6.1 The halting problem and its proof

The halting problem asks if a computer program will ever finish running or go on forever. It's a key concept in computer science that shows some problems can't be solved by any algorithm, no matter how smart or powerful.

This idea has big implications for what computers can and can't do. It proves there are limits to computation, affecting areas like program verification and automated theorem proving. Understanding these limits is crucial in computer science.

The Halting Problem and Its Undecidability

Halting problem in computability theory

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• Asks whether a given Turing machine will halt or run forever on a given input
• Fundamental problem demonstrates the existence of undecidable problems (problems that cannot be solved by any algorithm)
• Undecidability has significant implications for the limitations of computation and the capabilities of computers (program verification, automated theorem proving)
• Proves there are problems that cannot be solved by any algorithm, regardless of the computational power available

Proof of halting problem undecidability

• Assume, for contradiction, that a hypothetical algorithm HALT exists that can solve the halting problem
  • HALT takes as input a Turing machine $M$ and an input string $w$, determines whether $M$ halts on $w$ or runs forever
• Construct a new Turing machine CONTRA that uses HALT as a subroutine
  • CONTRA takes a Turing machine $M$ as input and behaves as follows:
    1. If HALT determines $M$ halts on input $M$, then CONTRA enters an infinite loop
    2. If HALT determines $M$ does not halt on input $M$, then CONTRA halts
• Consider what happens when CONTRA is given itself as input ($M$ = CONTRA)
  • If HALT determines CONTRA halts on input CONTRA, then CONTRA must enter an infinite loop, contradicting HALT's determination
  • If HALT determines CONTRA does not halt on input CONTRA, then CONTRA must halt, again contradicting HALT's determination
• This contradiction proves the initial assumption, the existence of HALT, must be false
• Therefore, the halting problem is undecidable, no algorithm can solve it for all possible inputs

Implications for computational limitations

• Demonstrates inherent limitations to what can be computed by algorithms
• Shows there are problems for which no algorithm can provide a solution, regardless of computational resources available
• Proves not all mathematically well-defined problems are computable
• Establishes a fundamental boundary between what is computable and what is not
• Has implications for various areas of computer science (program verification, automated theorem proving, software analysis)

Halting problem vs Turing completeness

• A programming language or computational system is Turing complete if it can simulate any Turing machine
• Turing completeness implies a system can perform any computation that a Turing machine can perform
• The undecidability of the halting problem is closely related to Turing completeness
  • In a Turing-complete system, the halting problem is undecidable
  • If a system can solve the halting problem, it would not be Turing complete, as it could perform a task Turing machines cannot
• The halting problem serves as a litmus test for Turing completeness
  • If a system is powerful enough to encounter the halting problem, it is likely Turing complete
  • If a system is not Turing complete, it may avoid the halting problem by having limited computational capabilities
• Understanding the relationship helps assess the computational power and limitations of various systems and programming languages (lambda calculus, cellular automata)