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Stone Duality and the Substitution Principle

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

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Célia Borlido
Silke Czarnetzki
Mai Gehrke
Andreas Krebs

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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)
https://doi.org/10.4230/LIPIcs.CSL.2017.13

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.
Keywords
  • C-variety of languages
  • typed monoid
  • Boolean space with an internal monoid
  • substitution principle
  • semidirect product

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