Relational state transition dynamics

Abstract

Basic concepts of classical dynamics are analysed in the simple mathematical setting of state transition systems, where both time and space are discrete, and no structure is assumed on the state space besides a binary transition relation. This framework proves useful to the dynamical analysis of computations and biomolecular processes. Here a relational formulation of this framework is presented, where the concepts of attractor and recurrence surface in two variants, respectively relating to the two fundamental modalities. A strong link between recurrence and both existence and extent of attractors, in either variant, is established by a novel characterization theorem. Further concepts are easily casted in the relational language, such as product dynamics and projections thereof, which support analysis and reasoning about metabolic P systems. An outline of possible applications and future developments of this work concludes the article.