🤖 AI Summary
This work addresses foundational questions in circuit complexity by reconceptualizing its theoretical framework from an information-theoretic perspective: Boolean circuits are treated as compact encodings of truth tables, rather than conventional computational models. We introduce “circuit description complexity”—a novel paradigm inspired by Kolmogorov complexity—that unifies the characterization of circuit size and the intrinsic information content of truth tables. Our framework rigorously reproduces several classical circuit lower bounds (e.g., for AC⁰ and monotone circuits), provides the first information-theoretic explanation for why most Boolean functions admit a unique optimal circuit structure, and reveals structural properties of optimal circuits—including hierarchical organization and sparsity. Methodologically, we integrate tools from descriptive complexity, information theory, and circuit lower-bound techniques. By reframing computation as information compression rather than functional realization, our approach advances the fundamental understanding of computational resources.
📝 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.