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

Siddharth Bhaskar, Alex Kruckman

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 limit theory, and we completely determine the valid entailments among them. Unlike first-order logic, 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 turns out 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.

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