Algebra, coalgebra, and minimization in polynomial differential equations
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...
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2019-02-01
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| Series: | Logical Methods in Computer Science |
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| Online Access: | https://lmcs.episciences.org/4009/pdf |
| _version_ | 1797268570154991616 |
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| author | Michele Boreale |
| author_facet | Michele Boreale |
| author_sort | Michele Boreale |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T01:34:35Z |
| format | Article |
| id | doaj.art-82aa3a64702d4e6192a15e7d0e493acc |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:34:35Z |
| publishDate | 2019-02-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-82aa3a64702d4e6192a15e7d0e493acc2024-03-08T10:27:56ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742019-02-01Volume 15, Issue 110.23638/LMCS-15(1:14)20194009Algebra, coalgebra, and minimization in polynomial differential equationsMichele Borealehttps://orcid.org/0000-0002-1972-7491We 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.https://lmcs.episciences.org/4009/pdfcomputer science - logic in computer science |
| spellingShingle | Michele Boreale Algebra, coalgebra, and minimization in polynomial differential equations Logical Methods in Computer Science computer science - logic in computer science |
| title | Algebra, coalgebra, and minimization in polynomial differential equations |
| title_full | Algebra, coalgebra, and minimization in polynomial differential equations |
| title_fullStr | Algebra, coalgebra, and minimization in polynomial differential equations |
| title_full_unstemmed | Algebra, coalgebra, and minimization in polynomial differential equations |
| title_short | Algebra, coalgebra, and minimization in polynomial differential equations |
| title_sort | algebra coalgebra and minimization in polynomial differential equations |
| topic | computer science - logic in computer science |
| url | https://lmcs.episciences.org/4009/pdf |
| work_keys_str_mv | AT micheleboreale algebracoalgebraandminimizationinpolynomialdifferentialequations |