Duality for the existential fragment of first-order logic on words with numerical predicates of a fixed arity

Abstract: 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.