Context-Free Graph Properties via Definable Decompositions.

Monadic-second order logic (MSO-logic) is successfully applied in both language theory and algorithm design. In the former, definable by MSO-formulas are exactly the regular 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 Courcelleu0027s. 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 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 Schutzenberger 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.