On Bounded Advice Classes

TL;DR

This paper introduces bounded advice classes where a powerful machine generates advice for a weaker one, connecting these classes to unary languages and analyzing their relationships with quantum, randomized, and other complexity classes.

Contribution

It develops the concept of bounded advice classes, linking them to unary languages and analyzing their relationships with various complexity classes, including quantum and randomized classes.

Findings

Bounded advice classes relate to unary languages.

Advice from EXP, NP, BQP, PSPACE can be useful under certain conditions.

The study improves understanding of advice functions in computational complexity.

Abstract

Advice classes in computational complexity have frequently been used to model real-world scenarios encountered in cryptography, quantum computing and machine learning, where some computational task may be broken down into a preprocessing and deployment phase, each associated with a different complexity. However, in these scenarios, the advice given by the preprocessing phase must still be generated by some (albeit more powerful) bounded machine, which is not the case in conventional advice classes. To better model these cases we develop `bounded advice classes', where a more powerful Turing machine generates advice for another, less powerful, Turing machine. We then focus on the question of when various classes generate useful advice, to answer this we connect bounded advice to unary languages. This connection allows us to state various conditional and unconditional results on the utility of advice generated by $\mathsf{EXP}$, $\mathsf{NP}$, $\mathsf{BQP}$, $\mathsf{PSPACE}$, and more. We study the relations between bounded advice classes, quantum bounded advice classes, and randomised bounded advice. We also examine how each of these concepts interact with recently introduced classes, like $\mathsf{BPP/samp}$. Our results also improve the state of the art in existing research on the complexity of advice functions.