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}
Undecidable:
{<P> | P is a two-variable polynomial expression with an integral root}

3. The acceptance problem for DFAs is decidable
Giorgi Japaridze Theory of Computability
Let ADFA = {<B,w> | B is a DFA that accepts input string w}
Theorem 4.1: ADFA is a decidable language.
Proof idea: Here is a Turing machine M that decides ADFA:
M = “On input <B,w>, where B is a DFA and w is a string:
1. Simulate B on input w.
2. If the simulation ends in an accept state, accept.
If the simulation ends in a nonaccept state, reject.”

4. The acceptance problem for NFAs is decidable
Giorgi Japaridze Theory of Computability
Let ANFA = {<B,w> | B is an NFA that accepts input string w}
Theorem 4.2: ANFA is a decidable language.
Proof idea: Here is a Turing machine N that decides ANFA:
N = “On input <B,w>, where B is an NFA and w is a string:
1. Convert NFA B to an equivalent DFA C using the procedure
for this conversion that we learned.
2. Run TM M from Theorem 4.1 on input <C,w>.
3. If M accepts, accept.
If M rejects, reject.”

5. The emptiness problem for the language of a DFA is decidable
Giorgi Japaridze Theory of Computability
Let EDFA = {<A> | A is a DFA and L(A)=}
Theorem 4.4: EDFA is a decidable language.
Proof idea: Here is a Turing machine T that decides EDFA:
T = “On input <A>, where A is a DFA:
1. Mark the start state of A.
2. Repeat until no new states get marked:
Mark any state that has a transition coming into it from any
state that is already marked.
4. If no accept state is marked, accept; otherwise reject.”

6. Other decidable problems from the language theory
Each the following languages are also decidable:
EQDFA = {<A,B> | A and B are DFAs and L(A)=L(B)}
AREX = {<R,w> | R is a regular expression that generates string w}
ACFG = {<G,w> | G is a CFG that generates string w}
Theorem 4.9: Every context-free language is decidable.
Proof idea: Suppose L is a context-free language. Let G be the CFG
that generates L, and S be a TM that decides ACFG.
Here is a Turing machine H that decides L:
H = “On input w:
1. Run S on input <G,w>.
2. If the machine accepts, accept; if it rejects, reject.”

7. 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.”
Does U also decide ATM?

8. 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.”
accept if M does not accept <M>
reject if M accepts <M>
Thus, D(<M>) =

9. 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!
To summarize:
H accepts <M,w> exactly when M accepts w.
D rejects <M> exactly when M accepts M.
D rejects <D> exactly when D accepts <D>.

10. 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: A TM
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,
Let M = “On input w:
1. Run both U and U on input w in parallel;
2. If U accepts, accept; if U accepts, reject.”
It can bee seen that M decides ATM, which contradicts Theorem 4.11.

11. The language hierarchy summary
4.2.d
Giorgi Japaridze Theory of Computability
All languages
Turing-recognizable languages
Turing-decidable languages
Context-free languages
Regular languages
{anbn | n0}
ATM
ATM
{anbncn | n0}