Resource Bounded Reductions
Harry Buhrman

Abstract:
In this thesis Harry Buhrman studies reductions bounded in
polynomial time. Such a reduction induces an equivalence relation on sets
such that the equivalence classes, consisting of sets that mutually reduce,
can be ordened partially. Buhrman studies the structures that polynomial
reductions impose on time-bounded complexity classes.

The central open problem in computational complexity theory is the question
whether polynomialy bounded deterministic computation is as powerfull as 
non-deterministic computation that is polynomialy bounded. To get a
precise view of the power of deterministic and nondeterministic computations,
the classes $\C{P}$ and $\C{NP}$ are introduced: the problem mentioned is known
as the $\C{P}$ versus $\C{NP}$ problem. 

Just because it is not possible to solve this problem with our current
knowledge, attention has shifted to classes that are close: $\C{E}$ and 
$\C{NE}$. Although for these classes many problems remain unsolved, there is
some progress through polynomial time reduction on the structure of these
classes. In this thesis a small step is made in this direction. 

In chapter three it is shown that the structure of complete sets for
different type reductions for  $\C{NE}$ equals those of $\C{E}$.
It is quite striking that the 1-truth-table reduction induces the same
completeness notion as the many-one reduction, in particular, since it is
unknown whether $\C{NE}$  is closed under complementation.

In chapter four complete sets for $\C{E}$, $\C{NE}$ 
en $\C{NP}$ are discussed. Buhrman tries to obtain some clarity in the
consequences of having sparse subexponentially dense, complete sets for these
classes. There are strong suggestions that these classes can only contain
dense complete sets. This generalizes earlier results in the field.

In chapter five a complementary strategy is used: what sets reduce to 
sprase polynomially dense sets? Two open problems are solved and it is proved
that every polynomially dense set can be decoded in a simple way into a
so-called tally set. This is an unexpected and surprising result.

Chapter six is the continuation of chapter four, be it that other structural
properties than density of complete sets are discussed.
Buhrman compares how robust complete sets are to the addition or deletion
of elements. The results show that some elements are crucial for 
completeness of the set, but that it is hard to find these.
It is interesting that a generalization of the results obtained could solve
another open problem: exponential time has no polynomial circuits. 

Furthermore, this chapter discusses whether is it possible to split (complete)
sets in two parts that contain equal quantities of computational information. 
In case of spliting complete sets, this means that the parts are in turn 
complete. For many-one complete sets in $\C{E}$ this is always possible,
resulting in the fact that a complete set can be split up in infinitely many
pieces that are complete.

Chapter seven discusses structural properties of sets with the use of 
{\em selfreduction}, which does not order two sets, but assigns internal 
structure to a set. Buhrman discusses what sets in $\C{NP}$ 
are selfreducible, and gives a strong characterization of the class $\C{P}$ in
terms of sets, that are selfreducible and p-selective, a polynomialy bounded 
version of the notion semi-recursivity from recursion theory. 
Next, it is proved that the class of p-selective sets is closed under
positive Turing reductions. 
Finally, a detailled view of the notions of selfreducibility on $\C{NP}$
is presented