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Deterministic Function Computation with Chemical Reaction Networks

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising language for the design of artificial molecular control circuitry. Nonetheless, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. CRNs have been shown to be efficiently Turing-universal when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, have been shown to decide only the semilinear predicates. We introduce the notion of function, rather than predicate, computation by representing the output of a function f:N^k --> N^l by a count of some molecular species, i.e., if the CRN starts with n_1,...,n_k molecules of some "input" species X1,...,Xk, the CRN is guaranteed to converge to having f(n_1,...,n_k) molecules of the "output" species Y1,...,Yl. We show that a function f:N^k --> N^l is deterministically computed by a CRN if and only if its graph {(x,y) | f(x) = y} is a semilinear set. Furthermore, each semilinear function f can be computed on input x in expected time O(polylog(|x|)).