Unary Quantifiers, Transitive Closure, and Relations of Large Degree

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° over ordered structures. Thus, proving expressivity bounds for first-order with counting can be seen as an attempt to show TC° ⊂ ≠ 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. Our goal is to extend techniques from “pure” setting to that of auxiliary relations.