In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.
I IntroductionThe Multiresolutional approach for representation of Intelligent Systems with NIST-RCS Architecture leads t o a number of mathematical problems related to categorical models associated with the Multiresolutional represent at ion.One of these problems is investigation of quasi-commutative diagrams that are mathematical formalization of relations between levels of Multiresolutional representation -2 A brief overview of Multiresolutional approach Multiresolutional formalism is the generdization of the concept of Discrete Event System that is investigated in the papers of W. M-Wonham (see [S] -is]) and other researchers (see [IO], 1111).Our main difference from these approaches is the kading role of multiresoiutimal principie in the construction of hierarchical representations far intelligent systems. As opposed to the Wohnam's approach, that operates with a given hierarchical structure of a representation, we assume that all Ievels of the mnItiresolutionai representation are generated by a special algorithm of generalization (Filtration-Grouping-Synthesis) ~ Our constructive approach gives efficient methods for practical implementation of our resutts. We beIieve that the ability to generaliie is a distinctive property of Intelligent Systems.Also one of the main differences of our approach from another approaches is generalization of the structure of input and output events: we assume that the events can be uncertain, evolutionary, and can have hybrid nature (i.e. one component of an event can evohe in continuous time, and another -in discrete time).Multiresolutional representation of the Intelligent System consists of 1. the description of representation of IS 011 several levels of resolution (which we call briefly "representation" of levels), 2. the description of relations between these levels. Formal definitionWe assume that there is given some partially ordered set A, elements of which are called levels of resolution.If two leveh XI , Xz satisfy the inequality XI < XZ, then we say that the resolution of the level level A1 is greater than the resolution of the level A, .We will describe representation on a level of resolution using the formalism of object-oriented approach. ObjectsOur definition of the concept of object is a mathematical formalization of objects as the following entities:1. 2. 3. 4. 5.Objects are characterized by their features (attributes which can be considered their coordt nates).AH numerical data known €or objects are uncertain data.Objects can be decomposed into their parts (which are usually objects of higher level of resolution)-Objects can be parts of other objects.Objects at a particular levei of resolution are obtained (a) from objects af higher level of resolution as a result of using aggregation procedure, which consists of procedures of Filtration (or Focusing Attention), Grouping and Combinatorid Search, (b) or from objects of lower level of resolution as a result of using detalization procedure.These procedures describe linkages between adjacent levels of resolution.
The article deals with the problem of proving observational equivalence for the class of computational processes called processes with message passing. These processes can execute actions of the following forms: sending or receiving of messages, checking logical conditions and updating values of internal variables of processes. Our main result is a theorem that reduces the problem of proving observational equivalence of a pair of processes with message passing to the problem of finding formulas associated with pairs of states of these processes, satisfying certain conditions that are associated with transitions of these processes. This reduction is a generalization of Floyd's method of flowchart verification, which reduces the problem of verification of flowcharts to the problem of finding formulas (called intermediate assertions) associated with points in the flowcharts and satisfying conditions, corresponding to transitions in the flowcharts. The above method of proving observational equivalence of processes with message passing is illustrated by a sliding window protocol verification.
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