. We introduce a notion of a partial derivative of a regular expression. It is a generalization to the non-deterministic case of the known notion of a derivative invented by Brzozowski. We give a constructive definition of partial derivatives, study their properties, and employ them to develop a new algorithm for turning regular expressions into relatively small NFA and to provide certain improvements to Brzozowski's algorithm constructing DFA. We report on a prototype implementation of our algorithm constructing NFA and present some examples. Introduction In 1964 Janusz Brzozowski introduced word derivatives of regular expressions and suggested an elegant algorithm turning a regular expression r into a deterministic finite automata (DFA); the main point of the algorithm is that the word derivatives of r serve as states of the resulting DFA [5]. In the following years derivatives were recognized as a quite useful and productive tool. Conway [8] uses derivatives to present various comp...

We concider an extened algebra of regular events (languages) with intersection besides the usual operations. This algebra has the structure of a distributive lattice with monotonic operations; the latter property is crucial for some applications. We give a new complete Horn equational axiomatiztion of the algebra and develop some termrewriting techniques for constructing logical inferences of valid equations. A shorter version of this paper is to appear in the proceedings of Developments in Language Theory, Univ. of Turku, July 1993, published by World Scientific. The present version has been submitted for publication elsewhere. 1 Introduction In this paper we consider an extended algebra of regular events (languages) on a given alphabet with intersection besides the usual operations (union, concatenation, Kleene star, empty, and the regular unit). This algebra has the structure of a distributive lattice (join is union, meet is intersection) with only monotonic operations. The latte...