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 (i.e., able to simulate arbitrary algorithms) 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 (a multi-dimensional generalization of “eventually periodic” sets). We introduce the notion of function, rather than predicate, computation by representing the output of a function \({f:{\mathbb{N}}^k\to{\mathbb{N}}^l}\) by a count of some molecular species, i.e., if the CRN starts with \(x_1,\ldots,x_k\) molecules of some “input” species \(X_1,\ldots,X_k, \) the CRN is guaranteed to converge to having \(f(x_1,\ldots,x_k)\) molecules of the “output” species \(Y_1,\ldots,Y_l\). We show that a function \({f:{\mathbb{N}}^k \to {\mathbb{N}}^l}\) is deterministically computed by a CRN if and only if its graph \({\{({\bf x, y}) \in {\mathbb{N}}^k \times {\mathbb{N}}^l | f({\bf x}) = {\bf y}\}}\) is a semilinear set. Finally, we show that each semilinear function f (a function whose graph is a semilinear set) can be computed by a CRN on input x in expected time \(O(\hbox{polylog} \|{\bf x}\|_1)\).

中文翻译：

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化学反应网络的确定性函数计算。

化学反应网络（CRN）在混合均匀的溶液中对化学进行正式建模。CRN被广泛用于描述自然细胞调控网络中发生的信息处理，并且随着合成生物学的不断发展，CRN是用于设计人工分子控制电路的有前途的语言。尽管如此，尽管CRN在自然科学中得到了广泛的使用，但对CRN表现出的计算行为的范围却知之甚少。当允许出现错误的可能性很小时，CRN已被证明是有效的图灵通用的（即，能够模拟任意算法）。另一方面，已证明保证会收敛于正确答案的CRN仅决定半线性谓词（“最终周期性”集的多维概括）。\（{f：{\ mathbb {N}} ^ k \ to {\ mathbb {N}} ^ l} \）通过一些分子种类的计数，即，如果CRN以\（x_1，\ ldots，某些“输入”物质\（X_1，\ ldots，X_k，\）的x_k \）分子保证CRN收敛到具有“输出”物质\\ （f（x_1，\ ldots，x_k）\）个分子\ （Y_1，\ ldots，Y_l \）。我们表明，一个函数\（{F：{\ mathbb {N}} ^ķ\到{\ mathbb {N}} ^ L} \）被确定性地由CRN计算当且仅当其图表\（{\ { （{\ bf x，y}）\ in {\ mathbb {N}} ^ k \ times {\ mathbb {N}} ^ l | f（{\ bf x}）= {\ bf y} \}} \\ ）是一个半线性集。最后，我们证明每个半线性函数f（其图是半线性集的函数）可以由CRN在预期时间\（O（\ hbox {polylog} \ | {\ bf x} \ | _1）\）中对输入x进行计算。