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oai:drops-oai.dagstuhl.de:3439 2024-03-06T10:34:13Z ddc:004 open_access

Regular tree languages, cardinality predicates, and addition-invariant FO Harwath, Frederik Schweikardt, Nicole regular tree languages algebraic closure properties decidable characterisations addition-invariant first-order logic logical interpretations This paper considers the logic FOcard, i.e., first-order logic with cardinality predicates that can specify the size of a structure modulo some number. We study the expressive power of FOcard on the class of languages of ranked, finite, labelled trees with successor relations. Our first main result characterises the class of FOcard-definable tree languages in terms of algebraic closure properties of the tree languages. As it can be effectively checked whether the language of a given tree automaton satisfies these closure properties, we obtain a decidable characterisation of the class of regular tree languages definable in FOcard. Our second main result considers first-order logic with unary relations, successor relations, and two additional designated symbols < and + that must be interpreted as a linear order and its associated addition. Such a formula is called addition-invariant if, for each fixed interpretation of the unary relations and successor relations, its result is independent of the particular interpretation of < and +. We show that the FOcard-definable tree languages are exactly the regular tree languages definable in addition-invariant first-order logic. Our proof techniques involve tools from algebraic automata theory, reasoning with locality arguments, and the use of logical interpretations. We combine and extend methods developed by Benedikt and Segoufin (ACM ToCL, 2009) and Schweikardt and Segoufin (LICS, 2010). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Frederik Harwath and Nicole Schweikardt 2012 Is Part Of LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2012.489 urn:nbn:de:0030-drops-34394 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.489 eng https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode