1. Theory of Computability
Giorgi Japaridze
Theory of Computability
Decidability
Chapter 4

2. Examples of decidable languages
Giorgi Japaridze Theory of Computability
Decidable:
{1,3,5} 
{x | x is even}
{x | x is a perfect square}
{x | x2-10x = 0}
{x | x=y*z for some integers y,z>1 (i.e. x is not prime)}
{x | x is a prime (i.e. x is not divisible by anything except 1 and
itself)}
{<G> | G is a connected graph}
{<P> | P is a one-variable polynomial expression with an integral root}
Hilbert’s 10th problem (are there integral roots to polynomial expressions?)
Or better:
“x is a perfect square”
Now: \exists x x is a solution to
X^2 – 10x = 2000
Or the problem
\exists m,n,k 2^m+3^n=
Undecidable:
{<P> | P is a two-variable polynomial expression with an integral root}

3. The universal Turing machine
Giorgi Japaridze Theory of Computability
Let ATM = {<M,w> | M is a TM and M accepts string w}
ATM is misleadingly called the halting problem in the textbook.
Instead, we will call it the acceptance problem.
Theorem: ATM is Turing-recognizable.
Proof idea.
The following TM U, called the universal TM, recognizes ATM:
U = “On input <M,w>, where M is a TM and w is a string:
1. Simulate M on input w.
2. If M ever enters its accept state, accept;
if M ever enters its reject state, reject.”
The idea of U played an important role in stimulated the development of stored-program computers.
Does U also decide ATM?

4. Theorem 4.11: ATM is undecidable.
Giorgi Japaridze Theory of Computability
Theorem 4.11: ATM is undecidable.
Proof idea. Suppose, for a contradiction, that ATM is decidable.
That is, there is a TM H that decides ATM.
Thus, that machine H behaves as follows:
accept if M accepts w
reject if M does not accept w
H(<M,w>) =
Using H as a subroutine, we can construct the following TM D:
D = “On input <M>, where M is a TM:
1. Run H on input <M,<M>>.
2. Do the opposite of what H does.
That is, if H accepts, reject, and if H rejects, accept.”
Mention programs taking themselves as an input.
Say, a C complier written in C, can compile itself!
accept if M does not accept <M>
reject if M accepts <M>
Thus, D(<M>) =

5. accept if D does not accept <D> reject if D accepts <D>
ATM is undecidable
4.2.b.2
Giorgi Japaridze Theory of Computability
accept if D does not accept <D>
reject if D accepts <D>
But then D(<D>) =
Contradiction!
The idea of U played an important role in stimulated the development of stored-program computers.

6. A Turing-unrecognizable language
Giorgi Japaridze Theory of Computability
ATM = {<M,w> | M is a TM and M accepts string w}
ATM = {<M,w> | M is a TM and M does not accept string w}
Theorem 4.23: ATM is Turing-unrecognizable.
Proof idea. Suppose, for a contradiction, that ATM is Turing-recognizable.
That is, there is a TM U that recognizes ATM.
U recognizes ATM (slide 4.2.a)
U recognizes ATM
Thus,
The idea of U played an important role in stimulated the development of stored-program computers.
Let K = “On input <M,w>:
1. Run both U and U on input <M,w> in parallel;
2. If U accepts, accept; if U accepts, reject.”
It can bee seen that K decides ATM, which contradicts Theorem 4.11.

7. The language hierarchy summary
4.2.d
Giorgi Japaridze Theory of Computability
All languages
ATM
Turing-recognizable languages
ATM
Turing-decidable languages
{<G> | G is a connected graph}