Fuzzy cognitive maps (FCMs) are fuzzy-graph structures for representing causal reasoning. Their fuzziness allows hazy degrees of causality between hazy causal objects (concepts). Their graph structure allows systematic causal propagation, in particular forward and backward chaining, and it allows knowledge bases to be grown by connecting different FCMs. FCMs are especially applicable to soft knowledge domains and several example FCMs are given. Causality is represented as a fuzzy relation on causal concepts. A fuzzy causal algebra for governing causal propagation on FCMs is developed. FCM matrix representation and matrix operations are presented in the Appendix.

Fuzzy cognitive maps

The role of fuzzy logic in the management of uncertainty in expert systems

Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions

Fuzzy entropy and conditioning

Fuzzy Relational Systems

Fuzzy power sets and fuzzy implication operators

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.

Semantics of implication operators and fuzzy relational products

Increasingly complex interaction between man and nature, increasingly complex interaction between man and man-made artificial systems, make it increasingly difficult to comprehend the consequences of changes in all these systems, to analyze them, to understand their dynamics, to influence their behaviour.

Fuzzy Relational Products as a Tool for Analysis and Synthesis of the Behaviour of Complex Natural and Artificial Systems

Many diverse problems of compatibility of structures can be unified by generalizing tjie concept of a homomorphism. The paper introduces generalized proteromorphisms and amphimorphisms, forward and backward compatibility of relations, as well as other new useful concepts. These generalizations find application in solving relational equalions, investigation of compatibility of systems and data structures, in examining faithfulness of representations of knowledge structures in knowledge-based systems, in semiotics and elsewhere.

Wyllis Bandler

Papers

Fuzzy power sets and fuzzy implication operators

Semantics of implication operators and fuzzy relational products

Fuzzy Relational Products as a Tool for Analysis and Synthesis of the Behaviour of Complex Natural and Artificial Systems

On the general theory of relationalmorphisms

Use of fuzzy relations in knowledge representation, acquisition, and processing