Cycles and Global Attractors of Reaction Systems

Abstract

International audienceReaction systems are a recent formal model inspired by the chemical reactions that happen inside cells and possess many different dynamical behaviours. In this work we continue a recent investigation of the complexity of detecting some interesting dynamical behaviours in reaction system. We prove that detecting global behaviours such as the presence of global attractors is PSPACE - complete. Deciding the presence of cycles in the dynamics and many other related problems are also PSPACE - complete. Deciding bijectivity is, on the other hand, a coNP - complete problem

Complexity of model checking for reaction systems

“…On one hand, this is surprising because checking them in quantitative frameworks such as ODE-based reaction models or Petri-net models is in general easy and reduces to problems of linear algebra. On the other hand, our results are in line with [6,7] who show that checking properties such as fixed points, cycles, and attractors are also difficult problems for reaction systems. Moreover, our conclusions are in line with the recent results of [11] that introduce a temporal logic for reaction systems and show that model checking in this logic is PSPACEcomplete.…”

“…The problem is already PSPACE-hard for context-free reaction systems [6], which can be seen as having (∅) as their periodic sequence of contexts.…”

“…Two main lines of research have been investigated for reaction systems so far. In one line of research, the focus has been on investigating their mathematical properties, e.g., functions defined by reaction systems, state sequences, effect of limited resources, cycles and attractors, connections to propositional logic, see, e.g., [4,6,7,15,16]. On the other main line of research, the focus has been on the capabilities of reaction systems as a modeling framework, see, e.g., [1,3].…”

Dependency graphs and mass conservation in reaction systems

“…The first line comprises the research focusing on the mathematical properties of reaction systems: the set functions they can implement, their state sequences, connections to Boolean functions, etc. (e.g., [3][4][5][6][7]). The second main line of research regards reaction systems as an instrument for biological modeling (e.g., [8][9][10]).…”

A graph isomorphism condition and equivalence of reaction systems

“…It is further assumed that there is a universal set of elements that can enter the system from the outside environment and interact with the reactants at any given time. Several studies have addressed the question of the dynamics of the system (the step by step changes of the states of the system), such as reachability [5], convergence [9], fixed points and cycles [7,8]. It has been observed that the complexity of deciding existence of certain dynamical properties falls within PSPACE (reachability) or NP-completness (fixed points and fixed point attractors).…”