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 -bisimulation. Two states (variables) are proven -bisimilar if and only if they correspond to the same solution in the ODEs system. We then characterize -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 -bisimulation containing all valid identities that are instances of a user-specified template. A specific largest -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.

Algebra, coalgebra, and minimization in polynomial differential equations / Boreale, Michele*. - STAMPA. - (2017), pp. 71-87. [10.1007/978-3-662-54458-7_5]

Algebra, coalgebra, and minimization in polynomial differential equations

Boreale, Michele
2017

Abstract

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 -bisimulation. Two states (variables) are proven -bisimilar if and only if they correspond to the same solution in the ODEs system. We then characterize -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 -bisimulation containing all valid identities that are instances of a user-specified template. A specific largest -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.
2017
9783662544570
Foundations of Software Science and Computation Structures - 20th International Conference, {FOSSACS} 2017, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
71
87
Boreale, Michele*
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1113069
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