COT 5310 Homework 6 Key
Fall 2007
1. Using reduction from the complement of the Halting Problem, show the undecidability of the problem
to determine if an arbitrary partial recursive function,
f
, has a summation upper bound. This means
that there is an
M
, such that the sum of all values in the range of
f
(repeats are added in and divergence
just adds 0) is
≤
M
.
The set HALT =
{a
f,x
A
:
f
(
x
)
↓}
, therefore CoHALT =
{a
f,x
A
:
f
(
x
)
↑}
.
The set of partial func-
tions
f
with a summation upper bound can be described as
UB =
{
f
:
∃
x
∀
y
[
y > x
⇒
(
f
(
y
) = 0 or
f
(
y
)
↑
)]
}
.
or in other words, only Fnitely many of the outputs can be non-zero.
To show that CoHALT
≤
m
UB we deFne
g
(
a
f,x
A
) =
g
f,x
as
g
f,x
(
y
) =
μ t
[STP(
f,x,t
)] + 1
.
Then if
a
f,x
A ∈
CoHALT,
g
f,x
(
y
)
↑
for all
y
∈
N
so dom(
g
f,x
) =
∅
⇒
g
f,x
∈
UB.
If
a
f,x
A
/
∈
CoHALT,
g
f,x
(
y
)
≥
1 for all
y
∈
N
. This means that no summation upper bound exists and
g
f,x
/
∈
UB.
2. Use one of the versions of Rice’s Theorem to show the undecidability of the problem to determine if
an arbitrary partial recursive function,
f
, has a summation upper bound. This means that there is an