[Paper Review] A note on dimensions of polynomial size circuits

Xiaoyang Gu|Apr 22, 2004

Computability, Logic, AI Algorithms18 references3 citations

TL;DR

This paper uses resource-bounded dimension theory to refine the understanding of polynomial-size circuits, showing that P/poly has 0th-order scaled p3-strong dimension 0 and P/polyi.o. has p3-dimension 1/2 and p3-strong dimension 1. It improves prior measure and dimension results by Lutz and Hitchcock-Vinodchandran through a novel application of scaled dimension and a Measure Conservation Theorem argument.

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).

Motivation & Objective

• To refine the understanding of polynomial-size circuits using resource-bounded dimension theory.
• To improve upon previous measure and dimension results for P/poly and P/polyi.o. by Lutz (1992) and Hitchcock-Vinodchandran (2004).
• To establish tighter bounds on the scaled dimension of P/poly and its infinitely often variant.
• To demonstrate that the p3-dimension of P/polyi.o. is exactly 1/2 and its p3-strong dimension is exactly 1.
• To show that for all i ≥ 0, the i-th order scaled p3-dimension of P/poly is 0, improving on prior results.

Proposed method

• Applies resource-bounded dimension and strong dimension theory to analyze complexity classes within E3 and ESPACE.
• Uses the equivalence between KT complexity and circuit-size complexity to derive precise dimension bounds.
• Employs a modified supermartingale construction (d′) based on a base supermartingale d and a language L that excludes string lengths that are powers of two.
• Applies the Measure Conservation Theorem (Lutz, 1992) to derive lower bounds on dimension by contradiction.
• Introduces a recursive supermartingale d′ that tracks behavior on strings where lengths are not powers of two, preserving success sets under transformation.
• Uses the structure of infinite-length prefixes and the intersection-closure of language classes to prove dimension lower bounds.

Experimental results

Research questions

• RQ1 What is the exact p3-dimension of P/polyi.o.?
• RQ2 What is the exact p3-strong dimension of P/polyi.o.?
• RQ3 How does the i-th order scaled p3-dimension of P/poly behave for i ≥ 0?
• RQ4 Can the dimension of P/polyi.o. be bounded below by 1/2 using resource-bounded dimension theory?
• RQ5 Is the p3-dimension of P/poly strictly 0 for all orders i ≥ 0?

Key findings

• The p3-dimension of P/polyi.o. is exactly 1/2, improving upon the prior bound of 0 from Hitchcock and Vinodchandran.
• The p3-strong dimension of P/polyi.o. is exactly 1, which is a significant improvement over the prior result of 0.
• For all i ≥ 0, the i-th order scaled p3-dimension of P/poly is 0, and the i-th order scaled p3-strong dimension is also 0.
• The p3-dimension of P/poly is 0, and the p3-strong dimension of P/poly is 1, confirming that the class is dimensionally small but highly structured.
• The lower bound of 1/2 for the p3-dimension of P/polyi.o. is tight, as confirmed by a general theorem on infinitely often classes.
• The results hold for all c > 0, showing that dimp3(SIZEi.o.(nc)) = 1/2 and Dimp3(SIZE(nc)) = 0, extending to the full P/poly class.