Computing preimages and ancestors in reaction systems

In reaction systems, preimages and n-th ancestors are sets of reactants leading to the production of a target set of products in either one or n steps, respectively. Many computational problems on preimages and ancestors, such as finding all minimum-cardinality n-th ancestors, computing their size, or counting them, are intractable. In this paper we propose a characterization of n-th ancestors as a Boolean formula, and we define an operator able to compute such a formula in polynomial time. Our formula can be exploited to solve all preimage and ancestors problems and, therefore, it can be directly used to study their complexity. In particular, we focus on two problems: (i) deciding whether a preimage/n-th ancestor exists (ii) finding a preimage/n-th ancestor of minimal size. Our approach naturally leads to the definition of classes of systems for which such problems can be solved in polynomial time.