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

Speed faults in computation by chemical reaction networks

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. Assuming a fixed molecular population size and bimolecular reactions, CRNs are formally equivalent to population protocols, a model of distributed computing introduced by Angluin, Aspnes, Diamadi, Fischer, and Peralta (PODC 2004). The challenge of fast computation by CRNs (or population protocols) is to ensure that there is never a bottleneck "slow" reaction that requires two molecules (agent states) to react (communicate), both of which are present in low (O(1)) counts. It is known that CRNs can be fast in expectation by avoiding slow reactions with high probability. However, states may be reachable (with low probability) from which the correct answer may only be computed by executing a slow reaction. We deem such an event a speed fault. We show that the problems decidable by CRNs guaranteed to avoid speed faults are precisely the detection problems: Boolean combinations of questions of the form "is a certain species present or not?". This implies, for instance, that no speed fault free CRN could decide whether there are at least two molecules of a certain species, although a CRN could decide this in "fast" expected time — i.e. speed fault free CRNs "can't count.