Extensional Uniformity for Boolean Circuits

Abstract

Imposing an extensional uniformity condition on a nonuniform circuit complexity class $\mathcal{C}$ means simply intersecting $\mathcal{C}$ with a uniform class $\mathcal{L}$. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining $\mathcal{C}$. We say that $(\mathcal{C},\mathcal{L})$ has the uniformity duality property if the extensionally uniform class $\mathcal{C}\cap\mathcal{L}$ can be captured intensionally by means of adding so-called $\mathcal{L}$-numerical predicates to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining $\mathcal{C}$. This paper exhibits positive instances and negative instances of the uniformity duality property.