Algebraic Proof Complexity: Progress, Frontiers and Challenges

TL;DR

This survey explores recent advances in algebraic proof complexity, highlighting its links to algebraic circuit complexity and derandomization, and discusses new methods, open problems, and research directions.

Contribution

It provides a comprehensive overview of the interplay between proof complexity, algebraic circuits, and derandomization, emphasizing recent progress and future challenges.

Findings

Connections between proof complexity lower bounds and algebraic circuit lower bounds.

Insights into the Polynomial Identity Testing problem from a proof complexity perspective.

Identification of open problems and new research directions in algebraic proof complexity.

Abstract

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight connections between proof complexity lower bounds (namely, lower bounds on the size of proofs of certain tautologies), algebraic circuit lower bounds, and the Polynomial Identity Testing problem from derandomization theory.