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

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.