Using a randomized reduction, it is straightforward to show that
the distributional halting problem (version 2) is complete for
DistNP although it cannot be complete under a one-one and length
preserving p-time reduction.
Denote by 
the set of positive instances 
of
the distributional halting problem (version 2).
The probability distribution 
of instance (positive or negative) is proportional to
,
where 
, 
, and 
.
Recall that the size of is 
while the
size of instance 
of distributional halting problem
(version 1) is 
.
Clearly, is NP-complete and
is flat.
Proof.reduce 
to 
by a p-time randomized reduction
as follows.
On instance of 
, the reduction flips a
coin 
times to produce a random string 
, and then outputs
. More precisely,
we define a good-input domain 
by
.
Clearly, the rarity function 
and
is p-time computable and so is certifiable.
Let 
for 
.
For all 
, let 
.
Then 
is one-one and p-time computable. Clearly,
for all : 
iff 
.
To check the domination property, we note that
.
Thus, is the desired p-time randomized reduction.