Democratic, existential, and consensus-based output conventions in stable computation by chemical reaction networks

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 consensus-based output convention have equivalent computational power to (1) existence-based and (2) democracy-based output conventions. The CRNs using the former output convention have only “yes” voters, with the interpretation that the CRN’s output is yes if any voters are present and no otherwise. The CRNs using the latter output convention 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 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.

Appendix: Consensus-based CRDs with nonvoters

A slightly modified definition of a con-CRD is found in the literature (Chen et al. 2014a), in which only a subset of species is designated as voters, and nonvoting species do not affect the output. Unlike exi-CRDs, which also have only a subset of voting species, these CRDs treat “yes” and “no” votes symmetrically with respect to interpreting what is the “output” of a configuration. We refer to this as a delegating CRD (in analogy to delegates who vote on behalf of others).

Definition A.1

A delegating output-stable chemical reaction decider (del-CRD) is a gen-CRD $\mathcal {D}=(\mathcal {N},\mathcal {I},\mathcal {O}_0,\mathcal {O}_1)$ where $\mathcal {N}=(\varLambda ,R)$ is a CRN and there are $\varSigma \subseteq \varLambda$ and disjoint subsets of voting species $\varGamma _0,\varGamma _1 \subseteq \varLambda$ such that

• 1. $\mathcal {I}= \{ {\mathbf {c}}\in \mathbb {N}^\varLambda \mid {\mathbf {c}}\upharpoonright _{\varLambda \setminus \varSigma } = \mathbf {0} \} \setminus \{\mathbf {0}\}$,

• 2. $\mathcal {O}_i = \{ {\mathbf {c}}\in \mathbb {N}^\varLambda \mid \mathsf {post}({\mathbf {c}}) \subseteq \mathcal {L}_i \setminus \mathcal {L}_{1-i} \}$, with $\mathcal {L}_i = \{ {\mathbf {c}}\in \mathbb {N}^\varLambda \mid {\mathbf {c}}\upharpoonright _{\varGamma _i} \ne \mathbf {0} \}$ for $i \in \{0,1\}$.

• 3. There is a partition $\{\mathcal {I}_0,\mathcal {I}_1\}$ of $\mathcal {I}$ such that $\mathsf {post}(\mathcal {I}_i) \subseteq \mathsf {pre}(\mathcal {O}_i)$ for $i \in \{0,1\}$.

The only difference between a con-CRD and a del-CRD is that the latter omits the requirement that $\varGamma _0 \cup \varGamma _1 = \varLambda$, so each con-CRD is a del-CRD. To show they have equivalent computational power, it then suffices to show that any del-CRD can be turned into a con-CRD deciding the same set. This equivalence is simpler to establish than for exi-CRDs and dem-CRDs, using a direct simulation argument that does not require the machinery of gen-CRDs.

Lemma A.2

For each del-CRD, there is a con-CRD deciding the same set.

Proof

Let $\mathcal {D}= (\mathcal {N},\mathcal {I},\mathcal {O}_0,\mathcal {O}_1)$ be an del-CRD deciding X, with $\mathcal {N}=(\varLambda ,R)$ and voting species $\varGamma _0,\varGamma _1 \subseteq \varLambda$ as in Definition A.1. Let $\Delta = \varLambda \setminus (\varGamma _0 \cup \varGamma _1)$ be the nonvoting species. Intuitively, we define a CRN $\mathcal {N}'$ in which all nonvoting species $S \in \Delta$ of $\mathcal {N}$ have an additional bit that determines whether S is a 0-voter or a 1-voter. We add reactions so that species in $\varGamma _i$ flip this bit to i in any molecule in $\Delta$. More precisely, let $\mathcal {N}'$ be obtained from $\mathcal {N}$ by first replacing every species $S \in \Delta$ by two species $S_0$ and $S_1$. Let $\varLambda '$ be the obtained set of species of $\mathcal {N}'$. Replace every reaction $\alpha = ({\mathbf {r}},{\mathbf {p}})$ of $\mathcal {N}$ by reactions $\alpha ' = ({\mathbf {r}}',{\mathbf {p}}')$ with ${\mathbf {r}}',{\mathbf {p}}' \in \mathbb {N}^{\varLambda '}$ such that $\pi ({\mathbf {r}}')={\mathbf {r}}$ and $\pi ({\mathbf {p}}')={\mathbf {p}}$, where $\pi : \varLambda ' \rightarrow \varLambda$ sends every species $S_i$ to S and sends each $V_i \in \varGamma _i$ to itself (and $\pi$ is applied component-wise to vectors). Moreover, for $i \in \{0,1\}$, add reactions $V_i+S_{1-i} \rightarrow V_i + S_i$ for all $S \in \Delta$ and $V_i \in \varGamma _i$.

