Rogério Reis

Enumeration and generation with a string automata representation

In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata (ICDFAs) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFAs but, to some extent, the set of regular languages. An exact generation algorithm can be used to partition the set of ICDFAs in order to parallelize the counting of minimal automata (and thus of regular languages). We present also a uniform random generator for ICDFAs that uses a table of pre-calculated values. Based on the same table it is also possible to obtain an optimal coding for ICDFAs.

ON THE AVERAGE SIZE OF GLUSHKOV AND PARTIAL DERIVATIVE AUTOMATA

In this paper, the relation between the Glushkov automaton and the partial derivative automaton of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of . Here we present a new quadratic construction of that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of to the number of states of which is about ½ for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in , which we then use to get an average case approximation. In conclusion, assymptotically, and for large alphabets, the size of is half the size of the . This is corroborated by some experiments, even for small alphabets and small regular expressions.

ANTIMIROV AND MOSSES'S REWRITE SYSTEM REVISITED

Antimirov and Mosses proposed a rewrite system for deciding the equivalence of two (extended) regular expressions. They argued that this method could lead to a better average-case algorithm than those based on the comparison of the equivalent minimal deterministic finite automata. In this paper we present a functional approach to that method, prove its correctness, and give some experimental comparative results. Besides an improved functional version of Antimirov and Mossess algorithm, we present an alternative one using partial derivatives. Our preliminary results lead to the conclusion that, indeed, these methods are feasible and, most of the time, faster than the classical methods.

Symmetric Groups and Quotient Complexity of Boolean Operations

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L′ are binary regular languages with quotient complexities m and n, and that the subgroups of permutations in the transition semigroups of the minimal deterministic automata accepting L and L′ are the symmetric groups S m and S n of degrees m and n, respectively. Denote by ∘ any binary boolean operation that is not a constant and not a function of one argument only. For m,n ≥ 2 with \((m,n)\not \in \{(2,2),(3,4),(4,3),(4,4)\}\) we prove that the quotient complexity of L ∘ L′ is mn if and only either (a) \(m\not= n\) or (b) m = n and the bases (ordered pairs of generators) of S m and S n are not conjugate. For (m,n) ∈ {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.

Interactive manipulation of regular objects with FAdo

FAdo1 is an ongoing project which aims the development of an interactive environment for symbolic manipulation of formal languages. In this paper we focus in the description of interactive tools for teaching and assisting research on regular languages, and in particular finite automata and regular expressions. Those tools implement most standard automata operations, conversion between automata and regular expressions, and word recognition. We illustrate their use in training and automatic assessment. Finally we present a graphical environment for editing and interactive visualisation.

Series-Parallel Automata and Short Regular Expressions

Computing short regular expressions equivalent to a given finite automaton is a hard task. In this work we present a class of acyclic automata for which it is possible to obtain in time O(n

A Hitchhiker's Guide to descriptional complexity through analytic combinatorics

^{2}

Acyclic automata with easy-to-find short regular expressions

log n) an equivalent regular expression of size O(n). A characterisation of this class is made using properties of the underlying digraphs that correspond to the series-parallel digraphs class. Using this characterisation we present an algorithm for the generation of automata of this class and an enumerative formula for the underlying digraphs with a given number of vertices.

Incomplete Transition Complexity of Some Basic Operations

Nowadays, increasing attention is being given to the study of the descriptional complexity in the average case. Although the underlying theory for such a study seems intimidating, one can obtain interesting results in this area without too much effort. In this gentle introduction we take the reader on a journey through the basic analytical tools of that theory, giving some illustrative examples using regular expressions. Additionally, new asymptotic average-case results for several @e-NFA constructions are presented, in a unified framework. It turns out that, asymptotically, and in the average case, the complexity gap between the several constructions is significantly larger than in the worst case. Furthermore, one of the @e-NFA constructions approaches the corresponding @e-free NFA construction, asymptotically and on average.

Regular Ideal Languages and Synchronizing Automata

Computing short regular expressions equivalent to a given finite automaton is a hard task. We present a class of acyclic automata for which it is easy-to-find a regular expression that has linear size. We call those automata UDR. A UDR automaton is characterized by properties of its underlying digraph. We give a characterisation theorem and an efficient algorithm to determine if an acyclic automaton is UDR, that can be adapted to compute an equivalent short regular expression.