Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

  • Michele BorealeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)


We consider reasoning and minimization in systems of polynomial ordinary differential equations (odes). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow polynomials with a transition system structure based on the concept of Lie derivative, thus inducing a notion of Open image in new window -bisimulation. Two states (variables) are proven Open image in new window -bisimilar if and only if they correspond to the same solution in the odes system. We then characterize Open image in new window -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 Open image in new window -bisimulation containing all valid identities that are instances of a user-specified template. A specific largest Open image in new window -bisimulation can be used to build a reduced system of odes, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations.


Ordinary Differential Equations Bisimulation Minimization Gröbner bases 



The author has benefited from stimulating discussions with Mirco Tribastone.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Dipartimento di Statistica, Informatica, Applicazioni (DiSIA) “G. Parenti”Università di FirenzeFirenzeItaly

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