Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over . This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.
The notion of n-fold grisly deductive systems is introduced. Some conditions for a deductive system to be an n-fold grisly deductive system are provided. Extension property for n-fold grisly deductive system is established.
Mathematics Subject Classification: 26A33; 93C15, 93C55, 93B36, 93B35, 93B51; 03B42; 70Q05; 49N05This paper proposes a novel method to design an H∞ -optimal fractional order PID (FOPID) controller with ability to control the transient, steady-state response and stability margins characteristics. The method uses particle swarm optimization algorithm and operates based on minimizing a general cost function. Minimization of the cost function is carried out subject to the H∞ -norm; this norm is also...
Almost Distributive Lattices (ADL) are structures defined by Swamy and Rao [14] as a common abstraction of some generalizations of the Boolean algebra. In our paper, we deal with a certain further generalization of ADLs, namely the Generalized Almost Distributive Lattices (GADL). Our main aim was to give the formal counterpart of this structure and we succeeded formalizing all items from the Section 3 of Rao et al.’s paper [13]. Essentially among GADLs we can find structures which are neither V-commutative...
In this article, we formalize the Advanced Encryption Standard (AES). AES, which is the most widely used symmetric cryptosystem in the world, is a block cipher that was selected by the National Institute of Standards and Technology (NIST) as an official Federal Information Processing Standard for the United States in 2001 [12]. AES is the successor to DES [13], which was formerly the most widely used symmetric cryptosystem in the world. We formalize the AES algorithm according to [12]. We then verify...
Based on the Petri net definitions and theorems already formalized in the Mizar article [13], in this article we were able to formalize the definition of cell Petri nets. It is based on [12]. Colored Petri net has already been defined in [11]. In addition, the conditions of the firing rule and the colored set to this definition, that defines the cell Petri nets are further extended to CPNT.i further. The synthesis of two Petri nets was introduced in [11] and in this work the definition is extended...
Fourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient...
The purpose of this note is to show that a known and natural four-valued logic co-exists with classical two-valued logic in the familiar context of truth tables. The tool required is the power construction.
Fuzzy sets establish a mapping from the interval of values of a criterial function onto a system of subsets of a basic set. In the paper, a system of definitions and theorems is introduced, which is aimed at an adequate expression of this point of view. The criterial function, with an arbitrary interval of values, serves for expressing the really existing objective property, forming the basis for defining a fuzzy set.
Fuzzy data mining by means of the fuzzy decision tree method enables the construction of a set of fuzzy rules. Such a rule set can be associated with a database as a knowledge base that can be used to help answering frequent queries. In this paper, a study is done that enables us to show that classification by means of a fuzzy decision tree is equivalent to the generalized modus ponens. Moreover, it is shown that the decision taken by means of a fuzzy decision tree is more stable when observation...
Knowledge about the relation between faults and the observed symptoms is necessary for fault isolation. Such a relation can be expressed in various forms, including binary diagnostic matrices or information systems. The paper presents the use of fuzzy logic for diagnostic reasoning. This method enables us to take into account various kinds of uncertainties connected with diagnostic reasoning, including the uncertainty of the faults-symptoms relation. The presented methods allow us to determine the...
In the paper, three different ways of constructing distances between vaguely described objects are shown: a generalization of the classic distance between subsets of a metric space, distance between membership functions of fuzzy sets and a fuzzy metric introduced by generalizing a metric space to fuzzy-metric one. Fuzzy metric spaces defined by Zadeh’s extension principle, particularly to are dealt with in detail.