Tameness in least fixed-point logic and McColm's conjecture

TL;DR

This paper explores model-theoretic tameness properties in least fixed-point logic over finite structures, showing their dependence on the limit theory and confirming McColm's conjecture in tame cases.

Contribution

It characterizes the relationships among tameness properties in least fixed-point logic and proves McColm's conjecture holds for tame limit theories.

Findings

Tameness properties depend only on the elementary limit theory.

Order property and independence property are equivalent in this setting.

McColm's conjecture is true for families with tame limit theories.

Abstract

We investigate four model-theoretic tameness properties in the context of least fixed-point logic over a family of finite structures. We find that each of these properties depends only on the elementary (i.e., first-order) limit theory, and we completely determine the valid entailments among them. In contrast to the context of first-order logic on arbitrary structures, the order property and independence property are equivalent in this setting. McColm conjectured that least fixed-point definability collapses to first-order definability exactly when proficiency fails. McColm's conjecture is known to be false in general. However, we show that McColm's conjecture is true for any family of finite structures whose limit theory is model-theoretically tame.