Title: Tameness in least fixed-point logic and McColm's conjecture,
Abstract: We investigate fundamental model-theoretic dividing lines (the order
property, the independence property, the strict order property, and the tree
property 2) in the context of least fixed-point (LFP) logic over families of
finite structures. We show that, unlike the first-order (FO) case, the order
property and the independence property are equivalent, but all of the other
natural implications are strict. We identify the LFP strict order property with
proficiency, a well-studied notion in finite model theory.
Gregory McColm conjectured that FO and LFP definability coincide over a
family C of finite structures exactly when C is non-proficient. McColm's
conjecture is false in general, but as an application of our results, we show
that it holds under standard FO tameness assumptions adapted to families of
finite structures.