On the Incompressibility of Truth With Application to Circuit Complexity

TL;DR

This paper introduces an information-theoretic framework for understanding circuit complexity, providing new insights into the structure of optimal circuits, exponential lower bounds, and unifying existing results through a fresh perspective.

Contribution

It presents a novel approach linking circuit complexity with Kolmogorov Complexity, re-proves bounds, and offers new structural insights into boolean functions.

Findings

Re-proved existing circuit complexity bounds

Identified structural features of optimal circuits

Provided an explicit boolean function family requiring exponential circuits

Abstract

We revisit the fundamentals of Circuit Complexity and the nature of efficient computation from a fresh perspective. We present a framework for understanding Circuit Complexity through the lens of Information Theory with analogies to results in Kolmogorov Complexity, viewing circuits as descriptions of truth tables, encoded in logical gates and wires, rather than purely computational devices. From this framework, we re-prove some existing Circuit Complexity bounds, explain what the optimal circuits for most boolean functions look like structurally, give an explicit boolean function family that requires exponential circuits, and explain the aforementioned results in a unifying intuition that re-frames time entirely.