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Pattern Recognition with Fuzzy Objective Function Algorithms

The concept of a linguistic variable and its application to approximate reasoning—II☆

Fuzzy sets as a basis for a theory of possibility

The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

Outline of a New Approach to the Analysis of Complex Systems and Decision Processes

The MAGMA algebra system I: the user language

L-fuzzy sets

We assune that the reader is familiar with the ubiquity of information in the modern world and is sympathetic with the need for restricting rights to read, add, modify, or delete information in specific contexts. This need is particularly acute for systems having computers as significant components.

Security Policies and Security Models

There is a population explosion among the logical systems used in computing science. Examples include first-order logic, equational logic, Horn-clause logic, higher-order logic, infinitary logic, dynamic logic, intuitionistic logic, order-sorted logic, and temporal logic; moreover, there is a tendency for each theorem prover to have its own idiosyncratic logical system. The concept of institution is introduced to formalize the informal notion of “logical system.” The major requirement is that there is a satisfaction relation between models and sentences that is consistent under change of notation. Institutions enable abstracting away from syntactic and semantic detail when working on language structure “in-the-large”; for example, we can define language features for building large logical system. This applies to both specification languages and programming languages. Institutions also have applications to such areas as database theory and the semantics of artificial and natural languages. A first main result of this paper says that any institution such that signatures (which define notation) can be glued together, also allows gluing together theories (which are just collections of sentences over a fixed signature). A second main result considers when theory structuring is preserved by institution morphisms. A third main result gives conditions under which it is sound to use a theorem prover for one institution on theories from another. A fourth main result shows how to extend institutions so that their theories may include, in addition to the original sentences, various kinds of constraint that are useful for defining abstract data types, including both “data” and “hierarchy” constraints. Further results show how to define institutions that allow sentences and constraints from two or more institutions. All our general results apply to such “duplex” and “multiplex” institutions.

Institutions: abstract model theory for specification and programming

The 'hard' sciences, such as physics and chemistry, construct exact mathematical models of empirical phenomena, and then use these models to make predictions. Certain aspects of reality always escape such models, and we look hopefully to future refinements. But sometimes there is an elusive fuzziness, a readjustment to context, or an effect of observer upon observed. These phenomena are particularly indigenous to natural language, and are common in the 'soft' sciences, such as biology and psychology. This paper examines problems of f i t zz iness , i.e., vagueness, ambiguity, and ambivalence. Although the theory is called a 'logic of inexact concepts', we do not endorse belief in some Platonic ideal 'concepts' or logic embodying their essence. Rather we suggest a method for constructing and studying models of the way we use words; and we use the word 'concept' metaphorically in discussing meaning. 'Exact concepts' are the sort envisaged in pure mathematics, while 'inexact concepts' are rampant in everyday life. This distinction is complicated by the fact that whenever a human being interacts with mathematics, it becomes part of his ordinary experience, and therefore subject to inexactness. Ordinary logic is much used in mathematics, but applications to everyday life have been criticized because our normal language habits seem so different. Various modifications of orthodox logic have been suggested as remedies, particularly omission of the Law of the Excluded Middle. Ordinary logic represents exact concepts syntactically: that is a concept is given a name (such as 'man') which becomes an object for manipulation in a formal language. Aristotelian logic and the predicate calculus are both such systems, in which rules are given to distinguish valid from

The logic of inexact concepts

Many apparently divergent approaches to specifying formal semantics of programming languages are applications of initial algebra semantics. In this paper an overview of initial algebra semantics is provided. The major technical feature is an initial continuous algebra which permits unified algebraic treatment of iterative and recursive semantic features in the same framework as more basic operations.

Joseph A. Goguen

Papers

L-fuzzy sets

Security Policies and Security Models

Institutions: abstract model theory for specification and programming

The logic of inexact concepts

Initial Algebra Semantics and Continuous Algebras