Complexity Classes and Theories for the Comparator Circuit Value Problem

Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem. He proved that several other problems are complete for CC, including the stable marriage problem, and finding the lexicographical first maximal matching in a bipartite graph. We suggest alternative definitions of CC based on different reducibilities and introduce a twosorted theory VCC based on one of them. We sharpen and simplify Subramanian’s completeness proofs for the above two problems and show how to formalize them in VCC.