Ranks of Strictly Minimal Reaction Systems Induced by Permutations and Cartesian Product

TL;DR

This paper investigates the maximum size of reaction systems that are strictly minimal and can simulate any reaction system, focusing on permutation-induced systems and Cartesian products, with detailed results for small alphabets.

Contribution

It provides a detailed analysis of the ranks of strictly minimal reaction systems induced by permutations and Cartesian products, extending understanding of their computational capabilities.

Findings

Determined the maximum rank for permutation-induced reaction systems up to quaternary alphabet.

Established a general result on reaction system rank for Cartesian product functions.

Showed that strictly minimal reaction systems can simulate all reaction systems at certain sizes.

Abstract

Reaction system is a computing model inspired by the biochemical interaction taking place within the living cells. Various extended or modified frameworks motivated by biological, physical, or purely mathematically considerations have been proposed and received significant amount of attention, notably in the recent years. This study, however, takes after particular early works that concentrated on the mathematical nature of minimal reaction systems in the context-free basic framework and motivated by a recent result on the sufficiency of strictly minimal reaction systems to simulate every reaction system. This paper focuses on the largest reaction system rank attainable by strictly minimal reaction systems, where the rank pertains to the minimum size of a functionally equivalent reaction system. Precisely, we provide a very detailed study for specific strictly minimal reaction system induced by permutations, up to the quaternary alphabet. Along the way, we obtain a general result about reaction system rank for Cartesian product of functions specified by reaction systems.