Decidability — Undecidable Problems, Halting Problem

🎯 Learning Objectives

• Define decidable and undecidable languages
• Prove the halting problem is undecidable
• Reduce known undecidable problems to prove new ones
• Understand Rice's theorem

1. Decidable vs Undecidable

1.1 Intuition

1.2 Key Examples

2. The Halting Problem

2.1 Statement

2.2 Proof by Contradiction

python
Run Inline
def D(M):
    if H(M, M) == "accept":  # If M halts on input M...
        loop_forever()        # ...then D loops forever
    else:                     # If M loops on input M...
        return "accept"       # ...then D halts

• If $D$ halts on $\langle D \rangle$, then $H(D, D)$ says "accept" → $D$ loops forever (contradiction)
• If $D$ loops on $\langle D \rangle$, then $H(D, D)$ says "reject" → $D$ halts (contradiction) Neither case is possible → our assumption that $H$ exists is false → $HALT_{TM}$ is undecidable.

3. Rice's Theorem

• Non-trivial: Some TMs have it, some don't
• About the language, not the machine itself Examples of undecidable properties:
• Does a TM accept any string? (Emptiness)
• Does a TM accept all strings? (Universality)
• Does a TM accept a finite language?

4. 📝 Practice Questions

Q1: What is the halting problem?

Answer: The halting problem asks: given a Turing machine M and input w, does M halt on w? It's undecidable — no algorithm can determine this for all TMs. This was proven by Alan Turing in 1936 and is the foundational undecidable problem. Q2: Prove that the emptiness problem for TMs is undecidable.

Answer: Reduce from halting problem. Given (M, w), construct M' that: (1) ignores its input, (2) simulates M on w, (3) accepts if M halts. Then M' accepts some input (any input) iff M halts on w. If we could decide emptiness of M', we could decide halting. Contradiction. Q3: What does Rice's theorem state?

Answer: Rice's theorem states that any non-trivial property of the language of a Turing machine is undecidable. "Non-trivial" means there exists at least one TM with the property and at least one without it. This means almost all interesting questions about program behavior are undecidable. Q4: Is the problem "Does a TM accept any even-length string?" decidable?

Answer: No. This is a non-trivial property of the language (some TMs accept even-length strings, some don't). By Rice's theorem, it's undecidable. Q5: What is the difference between undecidable and non-computable?

Answer: These terms are equivalent for decision problems. An undecidable problem has no algorithm that always answers yes/no correctly. Undecidable is the standard term in recursion theory and automata theory; non-computable is more general (includes non-decision problems).

5. 🔗 Cross-References

• Week 9 - Reductions: Using reductions to prove undecidability
• Week 6 - Turing Machines: Model of computation for undecidability
• BSCS3021 - Week 12 - NP-Completeness: Decidability vs complexity Join Discord PreviousTM VariantsNextReductions