This paper concerns some of the theoretical complexity aspects of the reconfigurable network model. The computational power of the model is investigated under several variants, depending on the type of switches (or switch operations) assumed by the network nodes. Computational power is evaluated by focusing on the set of problems computable in constant time in each variant. A hierarchy of such problem classes corresponding to different variants is shown to exist and is placed relative to traditional classes of complexity theory. Department of Mathematics and Computer Science, The Haifa University, Haifa, Israel. E-mail: yosi@mathcs2.haifa.ac.il y Department of Computer Science, Technische Universitat Munchen, 80290 Munchen, Germany. E-mail: lange@informatik.tu-muenchen.de z Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel. E-mail: peleg@wisdom.weizmann.ac.il. Supported in part by an Allon Fellowship, by a Bantrell Fellowship an...

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:

This paper has the following goals: -- To survey some of the recent developments in the field of derandomization. -- To introduce a new notion of time-bounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting various results in context. -- To illustrate the usefulness of KT, by answering a question that has been posed in the literature, and -- To pose some promising directions for future research.

We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting the number of accepting paths of a nondeterministic logspace machine can be done in NL/poly, if the number of paths is small. This clarifies the complexity of the class LogFew (defined and studied in [BDHM91]). Using derandomization techniques, we then improve this to show that this counting problem is in NL. Determining if our other theorems hold in the uniform setting remains an The material in this paper appeared in preliminary form in papers in the Proceedings of the IEEE Conference on Computational Complexity, 1998, and in the Proceedings of the Workshop on Randomized Algorithms, Brno, 1998. y Supported in part by NSF grants CCR-9509603 and CCR-9734918. z Supported in part by the ...

We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degree-three graphs had been shown to be in SL [23, 20].

Abstract. We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previously-studied subclass of planar graphs known as grid graphs. We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs. A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “acyclic grid graph reachability problem”. We show that this problem lies in the complexity class UL. 1

problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.

In this report, we emphasize the differences of grammar families and their properties versus language families and their properties. To this end, we investigate grammar families from an abstract standpoint, developping a new framework of reasoning. In particular when considering decidability questions, special care must be taken when trying to use decidability results (which are, in the first place, properties of grammar families) in order to establish results (e.g. hierarchy results) on language families. We illustrate this by inspecting some theorems and their proofs in the field of regulated rewriting. In this way, we also correct the formulation of an important theorem of Hinz and Dassow. As an exercise, we show that there is no `effective' grammatical characterization of the family of recursive languages. Moreover, we show how to prove the strictness of the Chomsky hierarchy using decidability properties only.

. The paper's main contributions are a compendium of problems that are complete for symmetric logarithmic space (SL), a collection of material relating to SL, a list of open problems, and an extension to the number of problems known to be SL-complete. Complete problems are one method of studying SL, a class for which programming is nonintuitive. Our exposition helps make the class SL less mysterious and more accessible to other researchers. Key words. Completeness, SL, space complexity, symmetric logarithmic space. Subject classifications. 68Q17. 1. Introduction In this paper we describe problems that are logarithmic space many-one complete for symmetric logarithmic space (SL). Our hope in collecting these problems and extending this list is that more insight can be gained about the relationships between the complexity classes deterministic logarithmic space (DL), SL, and nondeterministic logarithmic space (NL). The symmetric Turing machine model introduced by Lewis & Papadimitriou ...

Fagin's theorem, the first important result of descriptive complexity, asserts that a property of graphs is in NP if and only if it is definable by an existential second-order formula. In this paper, we study the complexity of evaluating existential second-order formulas that belong to prefix classses of existential second-order logic, where a prefix class is the collection of all existential second-order formulas in prenex normal form such that the second-order and the first-order quantifiers obey a certain quantifier pattern. We completely characterize the computational complexity of prefix classes of existential second-order logic in three different contexts: (1) over directed graphs, (2) over undirected graphs with self-loops and (3) over undirected graphs without self-loops. Our main result is that in each of these three contexts a dichotomy holds, i.e., each prefix class of existential second-order logic either contains sentences that can express NP-complete problems or each of its sentences expresses a polynomial-time solvable problem. Although the boundary of the dichotomy coincides for the first two cases, it changes, as one moves to undirected graphs without self-loops. The key difference is that a certain prefix class, based on the well-known Ackermann class of first-order logic, contains sentences that can express NP-complete problems over graphs of the first two types, but becomes tractable over undirected graphs without self-loops. Moreover, establishing the dichotomy over undirected graphs without selfloops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics, including results about graphs of bounded tree-width and Ramsey's theorem.