Reducibility

1
Reducibility
Giorgi Japaridze Theory of
Computability
Chapter 5
2
The undecidability of the halting problem
5.1.a
Giorgi Japaridze Theory of Computability
Let HALTTM ltM,wgt M is a TM and M halts on
input w HALTTM is called the halting problem.
Theorem 5.1 HALTTM is undecidable.
Proof idea Assume, for a contradiction, that
HALTTM is decidable. I.e. there is a TM R that
decides HALTTM. Construct the following TM S
S On input ltM,wgt, an encoding of a TM M and a
string w 1. Run R on input ltM,wgt.
2. If R rejects, reject. 3. If R
accepts, simulate M on w until it halts.
4. If M has accepted, accept if M has rejected,
reject.

• If M works forever on w, what will S do on ltM,wgt?

• If M accepts w, what will S do on input ltM,wgt?

• If M explicitly rejects w, what will S do on
  ltM,wgt?

Thus, S decides the language
But this is impossible (Theorem 4.11)


3
Definition of mapping reducibility
5.3.a
Giorgi Japaridze Theory of Computability

We say that A is
mapping reducible to B, written A?mB, if
there is a computable function f ??? such
that, for every w??,
w?A iff f(w)?B. The function f is
called a mapping reduction of A to B.
Let A and B be languages over an alphabet ?.
?
?
A
B
f
f



4
An example of a mapping reduction
5.3.b
Giorgi Japaridze Theory of Computability
Let f be the function computed by the following
TMO M MOn input ltN,wgt, where N is an NFA and
w is a string, 1. Convert N into an
equivalent DFA D using the algorithm
we learned 2. Return ltD,wgt.
f is then a mapping reduction of what language to
what language?
?
?
ltN,wgt
ltD,wgt
ltD,wgt
ltN,wgt




5
Using mapping reducibility for proving
decidability/undecidability
5.3.c
Giorgi Japaridze Theory of Computability
Theorem 5.22 If A?mB and B is decidable, then A
is decidable.
Proof Let DB be a decider for B and f be a
reduction from A to B. We describe a decider DA
for A as follows.
DA On input w 1. Compute f(w).
2. Run DB on input f(w) and do whatever DB
does.
Corollary 5.23 If A?mB and A is undecidable,
then B is undecidable.
Theorem 5.22 remains valid with Turing
recognizable instead of decidable. So
does Corollary 5.23.



6
A mapping reduction of ATM to HALTTM
5.3.d
Giorgi Japaridze Theory of Computability
For a TM M, let M be the TM obtained from M
by adding a dead-end (infinite loop) state and
re-directing into it all transitions that go to
the reject state.
Thus,

• If M accepts input x, then M
• If M explicitly rejects x, then M
• If M never halts on x, then M
• To summarize, M accepts x iff M

Let then f be the function defined by
f(ltM,wgt)ltM,wgt.
Is f computable?
Obviously ltM,wgt?ATM iff f(ltM,wgt)? i.e. f is
a

So, since ATM is undecidable, HALTTM is
undecidable as well.