Probability 1 computation with chemical reaction networks

The computational power of stochastic chemical reaction networks (CRNs) varies significantly with the output convention and whether or not error is permitted. Focusing on probability 1 computation, we demonstrate a striking difference between stable computation that converges to a state where the output cannot change, and the notion of limit-stable computation where the output eventually stops changing with probability 1. While stable computation is known to be restricted to semilinear predicates (essentially piecewise linear), we show that limit-stable computation encompasses the set of predicates ϕ:N→{0,1} in Δ^0_2 in the arithmetical hierarchy (a superset of Turing-computable). In finite time, our construction achieves an error-correction scheme for Turing universal computation. We show an analogous characterization of the functions f:N→N computable by CRNs with probability 1, which encode their output into the count of a certain species. This work refines our understanding of the tradeoffs between error and computational power in CRNs

Probability 1 computation with chemical reaction networks

The computational power of stochastic chemical reaction networks (CRNs) varies significantly with the output convention and whether or not error is permitted. Focusing on probability 1 computation, we demonstrate a striking difference between stable computation that converges to a state where the output cannot change, and the notion of limit-stable computation where the output eventually stops changing with probability 1. While stable computation is known to be restricted to semilinear predicates (essentially piecewise linear), we show that limit-stable computation encompasses the set of predicates ϕ:N→{0,1} in Δ^0_2 in the arithmetical hierarchy (a superset of Turing-computable). In finite time, our construction achieves an error-correction scheme for Turing universal computation. We show an analogous characterization of the functions f:N→N computable by CRNs with probability 1, which encode their output into the count of a certain species. This work refines our understanding of the tradeoffs between error and computational power in CRNs

Computational Complexity of Atomic Chemical Reaction Networks

Informally, a chemical reaction network is "atomic" if each reaction may be interpreted as the rearrangement of indivisible units of matter. There are several reasonable definitions formalizing this idea. We investigate the computational complexity of deciding whether a given network is atomic according to each of these definitions. Our first definition, primitive atomic, which requires each reaction to preserve the total number of atoms, is to shown to be equivalent to mass conservation. Since it is known that it can be decided in polynomial time whether a given chemical reaction network is mass-conserving, the equivalence gives an efficient algorithm to decide primitive atomicity. Another definition, subset atomic, further requires that all atoms are species. We show that deciding whether a given network is subset atomic is in $\textsf{NP}$, and the problem "is a network subset atomic with respect to a given atom set" is strongly $\textsf{NP}$-$\textsf{Complete}$. A third definition, reachably atomic, studied by Adleman, Gopalkrishnan et al., further requires that each species has a sequence of reactions splitting it into its constituent atoms. We show that there is a $\textbf{polynomial-time algorithm}$ to decide whether a given network is reachably atomic, improving upon the result of Adleman et al. that the problem is $\textbf{decidable}$. We show that the reachability problem for reachably atomic networks is $\textsf{Pspace}$-$\textsf{Complete}$. Finally, we demonstrate equivalence relationships between our definitions and some special cases of another existing definition of atomicity due to Gnacadja

Modular and Robust Computation with Deterministic Chemical Reaction Networks

In this thesis, we present four results concerning the computing capabilities of chemical reaction networks (CRNs) under deterministic mass action semantics: (1) we introduce a modular method for computing concentration signals using CRN extension operators; (2) we present a thorough analysis of two CRN signal restoration algorithms that prevent certain concentration signals from degrading over time; (3) we introduce a new model called input/output chemical reaction networks (I/O CRNs) which generalizes the CRN model to allow receiving input signals over time; and (4) we investigate what I/O CRNs can compute robustly and prove that I/O CRNs are capable of robustly simulating any nondeterministic finite automaton (NFA). CRN extension operators are operations that can be applied to a CRN to add extra functionality without affecting its original behavior. We show that common operators such as addition, multiplication, integration, and many others can be characterized as CRN extension operators. By iteratively applying these extensions, complex concentration signals can be modularly constructed from simple CRNs. To explore the full generality of these extensions, we introduce a notion of weakly CRN-computable signals and show that any CRN that can weakly compute a signal can be extended to exactly compute it. The two signal restoration algorithms that we investigate are related to the approximate majority algorithm for population protocols originally developed by Angluin, Aspnes, and Eisenstat. Under deterministic semantics, these algorithms are commonly used to prevent discrete memory from deteriorating over time. We investigate the behavior of these algorithms in the presence of adversarial reactions and show that under modest conditions they are capable of maintaining discrete memory indefinitely. We also give tight analytical bounds on how these algorithms evolve over time. The I/O CRN model that we introduce has two important impacts. (1) Concentration signals are a more natural way to provide arbitrarily long inputs to a CRN. Classical CRNs are restricted to encoding inputs into the initial state of the system which makes providing an arbitrarily long input (e.g. a binary string) quite difficult. (2) It promotes modular design of CRNs. Designing chemical systems from modular components that communicate via concentration signals is now possible, and we demonstrate its effectiveness in multiple constructions including our I/O CRN implementation of NFAs. Our notion of robustness requires that an I/O CRN tolerate perturbations with respect to four things, namely, its initial state, rate constants, input signal, and the measurement of its output signal. We investigate what I/O CRNs are capable of computing under this notion of robustness, and we prove that I/O CRNs can robustly compute the regular languages by simulating an NFA in real time

Probability 1 Computation with Chemical Reaction Networks

The computational power of stochastic chemical reaction networks (CRNs) varies significantly with the output convention and whether or not error is permitted. Focusing on probability 1 computation, we demonstrate a striking difference between stable computation that converges to a state where the output cannot change, and the notion of limit-stable computation where the output eventually stops changing with probability 1. While stable computation is known to be restricted to semilinear predicates (essentially piecewise linear), we show that limitstable computation encompasses the set of predicates in Δ^0_2 in the arithmetical hierarchy (a superset of Turing-computable). In finite time, our construction achieves an error-correction scheme for Turing universal computation. This work refines our understanding of the tradeoffs between error and computational power in CRNs