{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6491","name":"Cardinality and counting quantifiers on omega-automatic structures","abstract":"We investigate structures that can be represented by\r\n omega-automata, so called omega-automatic structures, and prove\r\n that relations defined over such structures in first-order logic\r\n expanded by the first-order quantifiers `there exist at most\r\n $aleph_0$ many', 'there exist finitely many' and 'there exist $k$\r\n modulo $m$ many' are omega-regular. The proof identifies certain\r\n algebraic properties of omega-semigroups.\r\n\r\n As a consequence an omega-regular equivalence relation of countable\r\n index has an omega-regular set of representatives. This implies\r\n Blumensath's conjecture that a countable structure with an\r\n $omega$-automatic presentation can be represented using automata\r\n on finite words. This also complements a very recent result of\r\n Hj\"orth, Khoussainov, Montalban and Nies showing that there is an\r\n omega-automatic structure which has no injective presentation.","keywords":["$omega$-automatic presentations","$omega$-semigroups","$omega$-automata"],"author":[{"@type":"Person","name":"Kaiser, Lukasz","givenName":"Lukasz","familyName":"Kaiser"},{"@type":"Person","name":"Rubin, Sasha","givenName":"Sasha","familyName":"Rubin"},{"@type":"Person","name":"B\u00e1r\u00e1ny, Vince","givenName":"Vince","familyName":"B\u00e1r\u00e1ny"}],"position":34,"pageStart":385,"pageEnd":396,"dateCreated":"2008-02-06","datePublished":"2008-02-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kaiser, Lukasz","givenName":"Lukasz","familyName":"Kaiser"},{"@type":"Person","name":"Rubin, Sasha","givenName":"Sasha","familyName":"Rubin"},{"@type":"Person","name":"B\u00e1r\u00e1ny, Vince","givenName":"Vince","familyName":"B\u00e1r\u00e1ny"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2008.1360","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6204","volumeNumber":1,"name":"25th International Symposium on Theoretical Aspects of Computer Science","dateCreated":"2008-02-05","datePublished":"2008-02-05","editor":[{"@type":"Person","name":"Albers, Susanne","givenName":"Susanne","familyName":"Albers"},{"@type":"Person","name":"Weil, Pascal","givenName":"Pascal","familyName":"Weil"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6491","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6204"}}}