Let $\mathcal {D}' = (\mathcal {N}',\mathcal {I}',\mathcal {O}'_0,\mathcal {O}'_1)$, with $\mathcal {I}', \mathcal {O}'_0$, and $\mathcal {O}'_1$ defined as in Definition 2.2 and $\mathcal {I}'$ defined with respect to $\varSigma ' = \{ S_1 \mid S \in \varSigma \}$ where $\varSigma$ corresponds to $\mathcal {I}$. (The choice of 1 instead of 0 is arbitrary.) We observe that $\mathcal {D}'$ is a con-CRD. Indeed, once a configuration ${\mathbf {c}}\in \mathcal {O}_i$ in $\mathcal {D}$ is reached from an input configuration, we have that for each ${\mathbf {c}}' \in \mathsf {post}({\mathbf {c}}), {\mathbf {c}}'$ contains at least one molecule of species $V_i$ and none of $V_{1-i}$. A configuration $\mathbf {d}$ in $\mathcal {D}'$ corresponding to ${\mathbf {c}}$ will turn every molecule into a i-voter. In other words, we eventually reach a configuration $\mathbf {d}' \in \mathcal {O}'_i$. Hence $\mathcal {D}'$ is a con-CRD deciding X. $\square$

Although the converse is trivial since, in creating a del-CRD from a con-CRD, one can choose the voting species $\varGamma _0,\varGamma _1$ to be the same, in some cases it is preferable to have a strict subset. One case in particular, in which there are exactly two voting species, i.e., $|\varGamma _0| = |\varGamma _1| = 1$, merits mention since this is often a convenient assumption to make about a CRD. The following lemma shows that we can make this assumption without loss of generality.

Lemma A.3

For each con-CRD, there is a del-CRD with exactly two voting species deciding the same set.

Proof

Let $\mathcal {D}= (\mathcal {N},\mathcal {I},\mathcal {O}_0,\mathcal {O}_1)$ be a con-CRD that decides X, with voting species $\varGamma _0,\varGamma _1$ that partition $\varLambda$. Let $\mathcal {N}'$ be the CRN obtained from $\mathcal {N}$ by adding two new species $V_0,V_1$ to $\mathcal {D}$ and adding, for each $S \in \varGamma _i$, the reactions $S \rightarrow S + V_i$ and $S + V_{1-i} \rightarrow S$. Let $\mathcal {D}' = (\mathcal {N}',\mathcal {I}',\mathcal {O}'_0,\mathcal {O}'_1)$, with $\mathcal {I}', \mathcal {O}'_0$, and $\mathcal {O}'_1$ defined as in Definition A.1 and $\mathcal {I}'$ defined with respect to the same $\varSigma$. Indeed, once an output-stable configuration ${\mathbf {c}}\in \mathcal {O}_i$ in $\mathcal {D}$ is reached from an input configuration, we have that for each ${\mathbf {c}}' \in \mathsf {post}({\mathbf {c}})$, every molecule of ${\mathbf {c}}'$ is an i-voter and ${\mathbf {c}}'$ has at least one molecule. A configuration $\mathbf {d}$ in $\mathcal {D}'$ corresponding to ${\mathbf {c}}$ may have some additional molecules of species $V_0$ or $V_1$. The i-voters will eventually remove all molecules of species $V_{1-i}$ and will produce molecules of species $V_i$, but no molecules of species $V_{1-i}$. Hence, eventually we reach a configuration $\mathbf {d}'$ with no molecules of species $V_{1-i}$ and at least one molecule of species $V_i$. We have that each configuration in $\mathsf {post}(\mathbf {d}')$ has this property. In other words, $\mathbf {d}' \in \mathcal {O}'_i$. Hence $\mathcal {D}'$ is a del-CRD. $\square$