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Context-Free Graph Properties via Definable Decompositions Elberfeld, Michael finite model theory monadic second-order logic tree decomposition context-free languages expressive power Monadic-second order logic (MSO-logic) is successfully applied in both language theory and algorithm design. In the former, properties definable by MSO-formulas are exactly the regular properties on many structures like, most prominently, strings. In the latter, solving a problem for structures of bounded tree width is routinely done by defining it in terms of an MSO-formula and applying general formula-evaluation procedures like Courcelle's. The present paper furthers the study of second-order logics with close connections to language theory and algorithm design beyond MSO-logic. We introduce a logic that allows to expand a given structure with an existentially quantified tree decomposition of bounded width and test an MSO-definable property for the resulting expanded structure. It is proposed as a candidate for capturing the notion of "context-free graph properties" since it corresponds to the context-free languages on strings, has the same closure properties, and an alternative definition similar to the one of Chomsky and Schützenberger for context-free languages. Besides studying its language-theoretic aspects, we consider its expressive power as well as the algorithmics of its satisfiability and evaluation problems. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Michael Elberfeld 2016 Is Part Of LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2016.17 urn:nbn:de:0030-drops-65575 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.17 eng https://creativecommons.org/licenses/by/3.0/legalcode