Hennessy-Milner Property Research Articles

Depth-Bounded Fuzzy Bisimulation for Fuzzy Modal Logic

We introduce depth-bounded fuzzy bisimulation between fuzzy Kripke models. Roughly speaking, a depth-bounded fuzzy bisimulation is a decreasing sequence of fuzzy binary relations whose infimum is a fuzzy bisimulation. We provide logical characterizations of depth-bounded fuzzy bisimulations between fuzzy Kripke models w.r.t. a fuzzy multimodal logic over complete residuated lattices, including fuzzy invariance of formulas of with a modal depth bounded by under the th component of a depth-bounded fuzzy bisimulation, as well as the Hennessy-Milner property of depth-bounded fuzzy bisimulations. We also provide a polynomial-time algorithm for computing the th component of the greatest depth-bounded fuzzy bisimulation between two finite fuzzy Kripke models when the underlying complete residuated lattice is linear.

Depth-bounded fuzzy simulations and bisimulations between fuzzy automata

Simulations and bisimulations are well-established notions in crisp/fuzzy automata theory and are widely used to compare the behaviors of automata. Their main drawback is that they compare the behaviors of fuzzy automata in a crisp manner. Recently, fuzzy simulations and fuzzy bisimulations have been defined for fuzzy automata as a kind of approximate simulations and approximate bisimulations that compare the behaviors of fuzzy automata in a fuzzy manner. However, they still suffer from serious shortcomings. First, they still cannot correlate all fuzzy automata that are intuitively “more or less” (bi)similar. Second, the currently known algorithms for computing the greatest fuzzy simulation or bisimulation between two finite fuzzy automata have an exponential time complexity when the Łukasiewicz or the product structure of fuzzy values is used. This work deals with these problems, providing approximations of fuzzy simulations and fuzzy bisimulations. We define such approximations via a novel notion of decreasing sequences of fuzzy relations whose infima are, under some conditions, fuzzy simulations (respectively, bisimulations). We call such a sequence a depth-bounded fuzzy simulation (respectively, bisimulation), as the nth element from the sequence compares the behaviors of fuzzy automata, but only for words with a length bounded by n. We further provide a logical characterization of the greatest depth-bounded fuzzy simulation or bisimulation between two fuzzy automata by proving that it satisfies the corresponding Hennessy-Milner property. Finally, we provide polynomial-time algorithms for computing the nth component of the greatest depth-bounded fuzzy simulation (respectively, bisimulation) between two finite fuzzy automata.

THE ZHOU ORDINAL OF LABELLED MARKOV PROCESSES OVER SEPARABLE SPACES

AbstractThere exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, whereas the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal {O}$ , and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the Zhou ordinal of $\mathbb {S}$ , $\mathfrak {Z}(\mathbb {S})$ . When $\mathfrak {Z}(\mathbb {S})=0$ , $\mathbb {S}$ satisfies the Hennessy–Milner property. The second author proved the existence of an LMP $\mathbb {S}$ with $\mathfrak {Z}(\mathbb {S}) \geq 1$ and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb {S}$ over separable metrizable spaces having arbitrary large countable $\mathfrak {Z}(\mathbb {S})$ and that it is consistent with the axioms of $\mathit {ZFC}$ that there is such a process with an uncountable Zhou ordinal.

Computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a fuzzy graph-based structure under the Gödel semantics

The similarity measure based on fuzzy bisimulation has the Hennessy-Milner property as a strong logical foundation. It is useful for classification and clustering. In this work, we design an efficient algorithm with the complexity O((mlog⁡l+n)log⁡n) for computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a finite fuzzy labeled graph G under the Gödel semantics, where n, m and l are the number of vertices, the number of non-zero edges and the number of different fuzzy degrees of edges of G, respectively. Our notion of fuzzy partition is novel, defined for finite sets with respect to the Gödel t-norm, with the aim to facilitate the computation of the greatest fuzzy auto-bisimulation. By using that algorithm, we also provide an algorithm with the complexity O(m⋅log⁡l⋅log⁡n+n2) for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the Gödel semantics. This latter algorithm is better (has a lower complexity order) than the previously known algorithms for the considered problem. Our algorithms can be restated for other fuzzy graph-based structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretations in fuzzy description logics.

