First-Order logic and its Infinitary Quantifier Extensions over Countable Words
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.
Cite
@article{arxiv.2107.01468,
title = {First-Order logic and its Infinitary Quantifier Extensions over Countable Words},
author = {Bharat Adsul and Saptarshi Sarkar and A. V. Sreejith},
journal= {arXiv preprint arXiv:2107.01468},
year = {2021}
}