🤖 AI Summary
This paper investigates the monotonic reachability problem in constrained chemical reaction networks (CRNs): given an initial and a target state, determine whether the latter is reachable via reaction rules. Focusing on computational complexity characterization within molecular computing models, we establish the first complete complexity classification under multiple restrictions—including bounded rule size, limited species production/consumption, volume constraints, and acyclic production orders. We prove that the problem is PSPACE-complete for bimolecular CRNs with at most two reactants and two products per rule. For feed-forward CRNs, monotonic reachability is polynomial-time solvable when devoid of spurious reactions and autocatalytic rules; however, introducing either renders it NP-complete. Leveraging a novel reduction technique inspired by motion planning—and integrating vector addition systems with Petri net theory—we construct a systematic complexity mapping across restricted CRN classes. Our results resolve a long-standing open hardness question for population protocols and settle several other open problems in constrained reachability.
📝 Abstract
The popularity of molecular computation has given rise to several models of abstraction, one of the more recent ones being Chemical Reaction Networks (CRNs). These are equivalent to other popular computational models, such as Vector Addition Systems and Petri-Nets, and restricted versions are equivalent to Population Protocols. This paper continues the work on core emph{reachability} questions related to Chemical Reaction Networks; given two configurations, can one reach the other according to the system's rules? With no restrictions, reachability was recently shown to be Ackermann-complete, which resolved a decades-old problem. In this work, we fully characterize monotone reachability problems based on various restrictions such as the allowed rule size, the number of rules that may create a species ($k$-source), the number of rules that may consume a species ($k$-consuming), the volume, and whether the rules have an acyclic production order (emph{feed-forward}). We show PSPACE-completeness of reachability with only bimolecular reactions in two-source and two-consuming rules. This proves hardness of reachability in a restricted form of Population Protocols. This is accomplished using new techniques within the motion planning framework. We give several important results for feed-forward CRNs, where rules are single-source or single-consuming. We show that reachability is solvable in polynomial time as long as the system does not contain special emph{void} or emph{autogenesis} rules. We then fully characterize all systems of this type and show that with void/autogenesis rules, or more than one source and one consuming, the problems become NP-complete. Finally, we show several interesting special cases of CRNs based on these restrictions or slight relaxations and note future significant open questions related to this taxonomy.
