Janusz A. Brzozowski

Derivatives of Regular Expressions

Kleenes regular expressions, which can be used for describing sequential circuits, were defined using three operators (union, concatenation and iterate) on sets of sequences. Word descriptions of problems can be more easily put in the regular expression language if the language is enriched by the inclusion of other logical operations. However, il~ the problem of converting the regular expression description to a state diagram, the existing methods either cannot handle expressions with additional operators, or are made quite complicated by the presence of such operators. In this paper the notion of a derivative of a regular expression is introduced atld the properties of derivatives are discussed. This leads, in a very natural way, to the construction of a state diagram from a regular expression containing any number of logical operators.

On equations for regular languages, finite automata, and sequential networks☆

Abstract We consider systems of equations of the form X i = ⋃ α∈A α·F i,a ∪δ i i=1,…,n where A is the underlying alphabet, the Xi are variables, the Pi,a are boolean functions in the variables Xi, and each δi is either the empty word or the empty set. The symbols υ and ∪ denote concatenation and union of languages over A. We show that any such system has a unique solution which, moreover, is regular. These equations correspond to a type of automation, called boolean automation, which is a generalization of a nondeterministic automation. The equations are then used to determine the language accepted by a sequential network; they are obtainable directly from the network.

The dot-depth hierarchy of star-free languages is infinite☆

Let A be a finite alphabet and A∗ the free monoid generated by A. A language is any subset of A∗. Assume that all the languages of the form {a}, where a is either the empty word or a letter in A, are given. Close this basic family of languages under Boolean operations; let B(0) be the resulting Boolean algebra of languages. Next, close B(0) under concatenation and then close the resulting family under Boolean operations. Call this new Boolean algebra B(1), etc. The sequence B(0), B(1),…, Bk,… of Boolean algebras is called the dot-depth hierarchy. The union of all these Boolean algebras is the family A of star-free or aperiodic languages which is the same as the family of noncounting regular languages. Over an alphabet of one letter the hierarchy is finite; in fact, B(2) = B(1). We show in this paper that the hierarchy is infinite for any alphabet with two or more letters.

Signal Flow Graph Techniques for Sequential Circuit State Diagrams

This paper considers the application of signal flow graph techniques to the problem of characterizing sequential circuits by regular expressions. It is shown that the methods of signal flow graph theory, with the proper interpretation, apply to state diagrams of sequential circuits. The use of these methods leads to a simple algorithm for obtaining a regular expression describing the behavior of a sequential circuit directly from its state diagram.

Languages of R-trivial monoids

We consider the family of languages whose syntactic monoids are R-trivial. Languages whose syntactic monoids are J-trivial correspond to a congruence which tests the subwords of length n or less that appear in a given word, for some integer n. We show that in the R-trivial case the required congruence also takes into account the order in which these subwords first appear, from left to right. Characterizations of the related finite automata and regular expressions are summarized. Dual results for L-trivial monoids are also discussed.

Syntactic complexity of ideal and closed languages

The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn-1 is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity nn-1 + n - 1, and two-sided ideals and factor-closed languages of syntactic complexity nn-2 + (n - 2)2n-2 + 1.

OPEN PROBLEMS ABOUT REGULAR LANGUAGES

Publisher Summary The theory of regular languages and finite automata was developed in the early 1950s and is one of the oldest branches of theoretical computer science. Regular languages constitute the best known family of formal languages, and finite automata constitute the best known family of abstract machine models. The concepts of regular languages and finite automata appear frequently in theoretical computer science and have several important applications. There is a vast literature on these subjects. Despite the fact that many researchers have worked in this field, there remain several difficult open problems. The chapter discusses six of these problems. These problems are of fundamental importance and considerable difficulty. Most of them are intimately involved with the fundamental property of finite automata, namely finiteness. In a monograph published in 1971, McNaughton and Papert included a collection of open problems concerning regular languages. Their list is headed by the star height problem and until now, no progress has been made on such an intriguing question. The bounds on star height apply only to languages whose syntactic monoids are groups. In that case, the corresponding semiautomata are permutation semiautomata.

A Survey of Regular Expressions and Their Applications

This paper is an exposition of the theory of regular expressions and its applications to sequential circuits. The results of several authors are presented in a unified manner, pointing out the similarities and differences in the various treatments of the subject. Whenever possible, the terminology and notation of sequential circuit theory are used. The topics presented include: the relation of regular expressions to sequential circuits; algorithms for constructing sequential circuits and state diagrams corresponding to a given regular expression; methods for obtaining a regular expression from a state diagram of a sequential circuit, improper state diagrams, algebraic properties of regular expressions, and applications to codes.

Roots of Star Events

A regular event <italic>W</italic> is a star event if there exists another event <italic>V</italic> such that <italic>W</italic> = <italic>V</italic><supscrpt>*</supscrpt>. In that case, <italic>V</italic> is called a root of <italic>W</italic>. It is shown that every star event has a unique minimum root, which is contained in every other root. An algorithm for finding the minimum root of a regular event is presented, and the root is shown to be regular. The results have applications to languages, codes, canonical forms for regular expressions, simplification of expressions, decomposition of sequential machines, and semigroup theory.

de Morgan bisemilattices

We study de Morgan bisemilattices, which are algebras of the form (S, /spl cup/, /spl and/, /sup -/, 1, 0), where (S, /spl cup/, /spl and/) is a bisemilattice, 1 and 0 are the unit and zero elements of S, and /sup -/ is a unary operation, called quasi-complementation, that satisfies the involution law and de Morgans laws. de Morgan bisemilattices are generalizations of de Morgan algebras, and have applications in multi-valued simulations of digital circuits. We present some basic observations about bisemilattices, and provide a set-theoretic characterization for a subfamily of de Morgan bisemilattices, which we call locally distributive de Morgan bilattices.