International audienceWe investigate the computational complexity of deciding the occurrence of many different dynamical behaviours in reaction systems, with an emphasis on biologically relevant problems (i.e., existence of fixed points and fixed point attractors). We show that the decision problems of recognising these dynamical behaviours span a number of complexity classes ranging from FO-uniform AC^0 to Π_2^P-completeness with several intermediate problems being either NP or coNP-complete

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

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.

Summary. We investigate the influence that the flow of information in membrane systems has on their computational complexity. In particular, we analyse the behaviour of P systems with active membranes where communication only happens from a membrane towards its parent, and never in the opposite direction. We prove that these "monodirectional P systems" are, when working in polynomial time and under standard complexity-theoretic assumptions, much less powerful than unrestricted ones: indeed, they characterise classes of problems defined by polynomial-time Turing machines with NP oracles, rather than the whole class PSPACE of problems solvable in polynomial space.

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Polynomial-time P systems with active membranes characterise PSPACE by exploiting membranes nested to a polynomial depth, which may be subject to membrane division rules. When only elementary (leaf) membrane division rules are allowed, the computing power decreases to P PP = P #P , the class of problems solvable in polynomial time by deterministic Turing machines equipped with oracles for counting (or majority) problems. In this paper we investigate a variant of intermediate power, limiting membrane nesting (hence membrane division) to constant depth, and we prove that the resulting P systems can solve all problems in the counting hierarchy CH, which is located between P PP and PSPACE. In particular, for each integer k ≥ 0 we provide a lower bound to the computing power of P systems of depth k.

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