A Verifier Hierarchy

TL;DR

This paper establishes a hierarchy relating certificate length and verifier runtime, providing a fundamental trade-off theorem and applying it to complexity class separations and natural problems.

Contribution

It introduces a Verifier Trade-off Theorem linking verification time reduction to certificate length, creating a hierarchy based on certificate complexity.

Findings

Proves a lower bound on certificate length for faster verification.

Applies the hierarchy to analyze class separations like NP vs. EXPTIME.

Provides insights into the P vs. NP problem through certificate size perspectives.

Abstract

We investigate the trade-off between certificate length and verifier runtime. We prove a Verifier Trade-off Theorem showing that reducing the inherent verification time of a language from \(f(n)\) to \(g(n)\), where \(f(n) \ge g(n)\), requires certificates of length at least \(\Omega(\log(f(n) / g(n)))\). This theorem induces a natural hierarchy based on certificate complexity. We demonstrate its applicability to analyzing conjectured separations between complexity classes (e.g., \(\np\) and \(\exptime\)) and to studying natural problems such as string periodicity and rotation detection. Additionally, we provide perspectives on the \(\p\) vs. \(\np\) problem by relating it to the existence of sub-linear certificates.