Identifying an Honest ${\rm EXP}^{\rm NP}$ Oracle Among Many

TL;DR

This paper introduces a framework for removing short advice in probabilistic computations using a concept called selectors, and demonstrates their existence for ${ m EXP}^{ m NP}$-complete languages, expanding understanding beyond instance checkers.

Contribution

The paper defines selectors as a new tool to remove short advice in probabilistic computation and proves their existence for ${ m EXP}^{ m NP}$-complete languages, surpassing the limitations of instance checkers.

Findings

Selectors can remove short advice for ${ m EXP}^{ m NP}$-complete languages.

Existence of selectors is proven using techniques related to ${ m MIP} = { m NEXP}$.

Selectors are a weaker property than instance checkers, with broader applicability.

Abstract

We provide a general framework to remove short advice by formulating the following computational task for a function $f$: given two oracles at least one of which is honest (i.e. correctly computes $f$ on all inputs) as well as an input, the task is to compute $f$ on the input with the help of the oracles by a probabilistic polynomial-time machine, which we shall call a selector. We characterize the languages for which short advice can be removed by the notion of selector: a paddable language has a selector if and only if short advice of a probabilistic machine that accepts the language can be removed under any relativized world. Previously, instance checkers have served as a useful tool to remove short advice of probabilistic computation. We indicate that existence of instance checkers is a property stronger than that of removing short advice: although no instance checker for ${\rm EXP}^{\rm NP}$-complete languages exists unless ${\rm EXP}^{\rm NP} = {\rm NEXP}$, we prove that there exists a selector for any ${\rm EXP}^{\rm NP}$-complete language, by building on the proof of ${\rm MIP} = {\rm NEXP}$ by Babai, Fortnow, and Lund (1991).