϕ ∈ Λ. For a contradiction, suppose that, for every y ∈ ∆ M , there exists ϕ y ∈ Γ such that ̺ M ( x, y ) (cid:15) ϕ M y ( y ) a . Let ϕ be an arbitrary formula ofΓ and let Λ = { ϕ y | ̺ M ( x, y ) > } ∪ { ϕ } . Since M is image-finite, Λ is finite. For every y ∈ ∆ M ,if ̺ M ( x, y ) = 0, then by (5), ̺ M ( x, y ) (cid:15) ϕ M ( y ) = 0 a , else ϕ y ∈ Λ and ̺ M ( x, y ) (cid:15) ϕ M y ( y ) a .Hence, for every y ∈ ∆ M , there exists ϕ ∈ Λ such that ̺ M ( x, y ) (cid:15) ϕ M ( y ) a . This contradictsthe assumption. (cid:4) Let L be a complete residuated lattice. We say that the operator (cid:15) is continuous (w.r.t. infima)if, for every x ∈ L and Y ⊆ L , x (cid:15) inf Y = inf { x (cid:15) y | y ∈ Y } . Clearly, all the G¨odel, Lukasiewiczand product t-norms (specified when L is the unit interval [0 , L is a Heyting algebra, then (cid:15) is continuous. Theorem 5.2.
Let M and M ′ be fuzzy Kripke models that are witnessed w.r.t. fK and modallysaturated. Suppose that the underlying residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i is complete and (cid:15) is continuous. Then, the fuzzy relation Z : ∆ M × ∆ M ′ → L specified by Z ( x, x ′ ) = inf { ϕ M ( x ) ⇔ ϕ M ′ ( x ′ ) | ϕ is a formula of fK } is the greatest fuzzy bisimulation between M and M ′ . These conditions are satisfied, for example, when M and M ′ are image-finite. roof. By Lemma 4.3, it is sufficient to prove that Z is a fuzzy bisimulation between M and M ′ .By definition, Z satisfies the condition (22).We prove that Z satisfies the condition (23). Let ̺ ∈ Σ A , x, y ∈ ∆ M and x ′ ∈ ∆ M ′ .Let a = Z ( x, x ′ ) (cid:15) ̺ M ( x, y ). For a contradiction, suppose that, for every y ′ ∈ ∆ M ′ , a ̺ M ′ ( x ′ , y ′ ) (cid:15) Z ( y, y ′ ). Since (cid:15) is continuous, by the definition of Z ( y, y ′ ), it follows that,for every y ′ ∈ ∆ M ′ , there exists a formula ϕ y ′ of fK such that a ̺ M ′ ( x ′ , y ′ ) (cid:15) ( ϕ M y ′ ( y ) ⇔ ϕ M ′ y ′ ( y ′ )) . For every y ′ ∈ ∆ M ′ , let ψ y ′ = ( ϕ y ′ → ϕ M y ′ ( y )) ∧ ( ϕ M y ′ ( y ) → ϕ y ′ ) . Let Γ = { ψ y ′ | y ′ ∈ ∆ M ′ } . Observe that, for every y ′ ∈ ∆ M ′ , ψ M y ′ ( y ) = 1 (by (4)) and a ̺ M ′ ( x ′ , y ′ ) (cid:15) ψ M ′ y ′ ( y ′ ). Since M ′ is modally saturated, it follows that there exists a finite subset Ψof Γ such that, for every y ′ ∈ ∆ M ′ , there exists ψ ∈ Ψ such that a ̺ M ′ ( x ′ , y ′ ) (cid:15) ψ M ′ ( y ′ ) . (49)Let ϕ = h ̺ i V Ψ. It is a formula of fK . Thus, ϕ M ( x ) ≥ ̺ M ( x, y ) since ( V Ψ) M ( y ) = 1. Since M ′ is witnessed w.r.t. fK , by (49) and (2), we have that a ϕ M ′ ( x ′ ), which means Z ( x, x ′ ) (cid:15) ̺ M ( x, y ) ϕ M ′ ( x ′ ) . Since ϕ M ( x ) ≥ ̺ ( x, y ), by (2), it follows that Z ( x, x ′ ) (cid:15) ϕ M ( x ) ϕ M ′ ( x ′ ) . By (1), this implies that Z ( x, x ′ ) ( ϕ M ( x ) ⇒ ϕ M ′ ( x ′ )) , which contradicts the definition of Z ( x, x ′ ).Analogously, it can be proved that Z satisfies the condition (24). This completes the proof. (cid:4) Corollary 5.3.
