Monadic Second Order Finite Satisfiability and Unbounded Tree-Width

The finite satisfiability problem of monadic second order logic is decidable only on classes of structures of bounded tree-width by the classic result of Seese [24]. We prove that the following problem is decidable: Input: (i) A monadic second order logic sentence α, and (ii) a sentence β in the two-variable fragment of first order logic extended with counting quantifiers. The vocabularies of α and β may intersect. Output: Is there a finite structure which satisfies α ∧ β such that the restriction of the structure to the vocabulary of α has bounded tree-width? (The tree-width of the desired structure is not bounded.) As a consequence, we prove the decidability of the satisfiability problem by a finite structure of bounded tree-width of a logic MSO∃card extending monadic second order logic with linear cardinality constraints of the form |X1|+ · · ·+ |Xr| < |Y1| + · · · + |Ys| on the variables Xi, Yj of the outer-most quantifier block. We prove the decidability of a similar extension of WS1S.