The Halting Problem in Complexity Theory: A Review

Abstract

The Halting Problem is a pivotal concept in computational theory, questioning whether a general algorithm can determine if a program halts or runs indefinitely on a given input. While foundational works have established its undecidability, a critical comparative evaluation of how the problem has evolved across theoretical, educational, quantum, and dynamical domains remains lacking in the literature. This gap limits comprehensive understanding and cross-domain applicability for researchers. Therefore, this paper aims to address that void by analyzing and synthesizing five influential contributions: Turing’s 1937 foundational proof, Pavlotskaya’s 2002 mathematical formalization, Sipser’s 2013 pedagogical perspective, Aaronson’s 2016 exploration into quantum undecidability, and Cotler and Rezchikov’s 2024 application in computational dynamical systems. Through structured analysis of their methodologies, assumptions, and implications, this paper highlights the strengths, limitations, and evolving interpretations of the Halting Problem. Results confirm that its undecidability has far-reaching effects on algorithm design, software verification, and emerging areas like quantum computing and chaotic systems. By identifying key gaps such as lack of empirical validation and practical tools, this paper proposes future directions including approximation algorithms, interdisciplinary modeling, and AI safety applications. This study thus offers a consolidated foundation for researchers seeking to explore the Halting Problem’s role across both classical theory and modern computational frontiers.