Is there a need for fuzzy logic

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

This paper surveys the work of the qualitative spatial reasoning group at the University of Leeds. The group has developed a number of logical calculi for representing and reasoning with qualitative spatial relations over regions. We motivate the use of regions as the primary spatial entity and show how a rich language can be built up from surprisingly few primitives. This language can distinguish between convex and a variety of concave shapes and there is also an extension which handles regions with uncertain boundaries. We also present a variety of reasoning techniques, both for static and dynamic situations. A number of possible application areas are briefly mentioned.

Qualitative Spatial Representation and Reasoning with the Region Connection Calculus

Fuzzy Relational Systems

In discussing the philosophical system put forward in PR, we need to point out that not only does Whitehead introduce a novel terminology, but the work itself is somewhat amorphous in character, and this despite his attempt to state a categoreal scheme — a general scheme of ideas in terms of which all our experience is to be described. There is also a considerable amount of overlap between the various parts of this book. It might have been a clearer and more effective work if Whitehead had engaged in some judicious pruning before publication.

Process and Reality

The theme of this book is fuzzy logic in a narrow sense, a promising new chapter of formal logic. The basic ideas of formal logic were formulated by Lotfi Zadeh in 1975. The aim of this logic is to investigate the wonderful human capacity of reasoning with vague notions by attempting to formalize the "approximate reasoning" we use in everyday life. A peculiarity of this book is to propose a general framework based on three mathematical tools: the theory of fuzzy closure operators, an extension principle for crisp logics and the theory of recursively enumerable fuzzy subsets. This book is unique in that it treats fuzzy logics which are not truth-functional in nature (as an example, the logic of the necessities, probabilistic logics and similarity-based logics). The book is addressed to people interested in artificial intelligence, fuzzy control, formal logic, and philosophy. It can be used in special post-graduate university studies and in advanced courses. The book is completely self-contained.

Fuzzy Logic: Mathematical Tools for Approximate Reasoning

It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general).

Fuzzy logic, continuity and effectiveness

Unification plays a central rule in Logic Programming. We ”soften” the unification process by admitting that two first order expressions can be ”similar” up to a certain degree and not necessarly identical. An extension of the classical unification theory is proposed accordingly. Indeed, in our approach, inspirated by the unification algorithm of Martelli-Montanari, the systems of equations go through a series of ”sound” transformations until a solvable form is found yielding a substitution that is proved to be a most general extended unifier for the given system of equations.

Similarity-based unification

Whitehead, in his famous book Process and Reality, proposed a definition of point assuming the concepts of "region" and "connection relation" as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper, proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as prim- itive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.

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Giangiacomo Gerla

Papers

Fuzzy Logic: Mathematical Tools for Approximate Reasoning

Fuzzy logic, continuity and effectiveness

Similarity-based unification

Connection Structures: Grzegorczyk's and Whitehead's Definitions of Point

Closure systems and L-subalgebras