A note on dimensions of polynomial size circuits

TL;DR

This paper applies resource-bounded dimension theory to analyze the complexity of polynomial size circuits, revealing new dimension properties that improve upon previous measure and dimension results.

Contribution

It introduces novel dimension results for polynomial size circuits, specifically for $ ext{P/poly}$ and its subclasses, advancing the understanding of their structural complexity.

Findings

$ ext{P/poly}$ has $i$th order scaled $ ext{P}_3$-strong dimension 0 for all $i",

$ ext{P/poly}^ ext{io}$ has $ ext{P}_3$-dimension 1/2 and strong dimension 1

Abstract

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension 0. We also show that $\Ppoly^\io$ has $\pthree$-dimension 1/2, $\pthree$-strong dimension 1. Our results improve previous measure results of Lutz (1992) and dimension results of Hitchcock and Vinodchandran (2004).