Advancing Anticipatory Systems Analysis with Hyperincursive Processes, Parity Logic, and Fuzzy Logic

This paper outlines four major topics of tantamount importance to computing anticipatory systems. Section 1 introduces the reader to several historical facts regarding Daniel Dubois' hyperincursive modeling approach and its relationship with Gérard Langlet's work and the author's conception of parity logic systems. It provides the connection of Dubois' hyperincursivity theory and fractal machines with parity logic engines, a special class of binary integro-differential cellular automata. Section 2 on modeling anticipatory systems recalls first the essence of Dubois' anticipatory systems approach by comparing briefly recursivity, incursivity, and self-referentiality. Their impact on modeling cognitive anticipations is then discussed by rendering Piaget's recursive concept of anticipatory schemata into incursive schemata. Section 2 closes with an unresolved problem regarding anticipatory conflicts. Section 3 exhibits in a more formal way the difference between recursivity and incursivity by explicating Dubois' most important digital equations, how they apply to hyperincursive fractal machines, and how they are related to parity logic engines. This includes self-organized processing of parity intergrals and differentials, self-organized development of binary transforms, and several group theoretic implications of transforming parity matrices generated with fractal machines or parity logic engines. Finally, in section 4, further perspectives are outlined for the advancement of anticipatory systems by considering causal predictor systems in terms of fuzzy cognitive maps. This includes the law of concomitant variation, non-Aristotelian causality, the relationship between fuzzy causality, fuzzy subsethood and fuzzy causal cross-impact analysis.