  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 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, this resolving a
decades-old problem.
  Here, we fully characterize monotone reachability problems based on various
restrictions such as the rule size, the number of rules that may create a
species (k-source) or consume a species (k-consuming), the volume, and whether
the rules have an acyclic production order (feed-forward). We show
PSPACE-completeness of reachability with only bimolecular reactions with
two-source and two-consuming rules. This proves hardness of reachability in
Population Protocols, which was unknown. Further, this shows reachability in
CRNs is PSPACE-complete with size-2 rules, which was previously only known with
size-5 rules. This is achieved using techniques within the motion planning
framework.
  We provide many important results for feed-forward CRNs where rules are
single-source or single-consuming. We show that reachability is solvable in
polynomial time if the system does not contain special void or autogenesis
rules. We then fully characterize all systems of this type and show that if you
allow void/autogenesis rules, or have 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.
