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.
@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} }
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