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 %our framework recovers probabilistic bisimulation for Markov chains and more recent forward bisimulations for %systems of nonlinear ODEs. %ordinary differential equations. Using $(\mathbb{R}, \cdot)$ we get %When the monoid is $(\mathbb{R}, \cdot)$ we can obtain nonlinear reductions for discrete-time dynamical systems and ODEs %ordinary differential equations 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(ODE)等动力模型最小化的经典工具。在前向双模拟中，商模型中的每个状态表示一个等价类，其动力学演化给出原始模型中该类成员的总和。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)，并开发了一种划分细化算法来计算最粗的GFB。当幺半群为$(\mathbb{R}, +)$时，我们的框架可恢复马尔可夫链的概率双模拟以及非线性ODE系统的最新前向双模拟；当使用$(\mathbb{R}, \cdot)$时，可对离散时间动力系统和ODE进行非线性简化，其中商模型中的每个变量表示等价类中原始变量的乘积。当定义域为布尔集$\mathbb{B}$等有限集时，可将GFB应用于布尔网络(BN)——一种计算生物学中广泛使用的动力模型。通过原型实现GFB的最小化算法，我们从两个知名基因库中选取60个布尔网络进行了析取保持与合取保持的约简，并展示了所获得的分析加速效果。针对两个选定的布尔网络，我们给出了约简结果的生物学解释，并展示了GFB如何使一个原本无法分析的大型布尔网络得以分析。使用算法的随机化版本，我们在文献中的21个动力加权网络上找到了乘积保持（即非线性）约简，这些约简无法通过精确算法处理。