Quantitative notions of bisimulation are well-known tools for the minimization of dynamical models such as Markov chains and ordinary differential equations (ODEs). In \emph{forward bisimulations}, each state in the quotient model represents an equivalence class and the dynamical evolution gives the overall sum of its members in the original model. Here we introduce generalized forward bisimulation (GFB) for dynamical systems over commutative monoids and develop a partition refinement algorithm to compute the coarsest one. When the monoid is $(\mathbb{R}, +)$, we recover probabilistic bisimulation for Markov chains and more recent forward bisimulations for nonlinear ODEs. Using $(\mathbb{R}, \cdot)$ we get nonlinear reductions for discrete-time dynamical systems and ODEs where each variable in the quotient model represents the product of original variables in the equivalence class. When the domain is a finite set such as the Booleans $\mathbb{B}$, we can apply GFB to Boolean networks (BN), a widely used dynamical model in computational biology. Using a prototype implementation of our minimization algorithm for GFB, we find disjunction- and conjunction-preserving reductions on 60 BN from two well-known repositories, and demonstrate the obtained analysis speed-ups. We also provide the biological interpretation of the reduction obtained for two selected BN, and we show how GFB enables the analysis of a large one that could not be analyzed otherwise. Using a randomized version of our algorithm we find product-preserving (therefore non-linear) reductions on 21 dynamical weighted networks from the literature that could not be handled by the exact algorithm.

翻译：双模拟的量化概念是马尔可夫链和常微分方程等动力学模型极小化的经典工具。在*前向双模拟*中，商模型的每个状态代表一个等价类，其动力学演化给出原始模型中该等价类成员的总和。本文针对交换幺半群上的动力系统引入广义前向双模拟（GFB），并开发了一种划分精化算法来求解最粗的GFB。当幺半群为$(\mathbb{R}, +)$时，我们恢复了马尔可夫链的概率双模拟以及非线性常微分方程的近期前向双模拟。使用$(\mathbb{R}, \cdot)$，我们得到了离散时间动力系统和常微分方程的非线性约简，其中商模型的每个变量代表等价类中原始变量的乘积。当域为有限集（如布尔集$\mathbb{B}$）时，可将GFB应用于布尔网络——计算生物学中广泛使用的动力学模型。通过GFB极小化算法的原型实现，我们从两个知名存储库中的60个布尔网络获得了保持析取与合取的约简，并展示了由此带来的分析加速。我们给出了两个选定布尔网络的约简的生物学解释，并展示了GFB如何使原本无法分析的大型布尔网络得以分析。利用算法的随机化版本，我们从文献中的21个动力学加权网络获得了保持乘积（即非线性）的约简，而这些网络无法通过精确算法处理。