Leen Torenvliet

On the structure of complete sets

The many types of resource-bounded reductions that are both an object of study and a research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The study of its structure can reveal properties that are general in that the complexity class, and the study of the structure of complete sets in different classes, can reveal secrets about the relation between these classes. Research into all sorts of aspects and properties of complete sets has been and will be a major topic in structural complexity theory. In this expository paper, we review the progress that has been made in recent years on selected topics in the study of complete sets.<<ETX>>

The value of agreement a new boosting algorithm

We present a new generalization bound where the use of unlabeled examples results in a better ratio between training-set size and the resulting classifiers quality and thus reduce the number of labeled examples necessary for achieving it. This is achieved by demanding from the algorithms generating the classifiers to agree on the unlabeled examples. The extent of this improvement depends on the diversity of the learners-a more diverse group of learners will result in a larger improvement whereas using two copies of a single algorithm gives no advantage at all. As a proof of concept, we apply the algorithm, named AgreementBoost, to a web classification problem where an up to 40% reduction in the number of labeled examples is obtained.

P-selective self-reducibles sets: a new characterization of P

It is shown that any p-selective and self-reducible sets is in P. As the converse is also true, the authors obtain a new characterization of the class P. A generalization and several consequences of this theorem are discussed. Among other consequences, it is shown that under reasonable assumptions autoreducibility and self-reducibility differ on NP, and there are non-p-T-mitotic sets in NP.<<ETX>>

Separating Complexity Classes Using Autoreducibility

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from exponential space by showing that all Turing complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Posts program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic, and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.

Completeness for nondeterministic complexity classes

We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic-space bounded) reducibilities turn out to be different for any class containingNE. For space classes the completeness notions under logspace reducibilities can be separated for any class properly containingLOGSPACE. Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes.

Nonapproximability of the normalized information distance

A water geyser assembly comprises an inner pressurizable container and an outer shell both formed from a glass fibre reinforced plastics material. A polyurethane foam material fills the void defined between the inner container and the shell. A fitting mounting land formed with a number of apertures is provided at one end of the inner container for receiving fittings in the form of a pair of flanged spigot portions which are externally threaded and which are bridged at the flange. The flange locates against an inner surface of the fitting mounting land via sealing rings, and lock nuts are screwed down over each of the threaded spigot portions which project through the apertures so as to form a pressure tight fit. The fittings are thus locked mechanically to the mounting land without being laid in glass fibre during manufacture. Water inlet and outlet conduits, a heating element assembly and a pressure relief valve are in turn mounted in communication with the spigot portions.

Splittings, Robustness, and Structure of Complete Sets

We investigate the structure of EXP-complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. We study for various types of reductions the question of whether the set difference A-S for a hard set A and a sparse set S is still hard. We also address the question of which complete sets A can be split into sets A1 and A2 such that

Optimal advice

A\equiv^P_r A_1\equiv^P_r A_2

Twenty questions to a p-selector

for reduction type r, i.e., which complete sets are mitotic. We obtain both positive and negative answers to these questions depending on the reduction type and the structure of the sparse set.

Six hypotheses in search of a theorem

Ko proved that the P-selective sets are in the advice class P=quadratic, and Hemaspaandra, Naik, Ogihara, and Selman showed that they are in PP=linear. We strengthen the latter result by establishing that the P-selective sets are in NP=linear T coNP=linear. We show linear advice to be optimal.