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Parity games can be used to solve satisfiability, verification and controller synthesis problems. As part of an effort to better understand their nature, or the nature of the problems they solve, preorders on parity games have been studied. Defining these relations, and in particular proving their transitivity, has proven quite difficult on occasion. We propose a uniform way of lifting certain preorders on Kripke structures to parity games and study the resulting preorders. We explore their relation with parity game preorders from the literature and we study new relations. Finally, we investigate whether these preorders can also be obtained via modal characterisations.

Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.

Agent-oriented techniques represent a promising approach for engineering complex systems where interaction is probably the most important single characteristic. Accordingly, the recent years have witnessed the emergence of different approaches for the study of intelligent agent-based systems. One such architecture views the system as a rational agent having certain mental attitudes of Belief, Desire and Intention (BDI). This paper explores a particular type of rational agent, a Belief-Goal-Role agent. Unlike most previous work, our approach has been to characterize the mental state of the agents that leads them to take part in cooperative action. Hence, beliefs, goals and roles are relevant to our study of cooperation which have lead to the identification of communication concepts (beliefs and goals) and organization concepts (roles). The model is formalized by expressing it as a theory in a first-order, multi-modal, and linear-time logic. We use labeled transition systems to deal with the truth conditions of the formulae of our theory as well as with the behavioral semantics of our agents. We illustrate our work with the well-known prey/predator game.

In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

Some intuitionistic fuzzy interpretations of modal logic axioms are discussed.

Investigations pursued in this paper contribute to a research project introduced by Resconi, Klir, and St. Clair [1] whose purpose is to employ syntactic and semantic structures of modal logic as a unifying framework within which various uncertainty theories can be formalized, compared, and organized hierarchically. This paper focuses on the explicit use of modal logic semantics to formalize fuzzy sets, belief measures, plausibility measures, and Sugeno λ-measures.

Belief function theory is one of broadly used numerical formalisms for uncertainty processing. It was also studied in its qualitative form. In this paper we study relationship between this qualitative belief function theory and modal logic. We present a sound system of axioms and show its completeness with respect to sematics considered.

This paper is a continuation in a series of papers started by Resconi et. al. [8] in which we try to develop interpretations of various uncertainty theories within the framework of standard modal logic. In this paper, we deal with possibility theory. We suggest an interpretation of possibility measures and necessity measures and show that this interpretation is complete for rational-valued measures.

Modal logic interpretations of plausibility and belief measures are developed based on the observation that the accessibility relation in a model of modal logic, regarded as a multivalued mapping, induces a plausibility measure and a belief measure on the set of possible worlds.

Coalitional power in multistage processes is modeled using effectivity frames, which link an effectivity function to every possible state of the world. Effectivity frames are general enough to capture, e.g., what groups of agents can bring about in extensive games of perfect and almost perfect information. Coalition Logic is used to describe effectivity frames, and the question of generating an extensive game satisfying a given specification is formulated as a satisfiability problem in Coalition Logic. Using this logical reformulation, we show that the complexity of this implementation problem depends on two parameters: For coalitional specifications, the problem is shown to be PSPACE-complete. For individual specifications on the other hand, i.e., for specifications which only refer to the powers of individual agents, generating an implementation with perfect information is PSPACE-complete, whereas generating an implementation with almost perfect information is NP-complete.

The relationships between various classes of Petri nets and modal logic are studied in this paper. Petri nets enlogy provides some hints for these relationships. The concepts of cases and case class have been successfully used to investigate knowledge representation of Condition/Event nets. In this paper both concepts are introduced into Place/Transition nets and High-level Petri nets for knowledge representation. We define case variables and equivalent case variables for knowledge representation of High-level Petri Nets in order to increase the representation power of knowledge.

In this paper we consider the phenomenon of superpositions in quantum mechanics and suggest a way to deal with the idea in a logical setting from a syntactical point of view, that is, as subsumed in the language of the formalism, and not semantically. We restrict the discussion to the propositional level only. Then, after presenting the motivations and a possible world semantics, the formalism is outlined and we also consider within this scheme the claim that superpositions may involve contradictions, as in the case of the Schrödinger's cat, which (it is usually said) is both alive and dead. We argue that this claim is a misreading of the quantum case. Finally, we sketch a new form of quantum logic that involves three kinds of negations and present the relationships among them. The paper is a first approach to the subject, introducing some main guidelines to be developed by a ‘syntactical’ logical approach to quantum superpositions.

The study of the foundations of Quantum Mechanics, especially after the advent of Quantum Computation and Information, has benefited from the application of category-theoretic tools and modal logics to the analysis of Quantum processes: we witness a wealth of theoretical frameworks casted in either of the two languages. This paper explores the interplay of the two formalisms in the peculiar context of Quantum Theory.

After a review of some influential abstract frameworks, we show how different modal logic frames can be extracted from the category of finite dimensional Hilbert spaces, connecting the Categorical Quantum Mechanics approach to some modal logics that have been proposed for Quantum Computing. We then apply a general version of the same technique to two other categorical frameworks, the ‘topos approach’ of Doering and Isham and the sheaf-theoretic work on contextuality by Abramsky and Brandenburger, suggesting how some key features can be expressed with modal languages.

This paper is a continuation of the authors' attempts to deal with the notion of indistinguishability (or indiscernibility) from a logical point of view. Now we introduce a two-sorted first-order modal logic to enable us to deal with objects of two different species. The intended interpretation is that objects of one of the species obey the rules of standard S5, while the objects of the other species obey only the rules of a weaker notion of indiscernibility. Quantum mechanics motivates the development. The basic idea is that in the ‘actual’ world things may be indiscernible but in another accessible world they may be distinguished in some way. That is, indistinguishability needs not be seen as a necessary relation. Contrariwise, things might be distinguished in the ‘actual’ world, but they may be indiscernible in another world. So, while two quantum systems may be entangled in the actual world, in some accessible world, due to a measurement, they can be discerned, and on the other hand, two initially separated quantum systems may enter in a state of superposition, losing their individualities. Two semantics are sketched for our system. The first is constructed within a standard set theory (the ZFC system is assumed at the metamathematics). The second one is constructed within the theory of quasi-sets, which we believe suits better the purposes of our logic and the mathematical treatment of certain situations in quantum mechanics. Some further philosophically related topics are considered.

McKinsey–Tarski (1944), Shehtman (1999), and Lucero-Bryan (2011) proved completeness theorems for modal logics with modalities □, □ and ∀, and [∂] and ∀, respectively, with topological semantics over the real numbers. We give short proofs of these results using lexicographic sums of linear orders.

A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell's Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:Ann believes that Bob assumes thatAnn believes that Bob's assumption is wrongThis is formalized to show that any belief model of a certain kind must have a “hole.” An interpretation of the result is that if the analyst's tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.