Extensional Uniformity for Boolean Circuits

TL;DR

This paper explores the relationship between extensional and intensional uniformity in Boolean circuit complexity, introducing the concept of the 'Uniformity Duality Property' and analyzing when it holds or fails.

Contribution

It defines the 'Uniformity Duality Property' for circuit classes and provides examples of classes that satisfy or do not satisfy this property.

Findings

Positive instances of the Duality Property identified.

Negative instances of the Duality Property demonstrated.

Insights into the nature of uniformity in circuit complexity.

Abstract

Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class 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 C. We say that (C,L) has the "Uniformity Duality Property" if the extensionally uniform class C \cap L can be captured intensionally by means of adding so-called "L-numerical predicates" to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining C. This paper exhibits positive instances and negative instances of the Uniformity Duality Property.