Low-level dichotomy for quantified constraint satisfaction problems

Abstract

Building on a result of Larose and Tesson for constraint satisfaction problems ( CSPs), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP ( B ) , where B is a finite structure that is a core. Specifically, such problems are either in A L o g t i m e or are L -hard. This involves demonstrating that if CSP ( B ) is first-order expressible, and B is a core, then QCSP ( B ) is in A L o g t i m e . We show that the class of B such that CSP ( B ) is first-order expressible (indeed trivial) is a microcosm for all QCSPs. Specifically, for any B there exists a C — generally not a core — such that CSP ( C ) is trivial, yet QCSP ( B ) and QCSP ( C ) are equivalent under logspace reductions.