Computation with Large Advice

TL;DR

This paper explores the computational power of constant space Turing machines with large advice, revealing they can solve problems beyond traditional classes, and introduces the concept of algorithms with large advice.

Contribution

It establishes the equivalence of constant space computation with large advice to known complexity classes and proposes the idea of algorithms with large advice.

Findings

Constant space Turing machines with polynomial advice include nonuniform-{ f}^1.

Quasipolynomial advice equals nonuniform-{ } polyL.

Large advice significantly enhances computational power.

Abstract

In this paper, we consider a new direction of computation, which we call computation with large advice. We mainly consider constant space computation with large advice in Turing machines, and prove the following facts: (i) The class of decision problems solvable by a constant space Turing machine with polynomial-size advice includes nonuniform-{\sf NC}$^1$, (ii) The class of decision problems solvable by a constant space Turing machine with quasipolynomial-size advice equals nonuniform-{\sf polyL}. The facts mean constant space computation with large advice has unexpected computational power. On the other hand, we mention bounded time computation with large advice, and attempt to propose a concept of ``algorithms with large advice''. In the proposal, advice is precomputed data for a problem and a fixed instance size, and we expect efficient algorithms by large or huge advice.