Computation with Advice

TL;DR

This paper introduces a generalized framework for computation with advice, including random advice, explores its connection to Weihrauch reducibility, and discusses complexity and examples illustrating the interplay of uniform and non-uniform methods.

Contribution

It proposes a novel generalization of advice-based computation, including random advice, and links it to Weihrauch reducibility and hypercomputation complexity.

Findings

Computability with random advice corresponds to solutions guessable with positive probability.

The framework allows defining computational complexity for hypercomputational concepts.

Examples demonstrate the interaction of uniform and non-uniform techniques in advice computation.

Abstract

Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solutions being guessable with positive probability. In the framework of computation with advice, it is possible to define computational complexity for certain concepts of hypercomputation. Finally, some examples are given which illuminate the interplay of uniform and non-uniform techniques in order to investigate both computability with advice and the Weihrauch lattice.