一阶逻辑及其可数词上的无穷量词扩张
First-Order logic and its Infinitary Quantifier Extensions over Countable Words
摘要：
我们有助于语言逻辑代数相互作用的背景下，可数字的一阶属性的精细理解。通过加强Simon关于分段可测试语言的定理，我们建立了FO的单变量片段的可判定代数刻画以及FO的存在片段的布尔闭包.我们提出了一个新的扩展FO承认无穷量词的原因，固有的无穷性质的可数字。我们提供了一个非常自然的和分层的块产品为基础的表征新的扩展。我们还阐述了它的作用，鉴于其他自然和经典的逻辑系统，如WMSO和FO[切] -FO的扩展，其中量化超过戴德金削减是允许的。我们还排除了有限的基础上，这些逻辑系统的块产品为基础的表征的可能性。最后，我们报告简单，但新颖的代数表征的一个变量片段的层次结构的新建议的扩展FO。
英文摘要：
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.
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