The Temporal Logic of Coalitional Goal Assignments in Concurrent Multiplayer Games

We introduce and study a natural extension of the Alternating time temporal logic ATL , called Temporal Logic of Coalitional Goal Assignments (TLCGA). It features one new and quite expressive coalitional strategic operator, called the coalitional goal assignment operator ⦉ γ ⦊, where γ is a mapping assigning to each set of players in the game its coalitional goal , formalised by a path formula of the language of TLCGA, i.e., a formula prefixed with a temporal operator X , U , or G , representing a temporalised objective for the respective coalition, describing the property of the plays on which that objective is satisfied. Then, the formula ⦉ γ ⦊ intuitively says that there is a strategy profile Σ for the grand coalition Agt such that for each coalition C , the restriction Σ | C of Σ to C is a collective strategy of C that enforces the satisfaction of its objective γ (C) in all outcome plays enabled by Σ | C . We establish fixpoint characterizations of the temporal goal assignments in a μ-calculus extension of TLCGA, discuss its expressiveness and illustrate it with some examples, prove bisimulation invariance and Hennessy–Milner property for it with respect to a suitably defined notion of bisimulation, construct a sound and complete axiomatic system for TLCGA, and obtain its decidability via finite model property.

Moss' logic for ordered coalgebras

We present a finitary version of Moss' coalgebraic logic for $T$-coalgebras, where $T$ is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor $T_\omega^\partial$, and the semantics of the modality is given by relation lifting. For the semantics to work, $T$ is required to preserve exact squares. For the finitary setting to work, $T_\omega^\partial$ is required to preserve finite intersections. We develop a notion of a base for subobjects of $T_\omega X$. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic, and prove its completeness.

A Logic for Conditional Local Strategic Reasoning

We consider systems of rational agents who act and interact in pursuit of their individual and collective objectives. We study and formalise the reasoning of an agent, or of an external observer, about the expected choices of action of the other agents based on their objectives, in order to assess the reasoner’s ability, or expectation, to achieve their own objective. To formalize such reasoning we extend Pauly’s Coalition Logic with three new modal operators of conditional strategic reasoning, thus introducing the Logic for Local Conditional Strategic Reasoning mathsf {ConStR}. We provide formal semantics for the new conditional strategic operators in concurrent game models, introduce the matching notion of bisimulation for each of them, prove bisimulation invariance and Hennessy–Milner property for each of them, and discuss and compare briefly their expressiveness. Finally, we also propose systems of axioms for each of the basic operators of mathsf {ConStR} and for the full logic.

Logical Characterizations of Fuzzy Simulations

We provide and prove logical characterizations of fuzzy simulations between fuzzy Kripke models that use a general t-norm-based semantics. We also extend the results for fuzzy Kripke models defined over a general residuated lattice. Our logical characterizations of fuzzy simulations are formulated w.r.t. positive existential fragments of fuzzy propositional dynamic logic without tests and concern fuzzy preservation of positive existential modal formulas under fuzzy simulations as well as the Hennessy-Milner property of fuzzy simulations.

Logical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices

There are two kinds of bisimulation, namely crisp and fuzzy, between fuzzy structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models and fuzzy interpretations in description logics. Fuzzy bisimulations between fuzzy automata over a complete residuated lattice have been introduced by Ćirić et al. in 2012. Logical characterizations of fuzzy bisimulations between fuzzy Kripke models (respectively, fuzzy interpretations in description logics) over the residuated lattice [0,1] with the Gödel 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 between fuzzy graph-based structures over a general residuated lattice, as well as over the residuated lattice [0,1] with the Łukasiewicz or product t-norm. In this article, we provide and prove logical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices. The considered logics are the fuzzy propositional dynamic logic and its fragments. Our logical characterizations concern invariance of formulas under fuzzy bisimulations and the Hennessy-Milner property of fuzzy bisimulations. They can be reformulated for other fuzzy structures such as fuzzy labeled transition systems and fuzzy interpretations in description logics.

