10/4/11
Dec 2,
© UCF (Charles E.
11
How They relate
•
Let S be an arbitrary subset of some universe U.
The predicate
χ
S over U may be defined by:
χ
S(x) = true
if and only if
χ
S is called the characteristic function
of S.
•
Let K be some arbitrary predicate defined over
some universe U.
The problem PK associated
with K is the problem to decide of an arbitrary
member x of U, whether or not K(x) is true.
•
Let P be an arbitrary decision problem and let U
denote the set of questions in P (usually just the
set over which a single variable part of the
questions ranges).
The set SP associated with P
is
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© UCF (Charles E.
22
Categorizing Problems (Sets) #
1
•
Recursively enumerable
 A set S is
recursively enumerable (re)
if S is empty
(S = Ø) or there exists an algorithm F, over
the natural numbers &, whose range is
exactly S.
A problem is said to be re if the
set associated with it is re.
•
SemiDecidable
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 Spring '08
 Staff
 Natural number, Halting problem, Countable set, Georg Cantor

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