Boreale, Michele - Algebra, coalgebra, and minimization in polynomial differential equations

lmcs:4009 - Logical Methods in Computer Science, February 15, 2019, Volume 15, Issue 1 -
Algebra, coalgebra, and minimization in polynomial differential equations

Authors: Boreale, Michele

We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.

Volume: Volume 15, Issue 1
Published on: February 15, 2019
Submitted on: October 24, 2017
Keywords: Computer Science - Logic in Computer Science


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