Observational interpretations of hybrid dynamic logic with binders and silent transitions

We extend hybrid dynamic logic with binders (for state variables) by distinguishing between observable and silent transitions. This differentiation gives rise to two kinds of observational interpretations: The first one relies on observational abstraction from the ordinary model class of a specification Sp by considering its closure under weak bisimulation. The second one uses an observational satisfaction relation for the axioms of the specification Sp, which relaxes the interpretation of state variables and the satisfaction of modal formulæ by abstracting from silent transitions. We establish a formal relationship between both approaches and show that they are equivalent under mild conditions. For the proof we instantiate the previously introduced concept of a behaviour-abstractor framework to the case of dynamic logic with binders and silent transitions. As a particular outcome we provide an invariance theorem and show the Hennessy-Milner property for weakly bisimilar labelled transition systems and observational satisfaction. In the second part of the paper we integrate our results in a development methodology for reactive systems leading to two versions of observational refinement. We provide conditions under which both kinds of refinement are semantically equivalent, involving implementation constructors for relabelling, hiding, and parallel composition.

Characterizing fuzzy simulations for fuzzy labeled transition systems in fuzzy propositional dynamic logic

We formulate and prove logical characterizations of fuzzy simulations for fuzzy labeled transition systems under the Gödel semantics. Apart from the usual setting, we consider also the setting that allows graded modal operators, which are used to talk about the number of successors with a certain property of the current state. Our characterizations are formulated w.r.t. the existential fragment of fuzzy propositional dynamic logic and its extension with graded modal operators. The results concern fuzzy preservation of existential modal formulas under fuzzy simulations as well as the Hennessy-Milner property of fuzzy simulations.

Bisimulation and bisimilarity for fuzzy description logics under the Gödel semantics

Description logics (DLs) are a suitable formalism for representing knowledge about domains in which objects are described not only by attributes but also by binary relations between objects. Fuzzy extensions of DLs can be used for such domains when data and knowledge about them are vague and imprecise. One of the possible ways to specify classes of objects in such domains is to use concepts in fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs is characterized by bisimilarity. The bisimilarity relation of an interpretation is the largest auto-bisimulation of that interpretation. In DLs and their fuzzy extensions, such equivalence relations can be used for concept learning. In this paper, we define and study fuzzy bisimulation and bisimilarity for fuzzy DLs under the Gödel semantics, as well as crisp bisimulation and strong bisimilarity for such logics extended with involutive negation. The considered logics are fuzzy extensions of the DL ALCreg (a variant of PDL) with additional features among inverse roles, nominals, (qualified or unqualified) number restrictions, the universal role, local reflexivity of a role and involutive negation. We formulate and prove results on invariance of concepts under fuzzy (resp. crisp) bisimulation, conditional invariance of fuzzy TBoxes/ABoxes under bisimilarity (resp. strong bisimilarity), and the Hennessy-Milner property of fuzzy (resp. crisp) bisimulation for fuzzy DLs without (resp. with) involutive negation under the Gödel semantics. Apart from these fundamental results, we also provide results on using fuzzy bisimulation to separate the expressive powers of fuzzy DLs, as well as results on using strong bisimilarity to minimize fuzzy interpretations.

Bisimilarity in Fuzzy Description Logics Under the Zadeh Semantics

Fuzzy description logics (DLs) are extensions of DLs for dealing with imprecise and vague concepts. They found the logical basis for fuzzy ontologies, which are useful for practical applications. Bisimilarity is a natural notion of equivalence between individuals in DLs. In this paper, for the first time, we introduce the notion of bisimilarity in fuzzy DLs under the Zadeh semantics. It is defined using our notion of $p$ -cut simulation between fuzzy interpretations. The considered logics are fuzzy DLs that extend the fuzzy version of the DL $\mathcal {ALC}_{\bf reg}$ (a variant of propositional dynamic logic) with features among inverse roles, the universal role, qualified number restrictions, nominals, and local reflexivity of a role. We provide results on preservation of information by the mentioned simulations, conditional invariance of ABoxes and TBoxes by bisimilarity between witnessed interpretations, as well as the Hennessy–Milner property for fuzzy DLs under the Zadeh semantics.

Expressivity in chain-based modal logics

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and (appropriately defined) modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.

Bisimilarity for paraconsistent description logics

We introduce comparisons with respect to information between interpretations in paraconsistent description logics and use them to define bisimilarity for such logics. As bisimilarity is a natural notion for characterizing indiscernibility in modal and description logics, it is useful for concept learning in description logics also when inconsistencies occur. We give preservation results and the Hennessy-Milner property for comparisons with respect to information in paraconsistent description logics. As consequences, we also obtain invariance results and the Hennessy-Milner property for bisimilarity in paraconsistent description logics.

