AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

A formal analysis of Correspondence Theory

This paper provides a computational analysis of the complexity of GEN and Correspondence Theory in terms of the nature of the logic involved in their formulation. The first result of this analysis shows that the GEN function is not definable in Monadic Second Order (MSO) logic. Second, we show that the set of input-output Correspondence-theoretic candidates from a given underlying representation is definable in First Order (FO) logic, which is less complex than MSO-logic. Third, we present some case studies where the correct input-output Correspondence-theoretic candidate from a given underlying representation can be accomplished with FO-definable, language-specific, inviolable constraints without recourse to optimization.