Reachability in Restricted Chemical Reaction Networks

TL;DR

The paper investigates reachability in restricted Chemical Reaction Networks (CRNs), situating CRNs within the broader landscape of Vector Addition Systems and Petri nets while focusing on restricted variants such as feed-forward structures and limited source/consumer counts. It demonstrates a sharp complexity division: reachability becomes PSPACE-complete for bimolecular reactions in two-source/two-consuming CRNs, while feed-forward CRNs exhibit tractable behavior (polynomial time) when void and autogenesis rules are absent and reduced to single-source or single-consuming cases; introducing void/autogenesis or multiple sources/consumers yields NP-hardness. The authors also develop a polynomial-time algorithm for a class of single-source, single-consuming simple CRNs and extend NP-hardness results to various restricted settings, including (3,0) void rules and (0,3) autogenesis rules. Overall, the work maps a detailed taxonomy of reachability under multiple structural restrictions, clarifying the computational limits for restricted molecular computation models and outlining several open questions for future exploration.

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.

Figures (3)

• Figure 1: Example CRN rules to demonstrate the primary restrictions.
• Figure 2: (a) Unlocked state of a Toggle-Lock gadget represented by species $G$. (b) Locked state of a Toggle-Lock gadget represented by species $G'$. (c-d) Reactions which implement a single gadget. (c) represents a successful traversal and (d) represents the 'bound-back' reactions. The arrow is incoming or outgoing from port. (e) The rotate gadget. (f) Rules for the rotate gadget.
• Figure 3: Our starting configuration $c = \{ S^*_0, A, B, C, T\}$. Our goal configuration is $c' = \{ S^v, A^v, B^v, C^v, T^{4 } \}$. Each vertex must be changed to the visited state to reach the target, and the $T$ must be the last vertex.

Theorems & Definitions (88)

• Definition 2.1: Discrete Chemical Reaction Networks
• Definition 2.2: Reachability Problem.
• Definition 2.3: Production Problem
• Definition 2.4: Universal Reachability Problem.
• Definition 2.5: Feed-Forward
• Definition 2.6: $j$-source, $j$-consuming
• Definition 2.7: Void and Autogenesis rules
• Definition 2.8
• Definition 2.9: size-$(i,j)$ rules
• Definition 2.10: Volume Decreasing, Increasing, Preserving