Abstract
A countable graph can be considered as the value of a certain infinite expression, represented itself by an infinite tree. We establish that the set of finite or infinite (expression) trees constructed with a finite number of operators, the value of which is a graph satisfying a property expressed in monadic second-order logic, is itself definable in monadic second-order logic. From Rabin's theorem, the emptiness of this set of (expression) trees is decidable. It follows that the monadic second-order theory of an equational graph, or of the set of countable graphs of width less than an integerm, is decidable.
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This work has been supported by the “Programme de Recherches Coordonnées: Mathématiques et Informatique.” Reprints can be requested by electronic mail at mcvax!inria!geocub!courcell (on UUCP network) or courcell@geocub.greco-prog.fr.
Unité de Recherche associée au C.N.R.S. no. 726.
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Courcelle, B. The monadic second-order logic of graphs, II: Infinite graphs of bounded width. Math. Systems Theory 21, 187–221 (1988). https://doi.org/10.1007/BF02088013
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DOI: https://doi.org/10.1007/BF02088013