COT 6410
Fall 2010
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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dutton
 Halting problem, Recursion theory, Recursively enumerable set, computable function

Click to edit the document details