Abstract

This paper studies expressivity bounds for extensions of first-order logic with counting and unary quantifiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that first-order logic with counting quantifiers captures uniform TC 0 over ordered structures. Thus, proving expressivity bounds for first-order with counting can be seen as an attempt to show TC 0 $ DLOG using techniques of descriptive complexity. Second, the presence of auxiliary built-in relations (e.g., order, successor) is known to make a big impact on expressivity results in finite-model theory and database theory (where logics with counting and unary quantifiers have recently been used to model query languages with aggregation). For those logics, our goal is to extend techniques from "pure" setting to that of auxiliary relations. Until now, all known results on the limitations of expressive power of the counting and unary quantifier extensions of first-order...

Citations