Proof Complexity and Feasible Interpolation

Abstract

This is a survey on propositional proof complexity aimed at introducing the basics of the field with a particular focus on a method known as feasible interpolation. This method is used to construct "hard theorems" for several proof systems for both classical and non-classical logics. Here, a "hard theorem" refers to a theorem in the logic whose shortest proofs are super-polynomially long in the length of the theorem itself. To make this survey more accessible, we only assume a basic familiarity with propositional, modal, and first-order logic, as well as a basic understanding of the key concepts in computational complexity, such as the definitions of the classes $\mathbf{NP}$ and $\mathbf{PSPACE}$. Any additional concepts will be introduced and explained as needed.

This is a chapter of the book "Theory and Applications of Craig Interpolation", edited by Balder ten Cate, Jean Christoph Jung, Patrick Koopmann, Christoph Wernhard and Frank Wolter