Resource dependent undecidability: computability landscape of distinct Turing theories

TL;DR

This paper explores how the decidability of problems varies across different Turing theories, showing that resource availability can change the solvability of certain problems like the Halting problem, with implications for classical and quantum computation.

Contribution

It introduces a novel logical framework to analyze resource-dependent decidability across distinct Turing theories, including classical and quantum, revealing paradoxes and new computational insights.

Findings

Decidability of certain problems depends on available computational resources.

Quantum resources can enable solutions impossible with classical resources.

The Halting problem remains unsolvable across all theories, despite resource differences.

Abstract

Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories. The apparent paradoxical situation gets resolved once it is perceived that the reducibility of Halting problem changes with varying resources available for computations in different theories. In the end, we propose a multi-agent game where winnability of the player having access to only classical resources is undecidable while quantum resources provide a perfect winning strategy.