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Introduction to Undecidability:

Undecidability refers to the concept that certain problems lack an algorithmic solution, meaning that no computer program or algorithm can determine the correct answer for all possible inputs within a finite amount of time. The notion of undecidability was first introduced by Alan Turing in the 1930s through his work on the Halting Problem.

Key Points:

  1. Limitation of Algorithms: Undecidable problems reveal inherent limitations in the capabilities of algorithms. They are problems for which no algorithm can be guaranteed to produce a correct answer for all instances.
  2. Theoretical Significance: The concept of undecidability has profound implications for theoretical computer science, logic, and mathematics. It provides insights into the fundamental limits of computation and formal systems.
  3. Relation to Turing Machines: Many undecidable problems are identified and studied within the framework of Turing Machines, which serve as a theoretical model of computation. Turing’s work on the Halting Problem demonstrated the existence of undecidable problems in the context of Turing Machines.

Undecidable Problems about Turing Machines:

Several fundamental problems related to Turing Machines have been proven to be undecidable. These problems play a crucial role in understanding the limitations of computation and the boundaries of formal systems.

1. The Halting 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).

2. The Emptiness Problem: Given a description of a Turing Machine , determine whether the language recognized by (the set of strings accepted by ) is empty.

3. The Post Correspondence Problem (PCP): Given a finite set of dominoes, each with two strings written on it, determine whether there exists a sequence of dominoes that can be chosen and arranged to form identical strings when concatenated.

4. The Diophantine Equation Problem: Given a polynomial equation with integer coefficients, determine whether there exist integer solutions to the equation.

5. The Word Problem for Groups: Given a finitely presented group and a word in the generators of the group, determine whether the word represents the identity element in the group.

6. The Entscheidungsproblem (Decision Problem): Introduced by David Hilbert, this problem asks whether there exists a general algorithm that can determine the truth or falsehood of any mathematical statement expressed in a formal language.


  • The existence of undecidable problems demonstrates the existence of limits in our ability to automate decision-making and solve certain types of problems algorithmically.
  • Undecidability has profound consequences for the foundations of mathematics, logic, and computer science, providing insights into the nature of computation, formal systems, and the concept of computability