COT 5310
Spring 2006
Midterm#1
Name:
PAGE#1
Generally useful information.
•
The notation
z =
<x,y>
denotes the pairing function with inverses
x =
<z>
1
and
y =
<z>
2
.
•
The minimization notation
µ
y [P(…,y)]
means the least
y
(starting at
0
) such that
P(…,y)
is
true. The bounded minimization (acceptable in primitive recursive functions) notation
µ
y (u
≤
y
≤
v) [P(…,y)]
means the least
y
(starting at
u
and ending at
v
) such that
P(…,y)
is true.
Unlike the text, I find it convenient to define
µ
y (u
≤
y
≤
v) [P(…,y)]
to be
v+1
, when no
y
satisfies this bounded minimization.
•
The tilde symbol,
~,
means the complement. Thus, set
~S
is the set complement of set
S
, and
predicate
~P(x)
is the logical complement of predicate
P(x).
•
A function
P
is a predicate if it is a logical function that returns either
1
(
true
) or
0
(
false
). Thus,
P(x)
means
P
evaluates to
true
on
x
, but we can also take advantage of the fact that
true
is
1
and
false
is
0
in formulas like
y
×
P(x)
, which would evaluate to either
y
(if
P(x)
) or
0
(if
~P(x)
).
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 Spring '08
 Staff
 Recursion theory, Primitive recursive function

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