. 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...

State complexity is a descriptive complexity measure for regular languages. We investigate the problems related to the state complexity of regular languages and their operations. In particular, we compare the state complexity results on regular languages with those on finite languages.

The state complexity of basic operations on regular languages has been studied in [9--11]. Here we focus on finite languages. We show that the catenation of two finite languages accepted by an m- state and an n-state DFA, respectively, with m ? n is accepted by a DFA of (m \Gamma n + 3)2 n\Gamma2 \Gamma 1 states in the two-letter alphabet case, and this bound is shown to be reachable. We also show that the tight upperbounds for the number of states of a DFA that accepts the star of an n-state finite language is 2 n\Gamma3 + 2 n\Gamma4 in the two-letter alphabet case. The same bound for reversal is 3 \Delta 2 p\Gamma1 \Gamma 1 when n is even and 2 p \Gamma 1 when n is odd. Results for alphabets of an arbitrary size are also obtained. These upper-bounds for finite languages are strictly lower than the corresponding ones for general regular languages.

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...

A cover-automaton A of a finite language L ` \Sigma is a finite deterministic automaton (DFA) that accepts all words in L and possibly other words that are longer than any word in L. A minimal deterministic finite cover automaton (DFCA) of a finite language L usually has a smaller size than a minimal DFA that accept L. Thus, cover automata can be used to reduce the size of the representations of finite languages in practice. In this paper, we describe an efficient algorithm that, for a given DFA accepting a finite language, constructs a minimal deterministic finite coverautomaton of the language. We also give algorithms for the boolean operations on deterministic cover automata, i.e., on the finite languages they represent. Key words: Finite languages,Deterministic Finite Automata,Cover Language,Deterministic Cover Automata 1 Introduction Regular languages and finite automata are widely used in many areas such as lexical analysis, string matching, circuit testing, image compression,...

Abstract. Systems of language equations of the form {ϕ(X1,..., Xn) = ∅, ψ(X1,..., Xn) � = ∅} are studied, where ϕ, ψ may contain set-theoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X1,..., Xn) ⊂ L0. It is proved that the problem whether such an inequality has a solution is Σ2-complete, the problem whether it has a unique solution is in (Σ3 ∩Π3)\(Σ2 ∪Π2), the existence of a regular solution is a Σ1-complete problem, while testing whether there are finitely many solutions is Σ3-complete. The class of languages representable by their unique solutions is exactly the class of recursive sets, though a decision procedure cannot be algorithmically constructed out of an inequality, even if a proof of solution uniqueness is attached. 1

The most Human-Computer-Interfaces will be materialized by Graphical User Interfaces (GUI). With the growing complexity of the computer-based system, also their GUIs become more complex, accordingly making the test process more and more costly. The paper introduces a holistic view of fault modeling that can be carried out as a complementary step to system modeling, enabling a precise scalability of the test process, revealing many rationalization potential while testing. Appropriate formal notions and tools enable to introduce efficient algorithms to generate test cases systematically. Based on a basic coverage metric, test case selection can be carried out efficiently. The elements of the approach will be narrated by realistic examples which will be used also to validate the approach.

Many fuzzy automaton models have been introduced in the past. Here, we discuss two basic finite fuzzy automaton models, the Mealy and Moore types, for lexical analysis. We show that there is a remarkable difference between the two types. We consider that the latter is a suitable model for implementing lexical analysers. Various properties of fuzzy regular languages are reviewed and studied. A fuzzy lexical analyzer generator (FLEX) is proposed.

Language equations are equations where both the constants occurring in the equations and the solutions are formal languages. They have first been introduced in formal language theory, but are now also considered in other areas of computer science. In the present paper, we restrict the attention to language equations with one-sided concatenation, but in contrast to previous work on these equations, we allow not just union but all Boolean operations to be used when formulating them. In addition, we are not just interested in deciding solvability of such equations, but also in deciding other properties of the set of solutions, like its cardinality (finite, infinite, uncountable) and whether it contains least/greatest solutions. We show that all these decision problems are ExpTime-complete.

The paper investigates down-sets associated to well quasi orders. Of particular language-theoretic interest is the quasi order u s v (resp. u P v) of u