Rule Based Fuzzy Cognitive Maps: Fuzzy Causal Relations João Paulo Carvalho José A. B. Tomé INESC - Instituto de Engenharia de Sistemas e Computadores, IST – Instituto Superior Técnico R. Alves Redol, 9, 1000 Lisboa, PORTUGAL Phone: +351.1.3100262, Fax: +351.1.3145843 E-mail: uke@eniac.inesc.pt jbt@eniac.inesc.pt Abstract: Rule Based Fuzzy Cognitive Maps (RBFCM) are proposed as an evolution of Fuzzy Causal Maps (FCM) that allow a more complete representation of cognition, since relations other than monotonic causality are made possible. Their structure is based on traditional fuzzy systems with feedback. The main problem to solve while trying to implement a RBFCM is the causal relation itself, since traditional fuzzy operations can not implement causality as it is usually defined in causal maps. This paper introduces Rule Based Fuzzy Cognitive Maps and presents a method to implement Fuzzy Causal Relations. This method allows a great flexibility in the addition and removal of concepts and links among concepts, and introduces a new Fuzzy Operation that simulates the “accumulative” property of causal relations – the Fuzzy Carry Accumulation (FCA). Keywords: Rule Based Fuzzy Cognitive Maps (RBFCM); Causality; Fuzzy Causal Relations (FCR); Fuzzy Carry Accumulation (FCA); Fuzzy sets. 1. Introduction Decision makers usually face serious difficulties when approaching significant, real-world dynamic systems. Such systems are composed of a number of dynamic concepts or actors which are interrelated in complex ways, usually including feedback links which propagate influences in complicated chains. Axelrod work on Cognitive Maps (CMs)[1] introduced a way to represent these systems, and several methods and tools exist to analyse the structure of causal maps like [2] [3]. However, complete, efficient and practical mechanisms to analyse and predict the evolution of data in CMs are necessary [4] but not yet available for several reasons. System Dynamics tools like [5] could be a solution, but since numerical data may be uncertain or hard to come by, and the formulation of a mathematical model may be difficult, costly or even impossible, then efforts to introduce knowledge on these systems should rely on natural language arguments in the absence of formal models. Fuzzy Cognitive Maps (FCM), as introduced by Kosko [6], are a qualitative alternative approach to dynamic systems. However, a FCM is indeed a man-trained Neural Network (Multilayer perceptron) which is not Fuzzy in a traditional sense, and doesn’t explore usual Fuzzy capabilities. FCM don’t share the properties of other fuzzy systems and can not be mixed with traditional fuzzy rules and operations. Although this is not a problem by itself since FCM perform rather well when trying to represent the evolution of management, organisational or socio-economic problems, FCM are limited to the representation of simple monotonic causal relations between concepts. FCM are indeed Fuzzy Causal Maps. Causal Maps advantages [7] reside essentially in the fact that causal associations are the major way in which understanding about the world is organised. However, in the area of social sciences and/or psychology we can find that Cognitive Maps should use other kind of relations between concepts (like the Cognitive Base Schemes defined in [8] and [9]) in order to allow a better representation of real world systems that involve cognition.