Proofs, Circuits, and Communication

TL;DR

This paper surveys the interconnections between proof complexity, circuit complexity, and communication complexity, emphasizing the role of total search problems as a unifying framework for understanding lower bounds in computational complexity.

Contribution

It introduces the perspective of using total search problems (TFNP) as a unifying language to connect various complexity lower-bound results and suggests directions for future research.

Findings

Highlights the interconnections between different complexity theories.

Proposes TFNP as a unifying framework for complexity lower bounds.

Suggests a research program based on total search problems.

Abstract

We survey lower-bound results in complexity theory that have been obtained via newfound interconnections between propositional proof complexity, boolean circuit complexity, and query/communication complexity. We advocate for the theory of total search problems (TFNP) as a unifying language for these connections and discuss how this perspective suggests a whole programme for further research.