Let M be a fuzzy Kripke model that is witnessed w.r.t. fK and modally satu-rated. Suppose that the underlying residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i is complete and (cid:15) iscontinuous. Then, the greatest fuzzy auto-bisimulation of M exists and is a fuzzy equivalencerelation. This corollary follows immediately from Theorem 5.2 and Corollary 3.5.
Corollary 5.4.
Let M and M ′ be fuzzy Kripke models that are witnessed w.r.t. fPDL − Φ andmodally saturated. Suppose that the underlying residuated lattice L = h L, ≤ , (cid:15) , ⇒ , , i is complete,satisfies the conditions (32) and (33), and (cid:15) is continuous. Then, for every x ∈ ∆ M and x ′ ∈ ∆ M ′ , inf { ϕ M ( x ) ⇔ ϕ M ′ ( x ′ ) | ϕ is a formula of fK } = inf { ϕ M ( x ) ⇔ ϕ M ′ ( x ′ ) | ϕ is a formula of fPDL − Φ } . This corollary follows immediately from Theorem 5.2 and Lemma 4.2. These conditions are satisfied, for example, when M is image-finite. . Related Work The works [10, 11, 19] on fuzzy bisimulations have been briefly discussed in the introduction.We give below some additional remarks on these works before discussing other works related tological characterizations of fuzzy/crisp bisimulations or simulations.The results on fuzzy bisimulations of [10] are formulated only for finite social networks over theresiduated lattice [0 ,
1] using the G¨odel t-norm. In that work, Fan and Liau did consider extendingtheir results on fuzzy bisimulations to the settings with the Lukasiewicz and product t-norms.However, in [10, Example 2] they claimed that the extension does not work. The problem withthat claim is that the authors used the logical language with the additional conjunction &, whichis interpreted as (cid:15) . In [10] Fan and Liau also studied crisp bisimulations under the name “gen-eralized regular equivalence relations” for finite weighted social networks. They provided logicalcharacterizations for crisp bisimulations under the G¨odel, Lukasiewicz and product semantics. Thecharacterizations are formulated w.r.t. fuzzy multimodal logics possibly with converse, which areextended with involutive negation and/or the Baaz projection operator. They concern invariance ofmodal formulas under crisp bisimulations and the Hennessy-Milner property of crisp bisimulations.In [11] Fan also studied crisp bisimulations for fuzzy monomodal logics under the G¨odel seman-tics. She provided logical characterizations of such bisimulations in the basic fuzzy monomodallogics possibly with converse, which are extended with involutive negation and/or the Baaz projec-tion operator. The results of [11] on invariance of modal formulas and the Hennessy-Milner propertyfor both crisp and fuzzy bisimulations are formulated for image-finite fuzzy Kripke models overa signature with only one accessibility relation.In [19] Nguyen et al . also provided logical characterizations of crisp bisimulations for fuzzydescription logics under the G¨odel semantics. For the case with such bisimulations, the consid-ered logics are extended with the Baaz projection operator or involutive negation. Apart fromresults on invariance of concepts and the Hennessy-Milner property of crisp/fuzzy bisimulations,the work [19] also gives results on conditional invariance of TBoxes and ABoxes under crisp/fuzzybisimulations, separation of the expressive power of fuzzy description logics, and minimization offuzzy interpretations by using crisp bisimulations.In [28, 26, 27], Wu et al . provided logical characterizations of crisp bisimulations/simulations fora few variants of fuzzy transition systems. The results are formulated w.r.t. crisp Hennessy-Milnerlogics, which use values from the unit interval [0 ,
1] as thresholds for modal operators.In [23] Pan et at. provided logical characterizations of fuzzy simulations for finite fuzzy la-beled transition systems over finite residuated lattices. They are formulated w.r.t. an existentialHennessy-Milner logic. In [24] Pan et at. provided logical characterizations of simulations for finitequantitative transition systems over finite Heyting algebras. Quantitative transition systems aretransition systems without labels for states but extended with a fuzzy equality relation betweenactions. Simulations studied in [24] are either fuzzy simulations or crisp simulations parameterizedby a threshold used as a cut for the fuzzy equality relation between actions. The logical char-acterizations of simulations provided in [24] are formulated w.r.t. an existential cut-based crispHennessy-Milner logic for the case of crisp simulations, and w.r.t. an existential fuzzy Hennessy-Milner logic for the case of fuzzy simulations.In [18] we provided logical characterizations of crisp cut-based simulations and bisimilarity for alarge class of fuzzy description logics under the Zadeh semantics. The results concern preservationof information by such simulations, conditional invariance of ABoxes and TBoxes under bisimilaritybetween witnessed interpretations, as well as the Hennessy-Milner property for fuzzy description17ogics under the Zadeh semantics.
