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Documents authored by Woracek, Harald

Document
DOI: 10.4230/LIPIcs.CALCO.2025.12
Cancellative Convex Semilattices

Authors: Ana Sokolova and Harald Woracek

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract
Convex semilattices are algebras that are at the same time a convex algebra and a semilattice, together with a distributivity axiom. These algebras have attracted some attention in the last years as suitable algebras for probability and nondeterminism, in particular by being the Eilenberg-Moore algebras of the nonempty finitely-generated convex subsets of the distributions monad. A convex semilattice is cancellative if the underlying convex algebra is cancellative. Cancellative convex algebras have been characterized by M. H. Stone and by H. Kneser: A convex algebra is cancellative if and only if it is isomorphic to a convex subset of a vector space (with canonical convex algebra operations). We prove an analogous theorem for convex semilattices: A convex semilattice is cancellative if and only if it is isomorphic to a convex subset of a Riesz space, i.e., a lattice-ordered vector space (with canonical convex semilattice operations).
Cite as

Ana Sokolova and Harald Woracek. Cancellative Convex Semilattices. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sokolova_et_al:LIPIcs.CALCO.2025.12,
  author =	{Sokolova, Ana and Woracek, Harald},
  title =	{{Cancellative Convex Semilattices}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.12},
  URN =		{urn:nbn:de:0030-drops-235714},
  doi =		{10.4230/LIPIcs.CALCO.2025.12},
  annote =	{Keywords: convex semilattice, cancellativity, Riesz space}
}
Document
(Co)algebraic pearls
DOI: 10.4230/LIPIcs.CALCO.2021.23
Nawrotzki’s Algorithm for the Countable Splitting Lemma, Constructively ((Co)algebraic pearls)

Authors: Ana Sokolova and Harald Woracek

Published in: LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

Abstract
We reprove the countable splitting lemma by adapting Nawrotzki’s algorithm which produces a sequence that converges to a solution. Our algorithm combines Nawrotzki’s approach with taking finite cuts. It is constructive in the sense that each term of the iteratively built approximating sequence as well as the error between the approximants and the solution is computable with finitely many algebraic operations.
Cite as

Ana Sokolova and Harald Woracek. Nawrotzki’s Algorithm for the Countable Splitting Lemma, Constructively ((Co)algebraic pearls). In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{sokolova_et_al:LIPIcs.CALCO.2021.23,
  author =	{Sokolova, Ana and Woracek, Harald},
  title =	{{Nawrotzki’s Algorithm for the Countable Splitting Lemma, Constructively}},
  booktitle =	{9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-212-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{211},
  editor =	{Gadducci, Fabio and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.23},
  URN =		{urn:nbn:de:0030-drops-153781},
  doi =		{10.4230/LIPIcs.CALCO.2021.23},
  annote =	{Keywords: countable splitting lemma, distributions with given marginals, couplings}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2017.22
Termination in Convex Sets of Distributions

Authors: Ana Sokolova and Harald Woracek

Published in: LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)

Abstract
Convex algebras, also called (semi)convex sets, are at the heart of modelling probabilistic systems including probabilistic automata. Abstractly, they are the Eilenberg-Moore algebras of the finitely supported distribution monad. Concretely, they have been studied for decades within algebra and convex geometry. In this paper we study the problem of extending a convex algebra by a single point. Such extensions enable the modelling of termination in probabilistic systems. We provide a full description of all possible extensions for a particular class of convex algebras: For a fixed convex subset D of a vector space satisfying additional technical condition, we consider the algebra of convex subsets of D. This class contains the convex algebras of convex subsets of distributions, modelling (nondeterministic) probabilistic automata. We also provide a full description of all possible extensions for the class of free convex algebras, modelling fully probabilistic systems. Finally, we show that there is a unique functorial extension, the so-called black-hole extension.
Cite as

Ana Sokolova and Harald Woracek. Termination in Convex Sets of Distributions. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{sokolova_et_al:LIPIcs.CALCO.2017.22,
  author =	{Sokolova, Ana and Woracek, Harald},
  title =	{{Termination in Convex Sets of Distributions}},
  booktitle =	{7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-033-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{72},
  editor =	{Bonchi, Filippo and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2017.22},
  URN =		{urn:nbn:de:0030-drops-80434},
  doi =		{10.4230/LIPIcs.CALCO.2017.22},
  annote =	{Keywords: convex algebra, one-point extensions, convex powerset monad}
}
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