A graph isomorphism condition and equivalence of reaction systems

Abstract

Abstract. We consider global dynamics of reaction systems as introduced by Ehrenfeucht and Rozenberg. The dynamics is represented by a directed graph, the so-called transition graph, and two reaction systems are considered equivalent if their corresponding transition graphs are isomorphic. We introduce the notion of a skeleton (a one-out graph) that uniquely defines a directed graph. We provide the necessary and sufficient conditions for two skeletons to define isomorphic graphs. This provides a necessary and sufficient condition for two reactions systems to be equivalent, as well as a characterization of the directed graphs that correspond to the global dynamics of reaction systems.

Reaction Systems, Transition Systems, and Equivalences

“…This naturally leads to a novel line of research concerning properties of state sequences. The investigation of properties of state sequences of context-independent processes of reaction systems resulted in a rich and elegant theory (see, e.g., [9,15,20,30,31,33,34]). Reaction systems with context determined by context controllers form a natural next level of a systematic and thorough investigation of state sequences (clearly, not much can be said when context is arbitrary as in the general model of reaction systems).…”

“…It should be mentioned here that there is a direct relationship between reaction systems and transition systems that has been studied in the literature, see e.g., [16,20]. One can simulate finite transition systems by reaction systems and one can simulate reaction systems by finite transition systems, see e.g., [5,16].…”

“…Also the state space of a rs A = (S, A) is a finite directed graph (transition system) with states of A and its nodes and edges determined either by the result function res A (in case of context-independent interactive processes) or by the joint effect of res A and the context sets (in case of unrestricted interactive processes). The isomorphism problem for these graphs and its relationship to the functional equivalence was studied in [20].…”

“…Although the model of reaction systems was inspired by biology, the research on reaction systems is driven by both biological considerations (see, e.g., [2,3,6,7,10,18,19]) and considerations concerned with the understanding of computations taking place in reaction systems which formalise dynamic processes instigated by the interactions of biochemical reactions in the living cell (see, e.g., [8,11,13,14,16,20,[30][31][32]). As a matter of fact, reaction systems turned out to be an interesting novel model of interactive computation.…”

Reaction Systems and Enabling Equivalence

“…A direct relationship between reaction systems and transition systems has been studied in the literature, see e.g., [8,19]. In particular, one can simulate finite transition systems by reaction systems and one can simulate reaction systems by finite transition systems, see e.g., [2,8].…”