On dynamically non-trivial three-valued logics: oscillatory and bifurcatory species

We interpret three-valued logical systems as pools of abstract chemical species, logical variables, where chemical reactions are catalyzed by logical connectives; then we study global- and space-time dynamics of these ‘logical’ chemical reactors. We develop a family of combinatorial systems {T,F,*}, , where T, F and * are truth-values, is commutative binary operator, acts as a Boolean conjunction on {T,F};* *=*, T *=a and F *=b, a,b {T,F,*}. We consider nine combinatorial systems of the family, specified by values of a and b, and derive from each member ab of the family an artificial chemical system, where interactions between reactants T, F and * are governed by . Computational experiments with well-stirred reactors show that all systems but T* and *T exhibit a dull behaviour and converge to their only stable points. In reactor of the system T* , catalyzed by , concentrations of reactants oscillate while reactor of the system *T finishes its evolution in one of two stable points; thus we call the models of T* and *T oscillatory and bifurcatory systems. We enrich the systems with negation connective, define additional connectives via ¬ and and undertake a detailed study of integral dynamic of the artificial stirred chemical reactors and of space-time dynamic of thin-layer non-stirred chemical reactors derived from the logical connectives. The thin-layer reactors show rich space-time dynamic ranging from breathing patterns and mobile localizations to fractal structures. A primitive hierarchy of connectives’ phenomenological complexity is constructed.