A generalization of Spira's theorem and circuits with small segregators or separators

Abstract

Spira showed that any Boolean formula of size s can be simulated in depth O(log⁡s). We generalize Spira's theorem and show that any Boolean circuit of size s with segregators (or separators) of size f(s) can be simulated in depth O(f(s)log⁡s). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O(k2log⁡n) by Jansen and Sarma. Our results imply that the class of languages computed by non-uniform families of polynomial-size circuits with constant size segregators equals non-uniform NC1.As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators is in deterministic SPACE(log2⁡n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete, is in SPACE(nlog⁡n); and that the Layered Circuit Value and Synchronous Circuit Value problems, which are both P-complete, are in SPACE(n).