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Halting Problem of Turing Machine:

The Halting Problem is a fundamental problem in computer science, first introduced by Alan Turing in 1936. It asks whether it is possible to determine, in general, whether a given Turing Machine will eventually halt (terminate) when given a particular input or whether it will run indefinitely.

Statement of the Problem: Given a description of a Turing Machine

and an input , determine whether halts on input , i.e., whether it eventually enters a halting state and stops, or whether it runs indefinitely.


  1. Undecidability: Turing proved that the Halting Problem is undecidable, meaning that there is no algorithm that can correctly determine whether any arbitrary Turing Machine halts on any arbitrary input.
  2. Limits of Computation: The undecidability of the Halting Problem demonstrates the existence of limitations on what algorithms can achieve, highlighting the incompleteness of formal systems and the inherent complexity of computation.


  1. Provability and Computability: The undecidability of the Halting Problem has profound implications for mathematical logic and computability theory, contributing to the understanding of the limits of what can be proven and computed.
  2. Algorithmic Complexity: The Halting Problem serves as a foundational concept in complexity theory, helping to classify problems based on their computational difficulty and providing insights into the nature of algorithms.

Church-Turing Thesis:

The Church-Turing Thesis is a hypothesis in computer science and mathematical logic, proposed independently by Alonzo Church and Alan Turing in the 1930s. It asserts that any function that can be effectively computed by an algorithmic process can be computed by a Turing Machine, and vice versa.

Key Points:

  1. Computational Equivalence: The Church-Turing Thesis suggests that Turing Machines capture the intuitive notion of computability and that any algorithmic process can be formalized and executed by a Turing Machine.
  2. Formalization of Computation: The thesis provides a formal framework for understanding the notion of effective computability and serves as a theoretical foundation for the study of computability and complexity.
  3. Limitations: While the Church-Turing Thesis is widely accepted as a guiding principle in computer science, it is not a provable mathematical theorem. Instead, it is based on empirical evidence and the absence of counterexamples.


  1. Universal Computation: The Church-Turing Thesis implies that Turing Machines are capable of simulating any algorithmic process, providing a universal model of computation.
  2. Algorithmic Limits: The thesis helps to delineate the boundaries of computability, highlighting the existence of problems that are inherently unsolvable or undecidable.
  3. Computational Complexity: By equating different computational models, the Church-Turing Thesis provides insights into the complexity of algorithms and the classification of problems based on their computational difficulty.