This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints. The linearity has to be understood with respect to nonstandard algebraic structures, e.g. the 'max-plus algebra'. A calculus is developed based on such structures, which is followed by tools for computing the time behaviour to such systems. This algebraic vision lays the foundation of a bona fide 'discrete event system theory', which is shown to parallel the classical linear system theory in several ways.

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Synchronization and Linearity: An Algebra for Discrete Event Systems

The subject of this chapter is the study of formal languages (mostly languages recognizable by finite automata) in the framework of mathematical logic.

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Languages, automata, and logic

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.

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Elements of Finite Model Theory

On uniformity within NC 1

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.

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Limits to Parallel Computation: P-Completeness Theory

Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.

Finite Automata, Formal Logic, and Circuit Complexity

1 Abstract In order to study circuit complexity classes within NC 1 in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC 1 uniformity Ru81,Co85], have appeared in recent research: Immerman's families of circuits deened by rst-order formulas Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC 1 Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of sub-classes of the regular languages based on their logical expressibility, extending recent work of Straubing, Th erien, and Thomas STT88]. A preliminary version of this work appeared as BIS88]. 2 Introduction 2.1 Circuit Complexity Computer scientists have long tried to classify problems (deened as Boolean predicates or functions) by the size or depth of Boolean circuits needed to solve them. This eeort has

On uniformity within NC 1 .

Finite semigroup varieties of the form v,d

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).

Howard Straubing

Papers

On uniformity within NC 1

Finite Automata, Formal Logic, and Circuit Complexity

On uniformity within NC 1 .

Finite semigroup varieties of the form v,d

Non-uniform automata over groups