Probabilistic verification of all languages

TL;DR

This paper introduces new space-efficient protocols for verifying all languages, including interactive proof systems with minimal space and time, expanding the scope of language verification in computational complexity.

Contribution

It presents three novel protocols for language verification with minimal space, including constant-space and multi-prover systems, applicable to all languages.

Findings

Existence of log-space interactive proof systems for unary languages

Constant-space weak-IPS for all languages

Uncountably many languages verifiable in constant space

Abstract

We present three protocols for verifying all languages: (i) For any unary (binary) language, there is a log-space (linear-space) interactive proof system (IPS); (ii) for any language, there is a constant-space weak-IPS (the non-members may not be rejected with high probability); and, (iii) for any language, there is a constant-space IPS with two provers where the verifier reads the input once. Additionally, we show that uncountably many binary (unary) languages can be verified in constant space and in linear (quadratic) expected time.