Editing Open Problems (section)

=== Problem statement ===
Give a definition of reachability in real-valued CRNs based on continuous curves in $\mathbb{R}^k$, such that curves traced out by all known rate laws are special cases of it, and such that it respects the forward-invariant absence of siphons, as well as the intuitive idea that "motion along the curve should be interpretable as each reaction happening at some nonegative rate".

There is a definition of reachability in <ref name="CDS2014">Ho-Lin Chen, David Doty, David Soloveichik (2014). Rate-independent computation with continuous chemical reaction networks. Innovations in Theoretical Computer Science, 2014</ref>. It is based on finite numbers of straight-line segments. It obeys the "forward-invariant absence of siphons" constraint, and it also enforces that each straight line is interpretable as running some reactions. However, finite unions of straight line segments cannot describe smooth curves such as those produced by mass-action kinetics. We did prove that this notion of reachability is more general than mass-action: if $\vec{d}$ is mass-action reachable from $\vec{c}$ (under some setting of rate constants, and $\vec{d}$ could be reached either in finite time, or in the limit as time goes to infinity), then $\vec{d}$ is reachable from $\vec{c}$ via a finite number of straight-line segments under our definition. But it would be more elegant if the actual curves used in the definition had the curves traced out by mass-action and other rate laws as special cases (which would immediately imply that reachability under those rate laws implies reachability under the more generalized notion).