# cardinality predicates

## Monadic second-order logic with cardinality predicates ★★

Author(s): Courcelle

The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas: \begin{itemize} \item $\text{``}\,\mathrm{Card}(X) = \mathrm{Card}(Y)\,\text{''}$ \item $\text{``}\,\mathrm{Card}(X) \text{ belongs to } A\,\text{''}$ \end{itemize} where $A$ is a fixed recursive set of integers.

Let us fix $k$ and a closed formula $F$ in this language.

\begin{conjecture} Is it true that the validity of $F$ for a graph $G$ of tree-width at most $k$ can be tested in polynomial time in the size of $G$? \end{conjecture}

Keywords: bounded tree width; cardinality predicates; FMT03-Bedlewo; MSO