Halting Problem

 

🧠 What Is the Halting Problem?

The Halting Problem is one of the most fundamental and famous results in computability theory.
It asks a simple but deep question:

Given a Turing machine M and an input w, can we decide whether M halts (stops running) on w or runs forever?

Formally, define the language:

$$HALT = \{ \langle M, w \rangle \mid \text{Turing machine } M \text{ halts on input } w \}$$

Here:

• $\langle M, w \rangle$ is an encoding of the Turing machine $M$ and its input $w$ (as a binary string, for example).

• The question is whether $HALT$is decidable — can some Turing machine always give a definite “yes” or “no” answer?

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⚙️ Intuitive Meaning

• Imagine writing a program halts(M, w) that tells you whether another program M stops when run on input w.

• If such a program exists, you could check for infinite loops or hangs automatically!

The Halting Problem asks whether such a program (or equivalently, a Turing machine) can exist.

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🚫 The Result

Alan Turing proved (in 1936) that:

The Halting Problem is undecidable.

That is:

• No Turing machine can take $\langle M, w \rangle$ as input
  and correctly decide for all cases whether $M$ halts on $w$.

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🧩 Intuitive Proof (by Contradiction)

Let’s go step by step.

Step 1: Assume the opposite

Suppose that a Turing machine H exists that can solve the halting problem.

That is, $H(⟨M,w⟩)\; behaves\; as\; follows:$

$$H(\langle M, w \rangle) = \begin{cases} \text{“yes”} & \text{if } M \text{ halts on } w \\ \text{“no”} & \text{if } M \text{ loops forever on } w \end{cases}$$

and importantly, H always halts on all inputs.

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Step 2: Construct a new Turing machine D

We now define a new Turing machine D that uses H as a subroutine, as follows:

D(M):
    if H(M, M) = "yes" then
         loop forever
    else
         halt

That is, D checks if machine M halts when run on itself:

• If M halts on itself, then D goes into an infinite loop.

• If M does not halt on itself, then D halts.

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Step 3: Ask — What happens when D runs on itself?

Now consider the input $\langle D, D \rangle$:

$$D(D)$$

Two cases:

1. Suppose $H(D, D) = \text{“yes”}$
   → meaning $D$ halts on input $D$.
   Then, by definition of $D$, it loops forever — contradiction!

2. Suppose $H(D, D) = \text{“no”}$
   → meaning $D$ does not halt on input $D$.
   Then $D$ halts — another contradiction!

So in both cases, we reach a contradiction.

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Step 4: Conclusion

Therefore, the assumption that H exists must be false.
So the Halting Problem is undecidable.

That is, no Turing machine can always decide whether another Turing machine halts.

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📘 Formal Summary

The Halting Problem Language is:

$$HALT = \{ \langle M, w \rangle \mid M \text{ halts on } w \}$$

Result:

• $HALT$ is Turing-recognizable (recursively enumerable).

• $HALT$ is not Turing-decidable (not recursive).

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🔍 Why It’s Recognizable but Not Decidable

A Turing machine can recognize the halting problem by simulating $M$ on $w$:

• If $M$ halts → accept.

• If $M$ doesn’t halt → loop forever.

So, if $\langle M, w \rangle \in HALT$, the recognizer halts and accepts.
But if $M$ doesn’t halt, the recognizer never halts — hence, not decidable.

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💡 Examples and Analogies

Analogy	Description
🧮 Decidable problem	You can always get an answer (like checking if a number is even).
🔁 Halting problem	Sometimes you never get an answer — the process might run forever.

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⚖️ Relationship to Recursive and Recursively Enumerable Languages

Language Type	Description	Example
Recursive (decidable)	Turing machine halts on all inputs	Palindromes, {aⁿbⁿ}
Recursively enumerable (recognizable)	Turing machine halts on accepted inputs only	HALT, { ⟨M, w⟩

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🧩 Significance

1. Fundamental Limit: There exist well-defined computational problems that no algorithm can solve.

2. Practical Importance: Explains why programs cannot, in general, check themselves for infinite loops or guarantee termination.

3. Theoretical Basis: Forms the foundation of the concept of undecidability and computability limits.

4. Connection to Gödel: Related to Gödel’s incompleteness theorem — there are true mathematical statements that cannot be proven mechanically.

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✅ Summary Chart

Concept	Description
Problem	Does machine M halt on input w?
Formal language	HALT = {⟨M, w⟩
Decidable?	❌ No
Recognizable?	✅ Yes
Proof method	Contradiction using self-reference (D(D))
Importance	Shows limits of what machines can compute