On bisimulations for description logics

We study bisimulations for useful description logics. The simplest among the considered logics is ALCreg (a variant of PDL). The others extend that logic with the features: inverse roles, nominals, qualified number restrictions, the universal role, and/or the concept constructor for expressing the local reflexivity of a role. They also allow role axioms. Our contributions are as follows. We propose to treat named individuals as initial states and give an appropriate bisimulation condition for that. We also give bisimulation conditions for the universal role and the concept constructor ∃r.Self. We prove that all of the bisimulation conditions for the features can be combined together to guarantee invariance of concepts and the Hennessy-Milner property for the whole class of studied description logics. We address and give results on invariance or preservation of ABoxes, RBoxes and knowledge bases in description logics. Independently from Lutz et al. [26,27] we also give results on invariance of TBoxes. We introduce a new notion called QS-interpretation, which is needed for dealing with minimizing interpretations in description logics with qualified number restrictions and/or the concept constructor ∃r.Self. We formulate and prove results on minimality of quotient interpretations w.r.t. the largest auto-bisimulations. We adapt Hopcroft’s automaton minimization algorithm to give an efficient algorithm for computing the partition corresponding to the largest auto-bisimulation of a finite interpretation in any description logic of the considered family. Using the invariance results we compare the expressiveness of the considered description logics w.r.t. concepts, TBoxes and ABoxes. Our results about separating the expressiveness of description logics are naturally extended to the case when instead of ALCreg we have any sublogic of ALCreg that extends ALC.

Adequacy and Complete Axiomatization for Timed Modal Logic

In this paper we develop the metatheory for Timed Modal Logic (TML), which is the modal logic used for the analysis of timed transition systems (TTSs). We solve a series of long-standing open problems related to TML. Firstly, we prove that TML enjoys the Hennessy-Milner property and solve one of the open questions in the field. Secondly, we prove that the set of validities are not recursively enumerable. Nevertheless, we develop a strongly-complete proof system for TML. Since the logic is not compact, the proof system contains infinitary rules, but only with countable sets of instances. Thus, we can involve topological results regarding Stone spaces, such as the Rasiowa-Sikorski lemma, to complete the proofs.

Characterization, definability and separation via saturated models

Three important results about the expressivity of a modal logic L are the Characterization Theorem (that identifies a modal logic L as a fragment of a better known logic), the Definability Theorem (that provides conditions under which a class of L-models can be defined by a formula or a set of formulas of L), and the Separation Theorem (that provides conditions under which two disjoint classes of L-models can be separated by a class definable in L).We provide general conditions under which these results can be established for a given choice of model class and modal language whose expressivity is below first order logic. Besides some basic constraints that most modal logics easily satisfy, the fundamental condition that we require is that the class of ω-saturated models in question has the Hennessy–Milner property with respect to the notion of observational equivalence under consideration. Given that the Characterization, Definability and Separation theorems are among the cornerstones in the model theory of L, this property can be seen as a test that identifies the adequate notion of observational equivalence for a particular modal logic.

Approximating Bisimilarity for Markov Processes

In this paper we investigate bisimilarity for general Markov processes through the correspondence between sub-σ-algebras and equivalence relations. In particular, we study bisimulations from the perspective of fixed-point theory. Given a Markov process M=〈Ω,Σ,τ〉, we characterize its state bisimilarity as the greatest fixed point of a composition of two natural set operators between equivalence relations on Ω and sub-σ-algebras of Σ. Moreover, we employ a Smith-Volterra-Cantor-set-construction to obtain an example to show that state bisimilarity is beyond ω iterations of these two operators alternately from event bisimilarity and hence the composite operator is not continuous. This process of iteration illustrates the gap between event bisimilarity (or logical equivalence) and state bisimilarity, and hence provides insights about the Hennessy-Milner property for general Markov processes. At the end of this paper, we also study approximation of Markov processes related to filtration.

Rank-1 Modal Logics are Coalgebraic

Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatized in rank 1. Here we establish the converse, i.e. every rank-1 modal logic has a sound and strongly complete coalgebraic semantics. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the Hennessy–Milner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.