Algebraic dynamics of k-valued fuzzy cognitive maps and its stabilization

Abstract

Fuzzy Cognitive Maps (FCMs) as knowledge-based tools are widely implemented for knowledge representation and reasoning. To reach final equilibrium states is essential for the results of causal reasoning in practical applications by using FCMs. However, the dynamics and the stabilization problem of FCMs are still not adequately studied so far. In this article, the algebraic dynamics of k-valued fuzzy cognitive maps and its stabilization is analytically and quantitatively investigated. Based on the technique of semi-tensor product (STP), k-valued FCMs are firstly converted into discrete-time linear representation. General formulas are obtained to calculate the number of fixed points and limit cycles of finite-state FCMs, and a necessary and sufficient criterion to determine attractors is proposed, by means of which an algorithm for calculating all attractors and their basins are achieved. Furthermore, the stabilization problem of k-valued FCMs is discussed. By introducing controls to certain concepts in FCMs, the system can be globally stabilized in equilibrium states. The related necessary and sufficient criterion is proposed and the corresponding method to achieve the global stabilization of FCMs under constant controls is given. Examples are shown to demonstrate the effectiveness and feasibility of the proposed scheme.