COT 5310
Fall 2006
More Sample Final Exam Questions
Generally useful information.
•
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
STP
, such that
STP(x, g, t)
is
1
(true), just in case
g
, started on
x
, halts in
t
or fewer steps.
STP(x, g, t)
is
0
(false), otherwise.
Finally, there is another primitive recursive function
VALUE
, such that
VALUE(x, g, t)
is
g(x)
, whenever
STP(x, g, t)
.
VALUE(x, g, t)
is defined but meaningless if
~STP(x, g, t)
.
•
The notation
f(x)
↓
means that
f
converges when computing with input
x
, but we don’t care about
the value produced. In effect, this just means that
x
is in the domain of
f
.
•
The notation
f(x)
↑
means
f
diverges when computing with input
x
. In effect, this just means that
x
is
not
in the domain of
f
.
•
The Post Correspondence problem is known to be undecidable. This problem is characterized by
instances that are described by two n-ary sequences of non-empty words
<x
1
,x
2
,…,x
n
>, <y
1
,y
2
,…,y
n
>
. The question is whether or not there exists a sequence,