Ladislav J. Kohout

Fuzzy power sets and fuzzy implication operators

The theory of fuzzy power sets, which has hitherto been insufficiently developed, is shown very naturally to require the use of a fuzzy implication operator (Section 1). Six such operators are gathered from the literature on multiple-valued logic (Section 2), and their effects on fuzzy power-set theory are compared throughout the rest of the paper. After certain fundamental definitions of set characteristics (Section 3), the six operators are carried in parallel while working out basic aspects of power-set theory. Among these are the properties of the set-inclusion relation and the set-equivalence relation (Section 4), two distinct concepts of disjointness (Section 5), questions of consistency in the relations between a set and its complement (Section 6), and a very concrete theorem on a difference among the operators with regard to the derivation of crisp conclusions from fuzzy premises (Section 7). Finally (Section 8), emphasis is placed on the dependence of the choice of operators upon the purposes the user has in hand.

Semantics of implication operators and fuzzy relational products

After a brief discussion of the need for fuzzy relation theory in practical systems work, the paper explains the new triangle products of relations and the sort of results to be expected from them, starting from a crisp situation. The asymmetry of these products, in contrast to correlation, is noted as essential to the investigation of hierarchial dependencies. The panoply of multi-valued implication operators, with which the fuzzification of these products can be accomplished, is presented, and a few of their properties noted. Then, most importantly, a checklist paradigm is given, by which entirely new light is thrown upon the semantics of these operators, connecting them, in a unified way, with measures which might be made upon more refined data. Using a well-known psychological test in an actual situation, so that the finer structure is in fact available, a comparison is made between a checklist measure and several of the operator values, showing the interrelationship concretely. Finally, some products and their interpretations are presented, using further real-world data.

The fuzzy decade: a bibliography of fuzzy systems and closely related topics

The main part of the paper consists of a bibliography of some 1150 items, each keyword-indexed with some 750 being classified as concerned with fuzzy system theory and its applications. The remaining items are concerned with closely related topics in many-valued logic, linguistics, the philosophy of vagueness, etc. These background references are annotated in an initial section that outlines the relationship of fuzzy system theory to other developments and provides pointers to various possible fruitful interrelationships. Topics covered include: the philosophy and logic of imprecision and vagueness; other non-standard logics; foundations of set theory; probability theory; fuzzification of mathematical systems; linguistics and psychology; and applications.

THE LOGIC OF AUTOMATA

Automata are the prime example of general systems over discrete spaces, and yet the theory of automata is fragmentary and it is not clear what makes a general structure an automaton. This paper investigates the logical foundations of automata relating it to the semantics of our notions of uncertainty, state and state-determined. A single framework is established for the conventional spectrum of automata: deterministic, probabilistic, fuzzy, and non-deterministic, which shows this set to be, in some sense, complete. Counter-examples are then developed to show that this spectrum alone is inadequate to describe the behaviour of certain forms of uncertain system. Finally a general formulation is developed based on the fundamental semantics of our notion of a state that shows that the logical Structure of an automaton must be at least a positive ordered semiring. The role of probability logic, its relationship to fuzzy logic, the rotes of topological models of automata, and the symmetry between inputs and outp...

PROTECTION AS A GENERAL SYSTEMS PROBLEM

Abstract It is argued that the problem of protection, of controlling mutual access rights to shared resources, is a topic appropriately treated as a major component of general systems theory. Although most widely studied and developed in the context of computer systems, protection modelsare equally applicable to biological systems, such as those involved in movement control. The paper first establishes the nature of the problem of protection in computer systems, noting that it only reaches its full potential complexity in large data-base systems with processes automatically invoked by “data interrupts”. The Graham and Denning model of protection and the concept of a “capability” are then described and the appropriate mathematical tools for the analysis of such models discussed. A detailed model of protection is then developed with examples of the role of algebraic, automata-theoretic, topological and modai/mutti-valued logical, techniques in its analysis. Finally, biological applications, general systems ...

Probabilistic versus fuzzy production rules in expert systems

Abstract In the design of Expert Systems there is increasing recognition of the need for graded production rules, that is, the use of grades or degrees of strength of implication, leading to grades or degrees of certainty or possibility attached to the conclusions. This accords well with the desire to accommodate imprecise, incomplete and faulty input data, and nonetheless to arrive—as humans do—at meaningful results, however provisional. For historical reasons, probabilistic methods are the best-known and the oftenest attempted, although they form a small and not particularly suitable sub-set of the methods available. This paper displays the probabilistic rules in their proper perspective in the wider canvas, dwelling on some of the inherent relationships among the various operators which have come to light. For this purpose, the Checklist Paradigm is described in some detail, providing a unifying framework within which these relationships can be understood. Meanwhile, the valid modes of reasoning are increased from the classical two to a total of four.

Representation of functional hierarchies of movement in the brain

The article discusses the notion of a functional hierarchy of movement and the problem of segmentation of movement behaviour. It is shown that certain principles, implicitly contained in the work of N. A. Bernstein can be formalized and further developed into a consistent methodology for the identification of individual levels. In section 5 a formal development is given of some neuro-biological ideas informally discussed in the preceding sections of the article. The formal apparatus used is based on generalized topologies and abstract logics. Fuzzification of models is discussed.

The Logic of Protection

This paper presents a brief exposition of the role of various mathematical techniques in the development and utilization of resource protection structures for computers. The first section is concerned with the semantics of the problem — the distinction between protection problems in general and those whose complexity necessitates deeper theoretical treatment. The second section considers the roles of algebraic, topological, and modal/multi-valued logic, techniques in the analysis of protection. Finally we give an analysis of a current protection model to illustrate the problems and techniques.

THE ALGEBRAIC STRUCTURE OF THE SPENCER BROWN AND VARELA CALCULI

The paper deals with the algebraic structure of the Spencer Brown and Varela calculi from various points of view. In the first two sections the calculi are brought into more familiar contexts, including more conventional notation, so that their originality and merit may be appreciated in a more general framework. In the next section some of the points, introduced first in a more intuitive way, are treated again with more mathematical formalism and precision, introducing also some necessary notions from the theory of universal algebras. In this way the calculi under investigation are examined with respect to some other known calculi of logic, especially the π-algebras.

GENERALIZED TOPOLOGIES AND THEIR RELEVANCE TO GENERAL SYSTEM† Bibliography of Works of the č ech Topological School

A brief list of the applications of topology to various disciplines is given in the introduction (dynamical systems, automata theory, psychology, linguistics, biology, economics) It is argued that topology plays an important role in general systems theory as well. This has already been shown by Cornacchio and Hammer. Unfortunately, the topologies needed for this purpose have to be generalized topologies in which Kuratowski closure axioms do not hold Work involving generalized topologies is usually associated with Hammer who published numerous papers on this topic. There have been some earlier isolated attempts to develop some generalizations, e.g. by Moore, Tarski, Birkhoff, Ore, but the matter has not been treated systematically. It is virtually unknown that there exists a very systematic treatment of the topic of generalized topologies in a series of papers which originated from the topological seminar conducted by Eduard c ech in Brno, Czechoslovakia, in the years 1936-1939 The purpose of this article ...