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Graph Structure and Monadic Second-Order Logic by Joost Engelfriet, Bruno Courcelle

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5.3 Monadic second-order logic and recognizability

This section is devoted to the proof of the Recognizability Theorem, a corner-stone of the theory developed in this book. Informally, every CMS-definable set of graphs or of relational structures is recognizable with respect to an appropriate algebra.

The most technical part of the proof (in the next section) is that of the Splitting Theorem for image, the union of disjoint graphs or structures. This theorem says that, for every MS formula φ, the set of assignments that satisfy φ in the union of two disjoint structures S and T can be determined from the sets of those that satisfy auxiliary formulas ...

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