Chapter 4: Decidability

For Your Enjoyment

U = "On input <M, w>, where M is a TM and w is a string:

Simulate M on input w.
If M ever enters its accept state, ACCEPT; if M ever enters its reject state, REJECT.

Georg Cantor proposed that a set is countable if either (1) it is finite or (2) it has a correspondence with the set of natural numbers, N.
Definition: A function that is both one-to-one and onto is a correspondence.
The set of even numbers, E = {2, 4, 6, ...} is countable. What is the correspondence with the natural numbers, N?
Counterintuitive note: Infinite sets E and N are considered to have the same size if there is a correspondance between the two.
The set Q = { m/n | m, n ∈ N} is countable. What is the correspondence with the natural numbers?
The set of real numbers, R, is uncountable. The proof is by contradiction. Assume R is countable and then show the correspondence fails using the diagonalization method.

Proof: Assume that A_TM is decidable and find a contradiction.
Create Turing Machine H(<M, w>) to ACCEPT if M accepts w and REJECT if M does not accept w.
Create Turing Machine D that runs H on input <M, <M>>. D outputs REJECT if H accepts and ACCEPT if H rejects.
Consider running D on itself. If D accepts, that means that D rejected <D>. If D rejects, that means that D accepted <D>. Contradiction!
Figure 4.21 shows how this relates to the diagonalization technique.

Definition: A language is co-Turing-recognizable if it is the complement of a Turing-recognizable language.
Theorem: A language is decidable iff it is Turing-recognizable and co-Turing-recognizable.
Proof: A_TM is Turing-recognizable. Therefore, the complement of A_TM can not be Turing-recognizable. Otherwise A_TM would be decidable.