The Halting Problem and Its Paradoxical Cousins: When Logic Looks at Itself

Old Computer – Last Century

The Halting Problem: Can a Program Predict Itself?


Problem: Given a program P and input x, can you determine whether P(x) halts or loops forever?


• Introduced by Alan Turing in 1936
• Proved to be undecidable — no algorithm can solve it for all inputs
• Involves self-reference: Can a program analyze another program, including itself?

Turing’s Thought Experiment

Assume: H(P, x) tells whether P(x) halts

Define the Following Python Code:

def D(P):
    if H(P, P):
        while True: pass  # loop forever
    else:
        return 0  # halt

Then what happens with D(D)?

• If D(D) halts, then it must loop forever
• If D(D) loops forever, it must halt

Logical Parallels: Paradoxes of Self-Reference

1. The Barber Paradox

• The barber shaves all those who do not shave themselves
• Does the barber shave himself?
• Leads to contradiction — exactly like D(D)

2. The Liar Paradox

"This statement is false."

• If true → it’s false
• If false → it’s true
• Pure self-reference causes a loop in truth evaluation

3. Russell’s Paradox

R = { x | x is a set and x ∉ x }

• Is R ∈ R?
• If yes → it must not be in R
• If no → it must be in R

4. Grelling–Nelson Paradox

Heterological = an adjective that does not describe itself

• Is “heterological” heterological?
• If yes → it isn’t; if no → it is

5. Gödel’s Incompleteness Theorem

"This statement is not provable."

• If provable → contradiction
• If not provable → true but unprovable
• Some truths are beyond the system itself

6. Quine Programs

# Python quine example
s = 's = {!r}\nprint(s.format(s))'
print(s.format(s))

• Programs that output their own source
• Illustrates controlled self-reference

7. The Entscheidungs problem

• Hilbert asked: Can we decide truth of all mathematical statements mechanically?
• Answer (by Turing & Church): No
• The Halting Problem is a key part of this result

Summary Table

Final Thought

• All these paradoxes show the limits of formal systems when they try to represent or reason about themselves
• Self-reference is powerful, but dangerous
• The Halting Problem isn’t just about computers — it’s a gateway into the heart of logic itself

Computer Science

• P versus NP problem (NP Complete, NP Hard)
• The Halting Problem and Its Paradoxical Cousins: When Logic Looks at Itself

–EOF (The Ultimate Computing & Technology Blog) —