What is the computational complexity of the halting problem?

Correct option is C

The  halting problem is a fundamental concept in computer science that asks whether a given program will eventually halt (terminate) or continue to run forever. Alan Turing proved that the halting problem is  undecidable, meaning there is no general algorithm to solve it for all possible program-input pairs. As a result, the computational complexity of the halting problem is  not computable. Therefore, the correct answer is (c) Not computable.
Important Key Points:
1. Halting Problem Definition:
· The halting problem determines whether a program will halt or run indefinitely for a given input.
· It is a classic example of an undecidable problem in computability theory.
2. Undecidability Proof:
· Alan Turing proved that no algorithm can solve the halting problem for all possible cases.
· The proof uses a contradiction approach, assuming such an algorithm exists and showing it leads to a logical inconsistency.
3. Computational Complexity:
· Since the halting problem is undecidable, it does not have a computational complexity in the traditional sense (e.g., O(1), O(n)).
· The problem is  not computable, meaning it cannot be solved by any algorithm.
Knowledge Booster:
· The halting problem is a key example of  undecidable problems, which cannot be solved by any algorithm.
· Other undecidable problems include:
· The  Entscheidungsproblem (decision problem for first-order logic).
· The  Post Correspondence Problem.
· While the halting problem is undecidable in general, it can be solved for specific cases or restricted classes of programs.
· The concept of undecidability has profound implications for the limits of computation and the design of programming languages and compilers.
· Turing's work on the halting problem laid the foundation for modern computer science and the study of computability.