Two-Step Simulations of Reaction Systems by Minimal Ones

Abstract

Reaction systems were introduced by Ehrenfeucht and Rozenberg with biochemical applications in mind. The model is suitable for the study of subset functions, that is, functions from the set of all subsets of a finite set into itself. In this study the number of resources of a reaction system is essential for questions concerning generative capacity. While all functions (with a couple of trivial exceptions) from the set of subsets of a finite set S into itself can be defined if the number of resources is unrestricted, only a specific subclass of such functions is defined by minimal reaction systems, that is, the number of resources is smallest possible. On the other hand, minimal reaction systems constitute a very elegant model. In this paper we simulate arbitrary reaction systems by minimal ones in two derivation steps. Various techniques for doing this consist of taking names of reactions or names of subsets as elements of the background set. In this way also subset functions not at all definable by reaction systems can be generated. We follow the original definition of reaction systems, where both reactant and inhibitor sets are assumed to be nonempty

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).…”

Facilitation in reaction systems

“…• Properties of their state sequences, and in particular comparing those properties with the properties of state sequences of general reaction systems, which are well understood by now (see, e.g. [10,11,20,28,[30][31][32]). • Complexity properties, and, again, comparing them with the complexity properties of general reaction systems, which are well understood by now (see, e.g.…”

“…[1, 3-5, 11, 12, 19, 22]) and considerations concerned with the understanding of computations resulting from the interactions of biochemical reactions within a reaction system as well as from the interactions of a reaction system with its environment (see, e.g. [2,6,8,13,14,16,17,23,24,26,28,[30][31][32]). As a matter of fact, reaction systems turned out to be a novel and interesting foundational model of interactive computations.…”

Simulation of reaction systems by the strictly minimal ones

“…However, the hybrid reaction system C as in the theorem depends on A such that res A = f . Therefore, the next theorem is a variation of Theorem 3 where the hybrid reaction system C is independent from f. This theorem is essentially implied by the proof of Theorem 4 in Salomaa [27], although over there any reaction system is required to be nondegenerate. The main idea is to give a name to each subset of the background set.…”