COT 5310
Fall 2006
Midterm#1 Sample
Name:_____________________
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)
).
•
A set
S
is recursive if
S
has a total recursive characteristic function
χ
S
, such that
x
∈
S
⇔
χ
S
(x)
.
Note
χ
S
is a predicate. Thus, it evaluates to
0
(
false
), if
x
∉
S
.
•
When I say a set
S
is re, unless I explicitly say otherwise, you may assume any of the following
equivalent characterizations:
1.
S
is either empty or the range of a total recursive function
f
S
.
2.
S
is the domain of a partial recursive function
g
S
.
•
If I say a function
g
is partially computable, then there is an index
g
(I know that’s overloading,
but that’s okay as long as we understand each other), such that
Φ
g
(x) =
Φ
(x, g) = g(x)
. Here
Φ
is
a universal partially recursive function.
Moreover, there is a primitive recursive function