Utilizing the connection between uniform constantdepth circuits and first-order logic with numerical predicates, we are able to provide a purely logical characterization of uniformity based on the intrinsic properties of these predicates. By requiring a numerical predicate R to satisfy a natural extensibility condition — that it can be translated to a polynomially magnified domain based on tupl...

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC0 appear as classes of this hierarchy. In this way, we unconditionally place #AC0 properly in a strict ...

Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting fi...

During the past decade, dependence logic has emerged as a formalism suitable for expressing and analyzing notions of dependence and independence that arise in different scientific areas. The sentences of dependence logic have the same expressive power as those of existential second-order logic, hence dependence logic captures NP on the class of all finite structures. In this paper, we identify ...

This paper introduces a new Ehrenfeucht-Fraı̈ssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai-Fagin game to the case when there are several alternating (coloring) moves played in different models. The game allows Duplicator to delay her choices of the models till (practically) the very end of the game, making it ...