1. CSC 4170
Theory of Computation
Mapping Reducibility
Section 5.3

2. Definition of mapping reducibility
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

3. 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:
M=“On input <N,w>, 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 <D,w>.”
f is then a mapping reduction of what language to what language? * *
<N,w>
<D,w>
<D,w>
<N,w>

4. Using mapping reducibility for proving decidability/undecidability
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 remains valid with “Turing recognizable” instead of
“decidable”. So does Corollary 5.23.

5. A mapping reduction of ATM to HALTTM
Giorgi Japaridze Theory of Computability
For every TM M, let M* be the following TM:
M* = “On input x:
1. Run M on x.
2. If M accepts, accept.
3. If M rejects, enter an infinite loop.”
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(<M,w>)=<M*,w>.
Is f computable?
Obviously <M,w>ATM iff f(<M,w>)
i.e. f is a
So, since ATM is undecidable, HALTTM is undecidable as well.