Complexity of model checking for reaction systems

Abstract

Reaction systems are a new mathematical formalism inspired by the living cell and driven by only two basic mechanisms: facilitation and inhibition. As a modeling framework, they differ from the traditional approaches based on ODEs and CTMCs in two fundamental aspects: their qualitative character and the nonpermanency of resources. In this article we introduce to reaction systems several notions of central interest in biomodeling: mass conservation, invariants, steady states, stationary processes, elementary fluxes, and periodicity. We prove that the decision problems related to these properties span a number of complexity classes from P to NP-and coNP-complete to PSPACE-complete.

Dependency graphs and mass conservation in reaction systems

“…This is in line with the observation that reactions that cannot be inhibited by the sets taken into consideration (e.g. subsets of the support set) are crucial for mass conservation [8].…”

“…For each source set T ∈ σ ( G cd ), compute the corresponding source set of G cd as T = cover( T ∪ P in ). 8. For each source set T computed in step 7, test whether T is a conserved set by checking that it is consistent with all connected components of the behavior graph G b .…”

“…(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]). Quite naturally, investigations along this second line led to the study of model checking for reaction systems.…”

“…They prove that the general model checking problem is PSPACE-complete. On the other hand, [8] starts with defining a series of biologically inspired properties for reaction systems and shows that checking some of them, while still intractable, is a problem of lower computational complexity.…”

“…In this paper we conduct a detailed study of the biologically inspired property of mass conservation in reaction systems, originally introduced and shown to be coNP-complete in [8]. We get a new insight into the connection between the internal structure of the reaction system and mass conservation by revealing a relation that the latter induces between the species, and we capture this relation by defining the conservation dependency graph.…”