Abstract
We show that some natural output conventions for error-free computation in chemical reaction networks (CRN) lead to a common level of computational expressivity. Our main results are that the standard definition of error-free CRNs have equivalent computational power to (1) asymmetric and (2) democratic CRNs. The former have only “yes” voters, with the interpretation that the CRN’s output is yes if any voters are present and no otherwise. The latter define output by majority vote among “yes” and “no” voters.
Both results are proven via a generalized framework that simultaneously captures several definitions, directly inspired by a recent Petri net result of Esparza, Ganty, Leroux, and Majumder [CONCUR 2015]. These results support the thesis that the computational expressivity of error-free CRNs is intrinsic, not sensitive to arbitrary definitional choices.
The first author is a postdoctoral fellow of the Research Foundation – Flanders (FWO). The second author was supported by NSF grant 1619343, and the third author by NSF grant 1618895.
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Notes
- 1.
We use the term “error-free” in this section to refer to a specific requirement of “stability” defined formally in Sect. 2.2. When the set of configurations reachable from an initial configuration is always finite (for instance, with population protocols, or more generally mass-conserving CRNs), then stability coincides with probability 0 of error. See [9] for an in-depth discussion of how these notions can diverge when the set of configurations reachable from an initial configuration is infinite.
- 2.
The only difference is catalysts: reactants that are also products, e.g., \(C+X \rightarrow C+Y\), are allowed in CRNs and Petri nets but not in vector addition systems. Most results for these models are insensitive to this difference.
- 3.
Notation \(\varnothing \) indicates that this reaction has no products.
- 4.
The definition of [8] allows only a subset of \(\varLambda \) to be voters, i.e., \(\varGamma _0 \cup \varGamma _1 \subseteq \varLambda \). This convention is more easily shown to define equivalent computational power than our main results about asymmetric and democratic voting.
- 5.
Indeed, the negative result of [4] that sym-CRDs decide only semilinear sets is more general than stated in Theorem 2.8, applying to any reachability relation \(\Rightarrow ^*\) on \(\mathbb {N}^\varLambda \) that is reflexive, transitive, and “additive” (\({{\varvec{x}}}\Rightarrow ^*{{\varvec{y}}}\) implies \({{\varvec{x}}}+ {{\varvec{c}}}\Rightarrow ^*{{\varvec{y}}}+ {{\varvec{c}}}\)). Also, the negative result of [4] implicitly assumes that the zero vector \(\mathbf{{0}}\) is not reachable (i.e., \(\mathsf {pre}(\mathbf{{0}}) = \{\mathbf{{0}}\}\)). This assumption is manifest for population protocols (if the population size is non-zero). For CRNs, this assumption can be readily removed; see Lemma 2.11.
- 6.
While Definition 3.1 appears almost too general to be useful, Corollary 3.3 says that if \(\mathcal {I}, \mathcal {O}_0, \mathcal {O}_1\) are semilinear, then so are \(\mathcal {I}_0,\mathcal {I}_1\), which implies that any CRD definition that can be framed as such a gen-CRD must decide only semilinear sets.
- 7.
In contrast, the proof of [4] crucially requires the hypothesis \(\mathsf {post}(\mathcal {I}_i) \subseteq \mathsf {pre}(\mathcal {O}_i)\).
- 8.
As noted, sym-CRDs could be defined by replacing the requirement \(\mathsf {post}(\mathcal {I}_i) \subseteq \mathsf {pre}(\mathcal {O}_i)\) with \(\mathcal {I}_i = \mathcal {I}\cap \mathsf {pre}(\mathcal {O}_i)\) and retain the same power, but for clarity we retain the original definition.
- 9.
Just as for sym-CRDs, \(\mathsf {post}(\mathcal {O}_i) = \mathcal {O}_i\). Note that \(\mathcal {V}_1\) above is the same as \(\mathcal {L}_1\) in Definition 2.2, but \(\mathcal {L}_0 \ne \mathcal {V}_0\), since \(\mathcal {L}_1\) and \(\mathcal {L}_0\) can have nonempty intersection if there are conflicting voters present in some configuration.
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Acknowledgements
R.B. thanks Grzegorz Rozenberg for interesting and useful discussions regarding chemical reaction networks. D.D. thanks Ryan James for suggesting the democratic CRD model. The authors are grateful to the anonymous reviewers for comments that have helped improve the presentation.
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Brijder, R., Doty, D., Soloveichik, D. (2016). Robustness of Expressivity in Chemical Reaction Networks. In: Rondelez, Y., Woods, D. (eds) DNA Computing and Molecular Programming. DNA 2016. Lecture Notes in Computer Science(), vol 9818. Springer, Cham. https://doi.org/10.1007/978-3-319-43994-5_4
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