7. Conclusions
We have provided and proved logical characterizations of fuzzy bisimulations in the fuzzy propo-sitional dynamic logic fPDL and its sublogics over residuated lattices. The results concern invari-ance of formulas under fuzzy bisimulations and the Hennessy-Milner property of fuzzy bisimula-tions. The first theorem is formulated for fuzzy Kripke models that are witnessed, whereas thesecond theorem is formulated for fuzzy Kripke models that are witnessed and modally saturated.Our results can be reformulated for other fuzzy structures such as fuzzy label transition systemsand fuzzy interpretations in description logics. It is worth emphasizing that our results concernfuzzy bisimulations over general residuated lattices . They are interesting from the theoretical pointof view, as the previous results on fuzzy bisimulations are formulated and proved only for theresiduated lattice [0 ,
1] using the G¨odel t-norm or Heyting algebras.In certain applications, the product t-norm is more suitable than the G¨odel t-norm. For exam-ple, the closeness of a person to his/her great-grandmother can be assumed to be smaller than thecloseness of that person to his/her mother. Furthermore, the product residuum is continuous w.r.t.both the arguments, whereas the G¨odel residuum is not. This causes that the product residuumis more resistant to noise than the G¨odel residuum. Our logical characterizations of fuzzy bisim-ulations open the way for studying logical similarity between individuals and concept learning infuzzy description logics under the product semantics by applying fuzzy bisimulations.On the technical matters, our results are formulated on a general level. Residuated latticesconsidered in this work may be infinite, whereas the work [23] considers only finite residuatedlattices. The class of fuzzy Kripke models that are witnessed and modally saturated is larger thanthe class of image-finite fuzzy Kripke models studied in [9, 11], the class of finite weighted socialnetworks studied in [10] and the class of finite fuzzy labeled transition systems studied in [23, 24].The considered fuzzy logic fPDL contains the program constructors of propositional dynamic logic,which are absent in [9, 10, 11, 23, 24]. They correspond to role constructors in description logics.
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Note that (cid:15) is commutative and associative. Let x, x ′ , y, y ′ , z ∈ L . • Since (cid:15) is commutative, to prove (2), it is sufficient to show that, if x ≤ x ′ , then x (cid:15) y ≤ x ′ (cid:15) y .Assume that x ≤ x ′ . By (1), x ′ ≤ ( y ⇒ ( x ′ (cid:15) y )). Hence, x ≤ ( y ⇒ ( x ′ (cid:15) y )). By (1), itfollows that x (cid:15) y ≤ x ′ (cid:15) y , which completes the proof of (2). • Consider the assertion (3) and assume that x ′ ≤ x and y ≤ y ′ . By (2), ( x ⇒ y ) (cid:15) x ′ ≤ ( x ⇒ y ) (cid:15) x . By (1), ( x ⇒ y ) (cid:15) x ≤ y . Hence, ( x ⇒ y ) (cid:15) x ′ ≤ y , which implies ( x ⇒ y ) ≤ ( x ′ ⇒ y ) by using (1). By (1), ( x ′ ⇒ y ) (cid:15) x ′ ≤ y . Since y ≤ y ′ , it follows that ( x ′ ⇒ y ) (cid:15) x ′ ≤ y ′ ,which implies ( x ′ ⇒ y ) ≤ ( x ′ ⇒ y ′ ) by using (1). We have proved that ( x ⇒ y ) ≤ ( x ′ ⇒ y )and ( x ′ ⇒ y ) ≤ ( x ′ ⇒ y ′ ), which together imply (3). • Consider the assertion (4). We have that x ≤ y iff 1 (cid:15) x ≤ y iff 1 ≤ ( x ⇒ y ), by using (1).The last inequality implies (4). 