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1. Reachability in Deletion-Only Chemical Reaction Networks

   https://doi.org/10.4230/lipics.dna.31.3  

   Fu, Bin; Gomez, Timothy; Knobel, Ryan; Luchsinger, Austin; Massie, Aiden; Rodriguez, Marco; Salinas, Adrian; Schweller, Robert; Wylie, Tim (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Schaeffer, Josie; Zhang, Fei (Ed.)

   For general discrete Chemical Reaction Networks (CRNs), the fundamental problem of reachability - the question of whether a target configuration can be produced from a given initial configuration - was recently shown to be Ackermann-complete. However, many open questions remain about which features of the CRN model drive this complexity. We study a restricted class of CRNs with void rules, reactions that only decrease species counts. We further examine this regime in the motivated model of step CRNs, which allow additional species to be introduced in discrete stages. With and without steps, we characterize the complexity of the reachability problem for CRNs with void rules. We show that, without steps, reachability remains polynomial-time solvable for bimolecular systems but becomes NP-complete for larger reactions. Conversely, with just a single step, reachability becomes NP-complete even for bimolecular systems. Our results provide a nearly complete classification of void-rule reachability problems into tractable and intractable cases, with only a single exception. 

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2. Computing Threshold Circuits with Bimolecular Void Reactions in Step Chemical Reaction Networks

   Anderson, Rachel; Fu, Bin; Massie, Aiden; Mukhopadhyay, Gourab; Salinas, Adrian; Schweller, Robert; Tomai, Evan; Wylie, Tim (June 2024, Springer Nature)

   Step Chemical Reaction Networks (step CRNs) are an augmentation of the Chemical Reaction Network (CRN) model where additional species may be introduced to the system in a sequence of “steps.” We study step CRN systems using a weak subset of reaction rules, void rules, in which molecular species can only be deleted. We demonstrate that step CRNs with only void rules of size (2,0) can simulate threshold formulas (TFs) under linear resources. These limited systems can also simulate threshold circuits (TCs) by modifying the volume of the system to be exponential. We then prove a matching exponential lower bound on the required volume for simulating threshold circuits in a step CRN with (2,0)-size rules under a restricted gate-wise simulation, thus showing our construction is optimal for simulating circuits in this way. 

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3. Synchronous Dynamical Systems on Directed Acyclic Graphs: Complexity and Algorithms

   https://doi.org/10.1145/3653723  

   Rosenkrantz, Daniel J; Marathe, Madhav V; Ravi, SS; Stearns, Richard E (June 2024, ACM Transactions on Computation Theory)

   Discrete dynamical systems serve as useful formal models to study diffusion phenomena in social networks. Several recent articles have studied the algorithmic and complexity aspects of some decision problems on synchronous Boolean networks, which are discrete dynamical systems whose underlying graphs are directed, and may contain directed cycles. Such problems can be regarded as reachability problems in the phase space of the corresponding dynamical system. Previous work has shown that some of these decision problems become efficiently solvable for systems on directed acyclic graphs (DAGs). Motivated by this line of work, we investigate a number of decision problems for dynamical systems whose underlying graphs are DAGs. We show that computational intractability (i.e.,PSPACE-completeness) results for reachability problems hold even for dynamical systems on DAGs. We also identify some restricted versions of dynamical systems on DAGs for which reachability problem can be solved efficiently. In addition, we show that a decision problem (namely, Convergence), which is efficiently solvable for dynamical systems on DAGs, becomesPSPACE-complete for Quasi-DAGs (i.e., graphs that become DAGs by the removal of asingleedge). In the process of establishing the above results, we also develop several structural properties of the phase spaces of dynamical systems on DAGs. 

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4. The Computational Power of Discrete Chemical Reaction Networks with Bounded Executions

   https://doi.org/10.4230/LIPIcs.DISC.2024.20  

   Doty, David; Heckmann, Ben (October 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Alistarh, Dan (Ed.)

   Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as A + B → 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:ℕ^d → {0,1} and integer-valued functions f:ℕ^d → ℕ; for instance X₁ + X₂ → Y computes the function min(x₁,x₂), since starting with x_i copies of X_i, eventually min(x₁,x₂) copies of Y are produced. 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 ⇌ 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 "non-collapsing" (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 a characterization of execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction [Czerner et al., 2024]. 

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5. Programming and training rate-independent chemical reaction networks

   https://doi.org/10.1073/pnas.2111552119  

   Vasić, Marko; Chalk, Cameron; Luchsinger, Austin; Khurshid, Sarfraz; Soloveichik, David (June 2022, Proceedings of the National Academy of Sciences)

   Embedding computation in biochemical environments incompatible with traditional electronics is expected to have a wide-ranging impact in synthetic biology, medicine, nanofabrication, and other fields. Natural biochemical systems are typically modeled by chemical reaction networks (CRNs) which can also be used as a specification language for synthetic chemical computation. In this paper, we identify a syntactically checkable class of CRNs called noncompetitive (NC) whose equilibria are absolutely robust to reaction rates and kinetic rate law, because their behavior is captured solely by their stoichiometric structure. In spite of the inherently parallel nature of chemistry, the robustness property allows for programming as if each reaction applies sequentially. We also present a technique to program NC-CRNs using well-founded deep learning methods, showing a translation procedure from rectified linear unit (ReLU) neural networks to NC-CRNs. In the case of binary weight ReLU networks, our translation procedure is surprisingly tight in the sense that a single bimolecular reaction corresponds to a single ReLU node and vice versa. This compactness argues that neural networks may be a fitting paradigm for programming rate-independent chemical computation. As proof of principle, we demonstrate our scheme with numerical simulations of CRNs translated from neural networks trained on traditional machine learning datasets, as well as tasks better aligned with potential biological applications including virus detection and spatial pattern formation. 

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