# Reachability problems for continuous chemical reaction networks

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## Abstract

Chemical reaction networks (CRNs) model the behavior of molecules in a well-mixed solution. The emerging field of molecular programming uses CRNs not only as a descriptive tool, but as a programming language for chemical computation. Recently, Chen, Doty and Soloveichik introduced rate-independent continuous CRNs (CCRNs) to study the chemical computation of continuous functions. A fundamental question for any CRN model is *reachability*, the question whether a given target state is reachable from a given start state via a sequence of reactions (a *path*) in the network. In this paper, we investigate CCRN-REACH, the reachability problem for rate-independent continuous chemical reaction networks. Our main theorem is that, for CCRNs, deciding reachability—and constructing a path if there is one—is computable in polynomial time. This contrasts sharply with the known exponential space hardness of the reachability problem for discrete CRNs. We also prove that the related problem Sub-CCRN-REACH, which asks about reachability in a CCRN using only a given number of its reactions, is NP-complete.

### Keywords

Reachability Continuous chemical reaction networks Analysis of algorithms## Notes

### Acknowledgements

We thank Tim McNicholl, Xiang Huang, Titus Klinge, and Jim Lathrop for useful discussions. We also thank three anonymous reviewers for detailed improvements to this paper.

### Funding

This research was supported in part by National Science Foundation Grants 1247051 and 1545028. Part of the second author’s work was carried out while participating in the 2015 Focus Semester on Computability and Randomness at Heidelberg University. A preliminary version of part of this work was presented at the Fifteenth International Conference on Unconventional Computation and Natural Computation (UCNC 2016, Manchester, UK, July 11–15, 2016).

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