The computational power of discrete chemical reaction networks with bounded executions

TL;DR

This work investigates the computational power of discrete chemical reaction networks under execution-boundedness, where only a finite number of reactions occur from the initial state. It shows that with an initial leader and a single-voter output, execution-bounded CRNs can stably compute exactly the semilinear predicates and functions in $O(n\log n)$ parallel time, matching the capabilities of unbounded CRNs while offering bounded runtime guarantees. Conversely, in leaderless, all-voting, non-collapsing settings, execution-bounded CRNs are severely limited to eventually constant predicates, with a linear-potential function providing a characterization of their behavior. The paper also develops a robust framework linking linear potentials to bounded executions and provides detailed constructions for modular components (mod/threshold predicates) and affine decompositions to realize semilinear computations, highlighting the necessity of the studied constraints for achieving positive results.

Abstract

Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as $A + B \to C$, representing that if a molecule of $A$ and $B$ collide, they disappear and a molecule of $C$ is produced. CRNs can compute Boolean-valued predicates $φ:\mathbb{N}^d \to \{0,1\}$ and integer-valued functions $f:\mathbb{N}^d \to \mathbb{N}$; for instance $X_1 + X_2 \to Y$ computes the function $\min(x_1,x_2)$. We study the computational power of execution bounded CRNs, in which only a finite number of reactions can occur from the initial configuration (e.g., ruling out reversible reactions such as $A \rightleftharpoons B$). The power and composability of such CRNs depend crucially on some other modeling choices that do not affect the computational power of CRNs with unbounded executions, namely whether an initial leader is present, and whether (for predicates) all species are required to "vote" for the Boolean output. If the CRN starts with an initial leader, and can allow only the leader to vote, then all semilinear predicates and functions can be stably computed in $O(n \log n)$ parallel time by execution bounded CRNs. However, if no initial leader is allowed, all species vote, and the CRN is "noncollapsing" (does not shrink from initially large to final $O(1)$ size configurations), then execution bounded CRNs are severely limited, able to compute only eventually constant predicates. A key tool is to characterize execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction, a result that may be of independent interest.

Figures (3)

• Figure 1: Geometric intuition of \ref{['cor:linear-alternative-integer']}. A matrix $\mathbf{M}$ has column vectors $\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3$. The cone of $\mathbf{M}$ is the nonnegative span of these vectors, shown as a faded gray region. Exactly one of two scenarios occurs: a) The cone of $\mathbf{M}$ intersects the first quadrant (nonnegative orthant in higher dimensions) away from the origin, i.e., some semipositive point ($\mathbf{M}\mathbf{u} \geq \mathbf{0}$ above) is a nonnegative linear combination of vectors $\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3$. b) The cone of $\mathbf{M}$ does not intersect the first quadrant except at the origin. In this case we can draw a dashed line (hyperplane in higher dimensions) separating the cone of $\mathbf{M}$ from the first quadrant. The orthogonal vector $\mathbf{v} \geq \mathbf{0}$ to this line lies in the first quadrant, but $\mathbf{v} \mathbf{M} < 0$ means each vector $\mathbf{x}_i$ has negative dot product $\mathbf{v} \cdot \mathbf{x}_i < 0$, i.e., all $\mathbf{x}_i$ vectors lie on the other side of the line.
• Figure 2: Boolean combinations of threshold predicates in 2D. In all cases, inputs of region $R_2$ output yes, while $R_1, R_3, R_4$ output no. In part (a), the no regions are finite, making the predicate trivially eventually constant. Part (b) features both infinite no and yes regions; setting $m_0 \geq 3$, the output within $\mathbb{N}^k_{\geq m_0}$ is constant, rendering the predicate eventually constant. In contrast, part (c) has two totally unbounded regions $R_2$ and $R_3$, both of which contain points that are arbitrarily large on all components. We further observe that this implies a separating line with a positive, non-infinite slope between the two regions (in 2D). Generalized to more dimensions: a hyperplane with a normal vector with at least one positive and at least one negative component (\ref{['lem:threshold-not-eventually-constant-regions-two-adjacent-infinite-opposite-output']}).
• Figure 3: A 2D periodic set $A$. The finite set $F = \{(0,0), (0,2), (1,1), (2,2)\}$, a subset of the $3 \times 3$ square $\{0,1,2\}^2$, is translated in both $x$ and $y$ directions by every nonnegative vector with entries that are multiples of 3. If the set is periodic but not constant, then there is some vector $\mathbf{v}$ such that the sequence $\mathbf{y}_0,\mathbf{y}_1,\dots$, each being the previous plus $\mathbf{v}$ ($\mathbf{v}=(2,1)$ in this example), is periodic but not constant, and then a regularly-spaced subsequence $\mathbf{x}_0,\mathbf{x}_1,\dots$, alternating points in $A$ and points not in $A$, satisfies \ref{['lem:semilinear-not-almost-constant-alternating-period-vector-and-offset']}. For brevity, the figure shows $\mathbf{y}_0$ as the origin, but in the proof of \ref{['lem:semilinear-not-almost-constant-alternating-period-vector-and-offset']}, $\mathbf{y}_0$ is chosen to be sufficiently far from the origin that, moving in direction $\mathbf{v}$ infinitely from that point will not cross any hyperplanes defined by the threshold sets of that proof.

Theorems & Definitions (29)

• Remark 1
• Definition 2
• Theorem 3: angluin2007computationalchen2023rate
• Definition 4
• Theorem 5: Ginsburg1966Semigroups
• Definition 6
• Lemma 8
• Lemma 9
• Definition 10
• Theorem 11