  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.
