Minimization of Dynamical Systems over Monoids

Quantitative notions of bisimulation are well-known tools for the minimization of dynamical models such as Markov chains and differential equations. In a forward-type bisimulation, 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 largest one. When the monoid is (R, +), our framework recovers probabilistic bisimulation for Markov chains and more recent forward bisimulations for systems of nonlinear ordinary differential equations. When the monoid is (R, product) we can obtain nonlinear model reductions for discrete-time dynamical systems and 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 B, we can apply GFB to Boolean networks, a widely used dynamical model in computational biology. Using a prototype implementation of our minimization algorithm for GFB, we find several disjunction- and conjuction-preserving reductions on 60 Boolean networks from two well-known model repositories.