First-Order logic and its Infinitary Quantifier Extensions over Countable Words

TL;DR

This paper explores the logical and algebraic properties of countable words using first-order logic extensions with infinitary quantifiers, providing decidable characterizations and hierarchical structures.

Contribution

It introduces a new infinitary extension of FO for countable words, with algebraic characterizations and connections to classical logics like WMSO and FO[cut].

Findings

Decidable algebraic characterizations of one-variable FO fragments.

Hierarchical block-product characterization of the new infinitary extension.

No finite basis exists for a block-product characterization of these logical systems.

Abstract

We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as boolean closure of existential fragment of FO via a strengthening of Simon's theorem about piecewise testable languages. We propose a new extension of FO which admits infinitary quantifiers to reason about the inherent infinitary properties of countable words. We provide a very natural and hierarchical block-product based characterization of the new extension. We also explicate its role in view of other natural and classical logical systems such as WMSO and FO[cut] - an extension of FO where quantification over Dedekind-cuts is allowed. We also rule out the possibility of a finite basis for a block-product based characterization of these logical systems. Finally, we report simple but novel algebraic characterizations of one variable fragments of the hierarchies of the new proposed extension of FO.