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DOI: 10.4230/LIPIcs.CONCUR.2021.19

Algebra and Coalgebra of Stream Products

Authors: Michele Boreale and Daniele Gorla

Published in: LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Abstract

We study connections among polynomials, differential equations and streams over a field 𝕂, in terms of algebra and coalgebra. We first introduce the class of (F,G)-products on streams, those where the stream derivative of a product can be expressed as a polynomial of the streams themselves and their derivatives. Our first result is that, for every (F,G)-product, there is a canonical way to construct a transition function on polynomials such that the induced unique final coalgebra morphism from polynomials into streams is the (unique) 𝕂-algebra homomorphism - and vice-versa. This implies one can reason algebraically on streams, via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G)-product. As an example of reasoning on streams, we focus on specific products (convolution, shuffle, Hadamard) and show how to obtain closed forms of algebraic generating functions of combinatorial sequences, as well as solutions of nonlinear ordinary differential equations.

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Michele Boreale and Daniele Gorla. Algebra and Coalgebra of Stream Products. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{boreale_et_al:LIPIcs.CONCUR.2021.19,
  author =	{Boreale, Michele and Gorla, Daniele},
  title =	{{Algebra and Coalgebra of Stream Products}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.19},
  URN =		{urn:nbn:de:0030-drops-143969},
  doi =		{10.4230/LIPIcs.CONCUR.2021.19},
  annote =	{Keywords: Streams, coalgebras, polynomials, differential equations}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2019.24

On the Coalgebra of Partial Differential Equations

Authors: Michele Boreale

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract

We note that the coalgebra of formal power series in commutative variables is final in a certain subclass of coalgebras. Moreover, a system Sigma of polynomial PDEs, under a coherence condition, naturally induces such a coalgebra over differential polynomial expressions. As a result, we obtain a clean coinductive proof of existence and uniqueness of solutions of initial value problems for PDEs. Based on this characterization, we give complete algorithms for checking equivalence of differential polynomial expressions, given Sigma.

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Michele Boreale. On the Coalgebra of Partial Differential Equations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{boreale:LIPIcs.MFCS.2019.24,
  author =	{Boreale, Michele},
  title =	{{On the Coalgebra of Partial Differential Equations}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.24},
  URN =		{urn:nbn:de:0030-drops-109689},
  doi =		{10.4230/LIPIcs.MFCS.2019.24},
  annote =	{Keywords: coalgebra, partial differential equations, polynomials}
}

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Documents authored by Boreale, Michele

Algebra and Coalgebra of Stream Products

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On the Coalgebra of Partial Differential Equations

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