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Documents authored by Gehrke, Mai

Document
DOI: 10.4230/LIPIcs.CSL.2017.13
Stone Duality and the Substitution Principle

Authors: Célia Borlido, Silke Czarnetzki, Mai Gehrke, and Andreas Krebs

Published in: LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Abstract
In this paper we relate two generalisations of the finite monoid recognisers of automata theory for the study of circuit complexity classes: Boolean spaces with internal monoids and typed monoids. Using the setting of stamps, this allows us to generalise a number of results from algebraic automata theory as it relates to Büchi's logic on words. We obtain an Eilenberg theorem, a substitution principle based on Stone duality, a block product principle for typed stamps and, as our main result, a topological semidirect product construction, which corresponds to the application of a general form of quantification. These results provide tools for the study of language classes given by logic fragments such as the Boolean circuit complexity classes.
Cite as

Célia Borlido, Silke Czarnetzki, Mai Gehrke, and Andreas Krebs. Stone Duality and the Substitution Principle. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{borlido_et_al:LIPIcs.CSL.2017.13,
  author =	{Borlido, C\'{e}lia and Czarnetzki, Silke and Gehrke, Mai and Krebs, Andreas},
  title =	{{Stone Duality and the Substitution Principle}},
  booktitle =	{26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
  pages =	{13:1--13:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-045-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{82},
  editor =	{Goranko, Valentin and Dam, Mads},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.13},
  URN =		{urn:nbn:de:0030-drops-77060},
  doi =		{10.4230/LIPIcs.CSL.2017.13},
  annote =	{Keywords: C-variety of languages, typed monoid, Boolean space with an internal monoid, substitution principle, semidirect product}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.112
The Schützenberger Product for Syntactic Spaces

Authors: Mai Gehrke, Daniela Petrisan, and Luca Reggio

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
Starting from Boolean algebras of languages closed under quotients and using duality theoretic insights, we derive the notion of Boolean spaces with internal monoids as recognisers for arbitrary formal languages of finite words over finite alphabets. This leads to recognisers and syntactic spaces in a setting that is well-suited for applying tools from Stone duality as applied in semantics. The main focus of the paper is the development of topo-algebraic constructions pertinent to the treatment of languages given by logic formulas. In particular, using the standard semantic view of quantification as projection, we derive a notion of Schützenberger product for Boolean spaces with internal monoids. This makes heavy use of the Vietoris construction - and its dual functor - which is central to the coalgebraic treatment of classical modal logic. We show that the unary Schützenberger product for spaces yields a recogniser for the language of all models of the formula EXISTS x.phi(x), when applied to a recogniser for the language of all models of phi(x). Further, we generalise global and local versions of the theorems of Schützenberger and Reutenauer characterising the languages recognised by the binary Schützenberger product. Finally, we provide an equational characterisation of Boolean algebras obtained by local Schützenberger product with the one element space based on an Egli-Milner type condition on generalised factorisations of ultrafilters on words.
Cite as

Mai Gehrke, Daniela Petrisan, and Luca Reggio. The Schützenberger Product for Syntactic Spaces. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 112:1-112:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gehrke_et_al:LIPIcs.ICALP.2016.112,
  author =	{Gehrke, Mai and Petrisan, Daniela and Reggio, Luca},
  title =	{{The Sch\"{u}tzenberger Product for Syntactic Spaces}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{112:1--112:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.112},
  URN =		{urn:nbn:de:0030-drops-62474},
  doi =		{10.4230/LIPIcs.ICALP.2016.112},
  annote =	{Keywords: Stone duality and Stone-Cech compactification, Vietoris hyperspace construction, logic on words, algebraic language theory beyond the regular setting}
}
Document
DOI: 10.4230/DagRep.5.10.66
Duality in Computer Science (Dagstuhl Seminar 15441)

Authors: Mai Gehrke, Achim Jung, Victor Selivanov, and Dieter Spreen

Published in: Dagstuhl Reports, Volume 5, Issue 10 (2016)

Abstract
This report documents the programme and outcomes of Dagstuhl Seminar 15441 `Duality in Computer Science'. This seminar served as a follow-up seminar to the seminar `Duality in Computer Science' (Dagstuhl Seminar 13311). In this seminar, we focused on applications of duality to semantics for probability in computation, to algebra and coalgebra, and on applications in complexity theory. A key objective of this seminar was to bring together researchers from these communities within computer science as well as from mathematics with the goal of uncovering commonalities, forging new collaborations, and sharing tools and techniques between areas based on their common use of topological methods and duality.
Cite as

Mai Gehrke, Achim Jung, Victor Selivanov, and Dieter Spreen. Duality in Computer Science (Dagstuhl Seminar 15441). In Dagstuhl Reports, Volume 5, Issue 10, pp. 66-88, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Article{gehrke_et_al:DagRep.5.10.66,
  author =	{Gehrke, Mai and Jung, Achim and Selivanov, Victor and Spreen, Dieter},
  title =	{{Duality in Computer Science (Dagstuhl Seminar 15441)}},
  pages =	{66--88},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2016},
  volume =	{5},
  number =	{10},
  editor =	{Gehrke, Mai and Jung, Achim and Selivanov, Victor and Spreen, Dieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.5.10.66},
  URN =		{urn:nbn:de:0030-drops-56999},
  doi =		{10.4230/DagRep.5.10.66},
  annote =	{Keywords: coalgebra, domain theory, probabilistic systems, recognizability, semantics of non-classical logics, Stone duality}
}
Document
DOI: 10.4230/DagRep.3.7.54
Duality in Computer Science (Dagstuhl Seminar 13311)

Authors: Mai Gehrke, Jean-Eric Pin, Victor Selivanov, and Dieter Spreen

Published in: Dagstuhl Reports, Volume 3, Issue 7 (2013)

Abstract
Duality allows one to move between the two worlds: the world of certain algebras of properties and a spacial world of individuals, thereby leading to a change of perspective that may, and often does, lead to new insights. Dualities have given rise to active research in a number of areas of theoretical computer science. Dagstuhl Seminar 13311 "Duality in Computer Science" was held to stimulate research in this area. This report collects the ideas that were presented and discussed during the course of the seminar.
Cite as

Mai Gehrke, Jean-Eric Pin, Victor Selivanov, and Dieter Spreen. Duality in Computer Science (Dagstuhl Seminar 13311). In Dagstuhl Reports, Volume 3, Issue 7, pp. 54-73, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{gehrke_et_al:DagRep.3.7.54,
  author =	{Gehrke, Mai and Pin, Jean-Eric and Selivanov, Victor and Spreen, Dieter},
  title =	{{Duality in Computer Science (Dagstuhl Seminar 13311)}},
  pages =	{54--73},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2013},
  volume =	{3},
  number =	{7},
  editor =	{Gehrke, Mai and Pin, Jean-Eric and Selivanov, Victor and Spreen, Dieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.7.54},
  URN =		{urn:nbn:de:0030-drops-43068},
  doi =		{10.4230/DagRep.3.7.54},
  annote =	{Keywords: Stone-Priestley duality, Point free topology, Infinite computations Exact real number computation, Computability in analysis, Hierarchies, Reducibilit Topological complexity, Domain theory, Semantics, Recognizability, Profinite topology}
}
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