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

Bharat Adsul, Saptarshi Sarkar, A.V. Sreejith
2021 arXiv   pre-print
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.
more » ... 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.
arXiv:2107.01468v1 fatcat:k5zie3jeibgkrfapfgqzdoazhe