Unary Quantiiers, Transitive Closure, and Relations of Large Degree

This paper studies expressivity bounds for extensions of rst-order logic with counting and unary quantiiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that rst-order logic with counting quantiiers captures uniform TC 0 over ordered structures. Thus, proving expressivity bounds for rst-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 nite-model theory and database theory (where logics with counting and unary quantiiers 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-quantiier extensions of rst-order logic dealt with auxiliary relations of \small" degree. For example , it is known that these logics fail to express some DLOG-queries in the presence of a sucessor relation. Our main result is that these extensions cannot deene the deterministic transitive closure (a DLOG-complete problem) in the presence of auxiliary relations of \large" degree, in particular, those which are \almost linear orders." They are obtained from linear orders by replacing them by \very thin" preorders on arbitrarily small number of elements. We also show that the technique of the proof (in a precise sense) cannot be extended to provide the proof of separation of TC 0 from DLOG. We also discuss a general impact of having built-in (pre)orders, and give some expressivity statements in the pure setting that would imply separation results for the ordered case. We also brieey discuss database applications.