Context-Dependent Functions: Narrowing the Realm of Turing's Halting Problem

TL;DR

This paper introduces context-dependent functions (CDFs) that vary behavior based on program context, challenging traditional views of Turing's Halting Problem and suggesting solutions may be CDFs rather than impossible.

Contribution

It re-examines the undecidability of the Halting Problem by proposing CDFs as potential solutions, thus narrowing the problem's scope.

Findings

CDFs exist and can solve certain instances of the Halting Problem

The classic proof of undecidability does not exclude CDFs as solutions

The proof's conclusion is overly broad, allowing for CDF-based solutions

Abstract

This paper describes Turing's Halting Problem (HP), and reviews the classic proof that no function exists that can solve HP. The concept of a "Context-Dependent Function" (CDF), whose behavior varies based on seemingly irrelevant changes to a program calling that function, is introduced, and the proof of HP's undecidability is re-examined in light of CDFs. The existence of CDFs is established via a pair of examples of such functions. The conclusion of the proof of HP's undecidability is thus shown to be overly strong, as it doesn't show that no solution to HP exists, but rather that a solution must be a CDF. A higher-level analysis of this work is given, followed by conclusions and comments on future work.