19 Since 0 ≤ ( x ⇒ (cid:15) x ≤
0. Hence, x (cid:15) (cid:15) x ≤ • Consider the assertion (6). By (2), x (cid:15) y ≤ x (cid:15) ( y ∨ z ) and x (cid:15) z ≤ x (cid:15) ( y ∨ z ). Hence, x (cid:15) y ∨ x (cid:15) z ≤ x (cid:15) ( y ∨ z ). It remains to prove the converse. By (1), y ≤ ( x ⇒ x (cid:15) y ). By (3),it follows that y ≤ ( x ⇒ x (cid:15) y ∨ x (cid:15) z ). Similarly, it can be shown that z ≤ ( x ⇒ x (cid:15) y ∨ x (cid:15) z ).Hence, y ∨ z ≤ ( x ⇒ x (cid:15) y ∨ x (cid:15) z ). By (1), it follows that x (cid:15) ( y ∨ z ) ≤ x (cid:15) y ∨ x (cid:15) z . Thiscompletes the proof of (6). • The assertion (7) follows from (1) and the commutativity of (cid:15) . • Consider the assertions (8) and (9). By (7) and (2), x (cid:15) ( x ⇒ y ) (cid:15) ( y ⇒ z ) ≤ y (cid:15) ( y ⇒ z ) ≤ z .Hence, x (cid:15) ( y ⇒ z ) (cid:15) ( x ⇒ y ) ≤ z . This implies (8), by using (1). The assertion (9) followsfrom (8), by using (2). • Consider the assertion (10). We need to prove that x (cid:15) ( y ⇔ z ) ≤ ( y ⇒ x (cid:15) z ) (A.1) x (cid:15) ( y ⇔ z ) ≤ ( x (cid:15) z ⇒ y ) (A.2)By (1), ( y ⇒ z ) (cid:15) y ≤ z . By (2), it follows that x (cid:15) ( y ⇒ z ) (cid:15) y ≤ x (cid:15) z . By (1), it follows that x (cid:15) ( y ⇒ z ) ≤ ( y ⇒ x (cid:15) z ). This implies (A.1), by using (2). Consider the assertion (A.2).By (1), ( z ⇒ y ) (cid:15) z ≤ y . By (2), it follows that x (cid:15) ( z ⇒ y ) (cid:15) x (cid:15) z ≤ y . By (1), it followsthat x (cid:15) ( z ⇒ y ) ≤ ( x (cid:15) z ⇒ y ). This implies (A.2), by using (2). • Consider the assertions (11)–(13). By (1), ( x ⇒ ( y ⇒ z )) (cid:15) x ≤ ( y ⇒ z ) and ( y ⇒ z ) (cid:15) y ≤ z .By (2), it follows that ( x ⇒ ( y ⇒ z )) (cid:15) y (cid:15) x ≤ z. (A.3)By applying (1) twice, this implies ( x ⇒ ( y ⇒ z )) ≤ ( y ⇒ ( x ⇒ z )), which in turn im-plies (11). The assertion (A.3) also implies (12), by using (1) and the commutativity of (cid:15) .The assertion (13) follows from (12), by using (3). • Consider the assertion (14). By (7) and (2), we have that x (cid:15) ( x ⇒ y ) (cid:15) ( y ⇒ z ) ≤ z . Theassertion (14) follows from this, using (1) and the commutativity of (cid:15) . • The assertion (15) follows from (14), using (2) and the commutativity of (cid:15) and ⇔ . • Due to the commutativity of ⇔ , to prove (16) it is sufficient to show that( x ⇔ x ′ ) ∧ ( y ⇔ y ′ ) ≤ ( x ∧ y ⇒ x ′ ∧ y ′ ) . By (1), this is equivalent to (( x ⇔ x ′ ) ∧ ( y ⇔ y ′ )) (cid:15) ( x ∧ y ) ≤ x ′ ∧ y ′ . We need to prove that(( x ⇔ x ′ ) ∧ ( y ⇔ y ′ )) (cid:15) ( x ∧ y ) ≤ x ′ (A.4)(( x ⇔ x ′ ) ∧ ( y ⇔ y ′ )) (cid:15) ( x ∧ y ) ≤ y ′ . (A.5)By (1), ( x ⇒ x ′ ) (cid:15) x ≤ x ′ . This implies (A.4), by using (2). Similarly, (A.5) also holds.20 Due to the commutativity of ⇔ , to prove (17) it is sufficient to show that( x ⇔ x ′ ) ∧ ( y ⇔ y ′ ) ≤ ( x ∨ y ⇒ x ′ ∨ y ′ ) . By (1), this is equivalent to (( x ⇔ x ′ ) ∧ ( y ⇔ y ′ )) (cid:15) ( x ∨ y ) ≤ x ′ ∨ y ′ . By (6), it is sufficientto prove that (( x ⇔ x ′ ) ∧ ( y ⇔ y ′ )) (cid:15) x ≤ x ′ (A.6)(( x ⇔ x ′ ) ∧ ( y ⇔ y ′ )) (cid:15) y ≤ y ′ . (A.7)By (1), ( x ⇒ x ′ ) (cid:15) x ≤ x ′ . This implies (A.6), by using (2). Similarly, (A.7) also holds. • To prove (18), it is sufficient to show that ( x ⇒ y ) ≤ (( z ⇒ x ) ⇒ ( z ⇒ y )). By (1), this isequivalent to ( x ⇒ y ) (cid:15) ( z ⇒ x ) (cid:15) z ≤ y . This latter inequality holds because, by (7) and (2), z (cid:15) ( z ⇒ x ) (cid:15) ( x ⇒ y ) ≤ x (cid:15) ( x ⇒ y ) ≤ y. • To prove (19), it is sufficient to show that ( y ⇒ x ) ≤ (( x ⇒ z ) ⇒ ( y ⇒ z )). By (1), this isequivalent to ( y ⇒ x ) (cid:15) ( x ⇒ z ) (cid:15) y ≤ z . This latter inequality holds because, by (7) and (2), y (cid:15) ( y ⇒ x ) (cid:15) ( x ⇒ z ) ≤ x (cid:15) ( x ⇒ z ) ≤ z. • Consider the assertion (20) and suppose that L is a Heyting algebra. Due to the commuta-tivity of ⇔ , to prove (20) it is sufficient to prove that( x ⇔ x ′ ) (cid:15) ( y ⇔ y ′ ) ≤ (( x ⇒ y ) ⇒ ( x ′ ⇒ y ′ )) . By (2) and (1), it is sufficient to prove that ( x ′ ⇒ x ) (cid:15) ( y ⇒ y ′ ) (cid:15) ( x ⇒ y ) (cid:15) x ′ ≤ y ′ . Thisholds because, by (7) and (2), x ′ (cid:15) ( x ′ ⇒ x ) (cid:15) ( x ⇒ y ) (cid:15) ( y ⇒ y ′ ) ≤ x (cid:15) ( x ⇒ y ) (cid:15) ( y ⇒ y ′ ) ≤ y (cid:15) ( y ⇒ y ′ ) ≤ y ′ . • Consider the assertion (21) and suppose that L is a Heyting algebra and x ≤ ( y ⇔ z ). Since x ≤ ( y ⇔ z ), we have that x ≤ ( y ⇒ z ). By (1), it follows that x ∧ y = x (cid:15) y ≤ z . Hence, x ∧ y ≤ x ∧ z since ∧ is idempotent. Similarly, it can also be shown that x ∧ z ≤ x ∧ y .Therefore, x ∧ y = x ∧ z , which means x (cid:15) y = x (cid:15) z . Appendix B. The Relationship with Fuzzy Bisimulations between Fuzzy Automata
Clearly, a fuzzy Kripke model can be treated as a fuzzy labeled transition system (FLTS) andDefinition 3.1 (which specifies fuzzy bisimulations) can be applied to FLTSs. In [6], ´Ciri´c et al .introduced a few kinds of fuzzy bisimulations (and simulations) for fuzzy automata over completeresiduated lattices. Among them the one that researchers would have in mind as the default iscalled “forward bisimulation”. We recall it below and simply refer to it as fuzzy bisimulationbetween fuzzy automata. After that we relate it to the notion of fuzzy bisimulation between fuzzyKripke models.In this appendix, suppose that the underlying residuated lattice L is complete.21iven fuzzy sets R : X → L , S : Y → L and Z : X × Y → L , we define ( R ◦ Z ) : Y → L and( Z ◦ S ) : X → L to be the fuzzy sets such that( R ◦ Z )( y ) = sup { R ( x ) (cid:15) Z ( x, y ) | x ∈ X } for y ∈ Y ;( Z ◦ S )( x ) = sup { Z ( x, y ) (cid:15) S ( y ) | y ∈ Y } for x ∈ X. A fuzzy automaton over an alphabet Σ (and L ) is a tuple A = h A, δ A , σ A , τ A i , where A is anon-empty set of states, δ A : A × Σ × A → L is the fuzzy transition function, σ A : A → L is thefuzzy set of initial states, and τ A : A → L is the fuzzy set of terminal states. For ̺ ∈ Σ, by δ A ̺ wedenote the fuzzy relation on A such that δ A ̺ ( x, y ) = δ A ( x, ̺, y ) for x, y ∈ A . A fuzzy automaton A is image-finite if, for every ̺ ∈ Σ and every x ∈ A , the set { y | δ A ( x, ̺, y ) > } is finite.Given fuzzy automata A = h A, δ A , σ A , τ A i and B = h B, δ B , σ B , τ B i over an alphabet Σ,a fuzzy bisimulation (called “forward bisimulation” in [6]) between A and B is a fuzzy relation Z : A × B → L satisfying the following conditions for all ̺ ∈ Σ: σ A ≤ σ B ◦ Z − σ B ≤ σ A ◦ ZZ − ◦ δ A ̺ ≤ δ B ̺ ◦ Z − Z ◦ δ B ̺ ≤ δ A ̺ ◦ ZZ − ◦ τ A ≤ τ B Z ◦ τ B ≤ τ A . Given a fuzzy automaton A = h A, δ A , σ A , τ A i over an alphabet Σ, we define the fuzzy Kripkemodel corresponding to A to be the fuzzy Kripke model M over the signature h Σ A , Σ P i withΣ A = Σ and Σ P = { i, f } such that: • ∆ M = A ∪ { s i , s f } , where s i and s f are new states; • i M = { s i : 1 } and f M = { s f : 1 } ; • for every ̺ ∈ Σ A , x, y ∈ A and z ∈ ∆ M : – ̺ M ( x, y ) = δ A ( x, ̺, y ), – ̺ M ( s i , x ) = σ A ( x ) and ̺ M ( x, s f ) = τ A ( x ), – ̺ M ( z, s i ) = ̺ M ( s f , z ) = ̺ M ( s i , s f ) = 0.Thus, s i (resp. s f ) stands for the new unique initial (resp. terminal) state; the propositions i and f are used to identify s i and s f , respectively. The given definition is a counterpart of the definitionof the fuzzy interpretation (in description logic) that corresponds to a fuzzy automaton [22].Recall that, in this appendix, the underlying residuated lattice is assumed to be complete.The following proposition relates our notion of fuzzy bisimulation between fuzzy Kripke modelsto the notion of fuzzy bisimulation between fuzzy automata, which is defined and called “forwardbisimulation” by ´Ciri´c et al. [6]. Its proof is straightforward. Proposition Appendix B.1.
Let A = h A, δ A , σ A , τ A i and A ′ = h A ′ , δ A ′ , σ A ′ , τ A ′ i be fuzzy au-tomata over the same alphabet, M and M ′ the fuzzy Kripke models corresponding to A and A ′ ,respectively. Let s i , s f ∈ ∆ M and s ′ i , s ′ f ∈ ∆ M ′ be the states such that i M ( s i ) = f M ( s f ) = i M ′ ( s ′ i ) = f M ′ ( s ′ f ) = 1 . Let Z be a fuzzy relation between A and A ′ , Z the fuzzy relation between ∆ M and ∆ M ′ such that Z = Z ∪ {h s i , s ′ i i : 1 , h s f , s ′ f i : 1 } . If Z is a fuzzy bisimulation between M and M ′ , then Z is a fuzzy bisimulation between A and A ′ . • Conversely, if the underlying residuated lattice is also linear, A and A ′ are image-finite and Z is a fuzzy bisimulation between A and A ′ , then Z is a fuzzy bisimulation between M and M ′ ..