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CSCI 2670
Introduction to Theory of
Computing
November 3, 2005
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Agenda
•
Yesterday
–
Reductions (Section 5.3)
•
Today
–
More on reductions
–
Section 6.3
–
We will not be covering section 6.4
•
I will discuss some basic issues of this section when covering
chapter 7
November 3, 2005
Announcement
•
Remember to let me know if you want to come to
my pizza party next Wednesday
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Mapping reducibility
Definition:
Language A is
mapping reducible
to language B,
written A
≤
m
B, if there is a computable function f:
∑
*
→
∑
*
, where for every w,
w
∈
A iff f(w)
∈
B
f
f
November 3, 2005
Why
≤
m
?
•
In some sense, if A
≤
m
B, then A is “less powerful”
than B
–
For example, we can map from CFG’s to decidable
languages that aren’t CF, but not vice versa
•
Example
–
We can easily map any CFG C to A
CFG
•
f(w) = <C,w>
•
w
∈
C iff <C,w>
∈
A
CFG
•
You can use this example to help you remember
how to use reductions
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Theorem:
If A
≤
m
B and B is decidable, then A is
decidable.
Proof:
Let M be a decider for B and let f be a
reduction from A to B.
Consider the following TM, N:
N = “On input w:
1.
Compute f(w)
2.
Run M on f(w) and report M’s output
”
Then N decides A
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This note was uploaded on 02/07/2011 for the course CS 501 taught by Professor Sm during the Spring '11 term at Indiana.
 Spring '11
 sm

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