Noname manuscript No. (will be inserted by the editor) On the complexity of occurrence and convergence problems in reaction systems Enrico Formenti · Luca Manzoni · Antonio E. Porreca the date of receipt and acceptance should be inserted later Abstract Reaction systems are a model of computation in- spired by biochemical reactions introduced by Ehrenfeucht and Rozenberg. Two problems related to the dynamics (the evolution of the state with respect to time) of reaction sys- tems, namely, the occurrence and the convergence problems, were recently investigated by Salomaa. In this paper, we prove that both problems are PSPACE-complete when the numerical parameter of the problems (i.e., the time step when a specified element must appear) is given as input. Moreover, they remain PSPACE-complete even for minimal reaction systems. Keywords Reaction systems · Computational complexity · Discrete dynamical systems 1 Introduction Reaction systems have been recently introduced as a formal model inspired to the reactions taking place in biochemical systems (Ehrenfeucht and Rozenberg 2007). Each reaction is described by means of a set of reactants, a set of inhibitors and a set of products. At each time step, the reactions trans- form a set of entities, generating a state sequence over time. E. Formenti Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, 06900 Sophia Antipolis, France E-mail: enrico.formenti@unice.fr Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, 06900 Sophia Antipolis, France L. Manzoni E-mail: luca.manzoni@i3s.unice.fr A.E. Porreca Dipartimento di Informatica, Sistemistica e Comunicazione, Universit ` a degli Studi di Milano-Bicocca, Viale Sarca 336/14, 20126 Milano, Italy E-mail: porreca@disco.unimib.it This note concerns some questions regarding the dynam- ics of reaction systems. In particular, it investigates the com- putational complexity of determining if a particular element appears at the m-th step in at least one state sequence (resp., all state sequences) consisting of at least m steps. Those two problems are called the occurrence and convergence prob- lems and were studied by Salomaa (2013b; 2013a) when the parameter m is fixed. We prove that, when m is allowed to vary, the complexity of the decision problem increases from coNP-completeness or NP-completeness to PSPACE- completeness, even when the reaction systems are minimal, i.e., when sets of reactants and inhibitors for all reactions in the system are singletons. Furthermore, we prove that the complexity results provided by Salomaa (2013b, Theorem 2; 2013a, Corollary 1) extend to the case of minimal reaction systems. The paper is structured as follows. In Section 2, the nec- essary basic notions are introduced. In Section 3 the main results are stated. The proofs are pretty technical and are given in a separate section (Section 4), which is organized in several subsections. We describe an implementation of finite cellular automata (4.1) and binary counters (4.2) using reaction systems. Section 4.3 shows the PSPACE-complete- ness of the two decision problems in the general case, and Section 4.4 provides a way to transport the complexity results to minimal reaction systems. Finally, Section 5 draws our conclusions. 2 Basic notions This section recalls the concepts of reaction, reaction systems, and of the state sequence generated by a reaction system starting from a state. The definitions of the occurrence and convergence problems for reaction systems are also stated.