THE HALTING PROBLEM PROOF
Review
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What makes a problem decidable?
3 properties of an efficient algorithm?
What is the meaning of “complete”, “mechanistic”,
and “deterministic”?
Is the Halting Problem decidable/undecidable?
How would you define the Universal Turing Machine?
Halting problem is undecidable Proof
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Proof by contradiction
 Assume
that there is a TM H that solves the halting
problem
 The computation of H can be depicted as follows:
M halts with input w
R(M)w
Accept
H
Reject
M does not halt
with input w
Halting problem is undecidable Proof
 We
modify H to construct a new TM H’, which behaves
very much as H:
 The
computations of H’ are the same as H, except that H’
loops indefinitely whenever H terminates in an accepting
state, i.e., whenever H halts on input w, and halts otherwise
M halts with input w
R(M)w
Loop
H’
Halt
M does not halt
with input w
Halting problem is undecidable Proof
 H’
is combined with a copy machine to construct a new TM
D:
 The
input to D is a TM representation R(M)
 A computation of D begins with copying the string R(M) to yield
R(M)R(M)
 The computation continues by running H’ on R(M)R(M)
M halts with input
R(M)
Copy
R(M)
D
R(M) R(M)
H’
Loop
Halt
M does not halt with
input R(M)
Halting problem is undecidable Proof

Consider a computation of D with input R(D):

The input to D is the representation to any arbitrary TM
D halts with input
R(D)
Copy
R(D)
D
 Examining
D
R(D) R(D)
Loop
H’
Halt
D does not halt with
input R(D)
the preceding computation, we see that:
halt on input R(D) iff D does not halt on input R(D), which is a
contradiction
 Therefore, the original assumption is wrong and the halting
problem is undecidable