On the Incompressibility of Truth With Application to Circuit Complexity
Luke Tonon
TL;DR
The work reframes circuit complexity through an information-theoretic lens, treating circuits as compressed descriptions of truth tables and connecting this view to Kolmogorov complexity. It re-derives classical bounds (notably Shannon's) and clarifies the typical structure of optimal circuits as near-lookup-table implementations, especially for incompressible functions. The paper also analyzes MCSP under this framework, presenting it as a meta-description problem and arguing for intrinsic hardness in the face of incompressibility, including an explicit exponential lower bound via Chaitin's Omega. Overall, the approach offers a unifying intuition that time is secondary to description length and suggests fundamental limits on proving lower bounds and on constructing universal compressors for truth tables.
Abstract
We revisit the fundamentals of Circuit Complexity and the nature of efficient computation from a new 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 strong Circuit Complexity bounds, explain what the optimal circuits for most Boolean functions look like structurally, give insight into new circuit bounds, and explain the aforementioned results in a unifying intuition that re-frames time entirely.