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

this paper can be generalized to the optimal model? 8 Acknowledgment I wish to thank L. Hemachandra for his invitation to me to spend the spring semester at the University of Rochester and for his permanent attention to my research and helpfulness in all my problems and J. Seiferas for extensive comments on an earlier draft of this paper. The results of section 4.1 of the paper are the realization of J. Seiferas's advice to investigate the probabilistic complexity properties of almost all functions in comparison with Yao's [Y1] results. I wish also to thank P. Dietz for his comments, which helped to simplify the proof of lemma 4.1

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

We generalize the definition of first-order counting classes [SST92] to use !, SUCC, and + as linear orderings. It turns out that #\Pi2 [!] = #\Pi1[SUCC] = #\Pi1[+]. We introduce certain classes of logtime counting functions and show that the classes of first-order definable counting functions are subclasses of the corresponding logtime counting classes. These logtime counting classes are itself subclasses of the corresponding one-way logspace counting classes. These logspace counting classes form a strict hierachy within #P: F1L` = #1L` = span-1L` = #P: Using the logical characterization of #P we obtain a characterization of #P via universally branching logtime Turing machines. 1 Introduction An important open question in complexity theory is whether the two classes NL and NP are equal. Although in the case of computing partial multivalued functions nondeterministically the corresponding classes can be separated [Bur89], a solution for the class of decision problems is not in sight. ...

This paper presents a perspective on the relationship between Lindström quantiers in model theory and oracle computations in complexity theory. We do not study this relationship here in full generality (indeed, there is much more work to do in order to obtain a full appreciation), but instead we examine what amounts to a thread of research in this topic running from the motivating results, concerning logical characterizations of nondeterministic polynomial-time, to the consideration of Lindström quantiers as oracles, and through to the study of some naturally arising questions (and subsequent answers). Our presentation follows the chronological progress of the thread and highlights some important techniques and results at the interface between nite model theory and computational complexity theory.

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

space compression is incomparable with

The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [11, 17], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. We also consider online compression algorithms that use at most polylogarithmic space (plogon). These algorithms correspond to compressors in the data stream model. We compare the performance of these two families of compressors with each other and with the general purpose Lempel-Ziv algorithm. This comparison is made without any a priori assumption on the data’s source and considering the asymptotic compression ratio for infinite sequences. We prove that in all cases they are incomparable.