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Chemical reaction networks (CRNs) formally model chemistry in a well-mixedsolution. CRNs are widely used to describe information processing occurring innatural cellular regulatory networks, and with upcoming advances in syntheticbiology, CRNs are a promising language for the design of artificial molecularcontrol circuitry. Nonetheless, despite the widespread use of CRNs in thenatural sciences, the range of computational behaviors exhibited by CRNs is notwell understood.  CRNs have been shown to be efficiently Turing-universal when allowing for asmall probability of error. CRNs that are guaranteed to converge on a correctanswer, on the other hand, have been shown to decide only the semilinearpredicates. We introduce the notion of function, rather than predicate,computation by representing the output of a function f:N^k --> N^l by a countof some molecular species, i.e., if the CRN starts with n_1,...,n_k moleculesof some "input" species X1,...,Xk, the CRN is guaranteed to converge to havingf(n_1,...,n_k) molecules of the "output" species Y1,...,Yl. We show that afunction f:N^k --> N^l is deterministically computed by a CRN if and only ifits graph {(x,y) | f(x) = y} is a semilinear set. Furthermore, each semilinearfunction f can be computed on input x in expected time O(polylog(|x|)).

Deterministic Function Computation with Chemical Reaction Networks