This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

We show that for most complexity classes of interest, all sets complete under first-order projections are isomorphic under first-order isomorphisms. That is, a very restricted version of the Berman-Hartmanis Conjecture holds.

The many types of resource bounded reductions that are both object of study and 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 complexity class, and the study of the structure of complete sets in different classes can reveal secrets about the relation between these classes. The 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 of the study of complete sets.

In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...

Growing context-sensitive grammars (GCSG) are investigated. The variable membership problem for GCSG is shown to be NP-complete. This solves a problem posed in [DW86]. It is also shown that the languages generated by GCSG form an abstract family of languages and several implications are presented. Institut fur Informatik, Universitat Wurzburg, D-8700 Wurzburg, Germany. y Instytut Informatyki, Uniwersytet Wroc/lawski, 51-151 Wroc/law, Poland (permanent address). This research was supported by the Humboldt Foundation. 1 Introduction It is well known that the class of languages generated by context-sensitive grammars is equal to NSPACE(n) and that, even for fixed grammars, the membership problem can be PSPACE-complete. On the other hand the context-free grammars are known to have, for many applications, too weak derivative power. While many modifications extending context-free grammars have been studied, only a few papers concern some restricted versions of context-sensitive gramm...

It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a non-trivial ≤1−NL m-degree that coincides with a ≤1−L m-degree. 3. For every class C closed under log-lin reductions, the ≤1−NL m coincides with the ≤1−L m-complete degree of C. 1

The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under log-lin reductions: -- All complete sets for the class C under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic. -- The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions.

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions.

We study the structure of the polynomial-time complete sets for NP and PSPACE under strong nondeterministic polynomial-time reductions (SNP-reductions). We show the following results. • If NP contains a p-random language, then all polynomial-time complete sets for PSPACE are SNP-isomorphic. • If NP ∩ co-NP contains a p-random language, then all polynomial-time complete sets for NP are SNP-isomorphic.

In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1