# Capturing PTime

## Fixed-point logic with counting ★★

Author(s): Blass

\begin{question} Can either of the following be expressed in fixed-point logic plus counting: \begin{enumerate} \item Given a graph, does it have a perfect matching, i.e., a set $M$ of edges such that every vertex is incident to exactly one edge from $M$? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant? \end{enumerate} \end{question}

Keywords: Capturing PTime; counting quantifiers; Fixed-point logic; FMT03-Bedlewo