Separating LREC from LFP

LREC = is an extension of first-order logic with a logarithmic recursion operator. It was introduced by Grohe et al. and shown to capture the complexity class L over trees and interval graphs. It does not capture L in general as it is contained in FPC-fixed-point logic with counting. We show that this containment is strict. In particular, we show that the path systems problem, a classic P-complete problem which is definable in LFP-fixed-point logic-is not definable in LREC = . This shows that the logarithmic recursion mechanism is provably weaker than general least fixed points. The proof is based on a novel Spoiler-Duplicator game tailored for this logic.