This article fits in the area of research that investigates the application of topological duality methods to problems that appear in theoretical computer science. One of the eventual goals of this approach is to derive results in computational complexity theory by studying appropriate topological objects which characterize them. The link which relates these two seemingly separated fields is logic, more precisely a subdomain of finite model theory known as logic on words. It allows for a description of complexity classes as certain families of languages, possibly non-regular, on a finite alphabet. Very few is known about the duality theory relative to fragments of first-order logic on words which lie outside of the scope of regular languages. The contribution of our work is a detailed study of such a fragment. Fixing an integer $k \geq 1$, we consider the Boolean algebra $\mathcal{B}\Sigma_1[\mathcal{N}^{u}_k]$. It corresponds to the fragment of logic on words consisting in Boolean combinations of sentences defined by using a block of at most $k$ existential quantifiers, letter predicates and uniform numerical predicates of arity $l \in \{1,...,k\}$. We give a detailed study of the dual space of this Boolean algebra, for any $k \geq 1$, and provide several characterizations of its points. In the particular case where $k=1$, we are able to construct a family of ultrafilter equations which characterize the Boolean algebra $\mathcal{B} \Sigma_1[\mathcal{N}^{u}_1]$. We use topological methods in order to prove that these equations are sound and complete with respect to the Boolean algebra we mentioned.
翻译:此文章适用于研究领域, 研究对理论计算机科学中出现的问题应用上层双重性方法 。 此方法的最终目的之一是通过研究适当的表层对象来得出计算复杂性理论的结果。 与这两个表面上似乎分离的字段相关的链接是逻辑, 更准确地说, 是一个称为文字逻辑的有限模型理论的子域。 它允许描述复杂等级, 可能是非常规的, 使用一个限定的字母。 很少有人知道与位于常规语言范围之外的单词上的一阶逻辑碎片[一阶逻辑的碎片] 的双重性理论。 我们的工作贡献是对这种碎片进行详细研究。 固定整数 $k\ geq 1, 我们考虑Bole argebra $\ $\ 的亚值值值值值值值 。 在布尔里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里加里。