Emphasizes the computer science aspects of the subject. Details applications in databases, complexity theory, and formal languages, as well as other branches of computer science.

/pdf/elements-of-finite-model-theory-3e3x2876gu.pdf

Elements of Finite Model Theory

This volume provides an ideal introduction to key topics in parallel computing. With its cogent overview of the essentials of the subject as well as lists of P -complete- and open problems, extensive remarks corresponding to each problem, a thorough index, and extensive references, the book will prove invaluable to programmers stuck on problems that are particularly difficult to parallelize. In providing an up-to-date survey of parallel computing research from 1994, Topics in Parallel Computing will prove invaluable to researchers and professionals with an interest in the super computers of the future.

/pdf/limits-to-parallel-computation-p-completeness-theory-2vkk1tcpy5.pdf

Limits to Parallel Computation: P-Completeness Theory

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k−1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.

An optimal lower bound on the number of variables for graph identification

In this paper we ask the question, “What must be added to first-order logic plus least-fixed point to obtain exactly the polynomial-time properties of unordered graphs?” We consider the languages L k consisting of first-order logic restricted to k variables and C k consisting of L k plus “counting quantifiers”. We give efficient canonization algorithms for graphs characterized by C k or L k . It follows from known results that all trees and almost all graphs are characterized by C 2.

Describing Graphs: A First-Order Approach to Graph Canonization

Introduction to Circuit Complexity

On uniformity within NC 1

A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC1, NUDFA characterizations of several important subclasses of NC1, and a new proof of the old result that the dot-dephth hierarchy is infinite, using M. Sipser's (1983, in “Proceedings, 15th ACM Symposium on the Theory of Computing,” Association for Computing Machinery, New York, pp. 61–69) work on constant depth circuits. Here we extend this theory to NUDFA's over solvable groups (NUDFA's over non-solvable groups have the maximum possible computing power). We characterize the power of NUDFA's over nilpotent groups and prove some optimal lower bounds for NUDFA's over certain groups which are solvable but not nilpotent. Most of these results appeared in preliminary form in ( D. A. Barrington and D. Therien, 1987 , in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).

David A. Mix-Barrington

Papers

On uniformity within NC 1

Non-uniform automata over groups