Chemical reaction networks (CRNs) are a standard formalism used in chemistry and biology to reason about the dynamics of molecular interaction networks. In their interpretation by ordinary differential equations, CRNs provide a Turing-complete model of analog computattion, in the sense that any computable function over the reals can be computed by a finite number of molecular species with a continuous CRN which approximates the result of that function in one of its components in arbitrary precision. The proof of that result is based on a previous result of Bournez et al. on the Turing-completeness of polyno-mial ordinary differential equations with polynomial initial conditions (PIVP). It uses an encoding of real variables by two non-negative variables for concentrations, and a transformation to an equivalent quadratic PIVP (i.e. with degrees at most 2) for restricting ourselves to at most bimolecular reactions. In this paper, we study the theoretical and practical complexities of the quadratic transformation. We show that both problems of minimizing either the number of variables (i.e., molecular species) or the number of monomials (i.e. elementary reactions) in a quadratic transformation of a PIVP are NP-hard. We present an encoding of those problems in MAX-SAT and show the practical complexity of this algorithm on a benchmark of quadratization problems inspired from CRN design problems.

中文翻译：

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关于多项式微分方程二次化的复杂性

化学反应网络 (CRN) 是化学和生物学中用于推理分子相互作用网络动力学的标准形式。在通过常微分方程的解释中，CRN 提供了模拟计算的图灵完备模型，从某种意义上说，实数上的任何可计算函数都可以通过有限数量的分子种类来计算，并具有近似该函数结果的连续 CRN在其任意精度的组件之一。该结果的证明基于 Bournez 等人先前的结果。关于具有多项式初始条件 (PIVP) 的多项式常微分方程的图灵完备性。它使用两个非负浓度变量对实变量进行编码，并转换为等效的二次 PIVP（即 度数至多 2) 用于限制我们至多双分子反应。在本文中，我们研究了二次变换的理论和实践复杂性。我们表明，在 PIVP 的二次变换中最小化变量数量（即分子种类）或单项式数量（即基本反应）的两个问题都是 NP-hard 问题。我们在 MAX-SAT 中提出了这些问题的编码，并在受 CRN 设计问题启发的二次化问题基准上展示了该算法的实际复杂性。分子种类）或 PIVP 二次变换中的单项式数量（即基本反应）是 NP 难的。我们在 MAX-SAT 中提出了这些问题的编码，并在受 CRN 设计问题启发的二次化问题基准上展示了该算法的实际复杂性。分子种类）或 PIVP 二次变换中的单项式数量（即基本反应）是 NP 难的。我们在 MAX-SAT 中提出了这些问题的编码，并在受 CRN 设计问题启发的二次化问题基准上展示了该算法的实际复杂性。