Jean-Marc Champarnaud

Canonical derivatives, partial derivatives and finite automaton constructions

Let E be a regular expression. Our aim is to establish a theoretical relation between two well-known automata recognizing the language of E, namely the position automaton PE constructed by Glushkov or McNaughton and Yamada, and the equation automaton EE constructed by Mirkin or Antimirov. We define the notion of c-derivative (for canonical derivative) of a regular expression E and show that if E is linear then two Brzozowskis derivatives of E are aci-similar if and only if the corresponding c-derivatives are identical. It allows us to represent the Berry-Sethis set of continuations of a position by a unique c-derivative, called the c-continuation of the position. Hence the definition of CE, the c-continuation automaton of E, whose states are pairs made of a position of E and of the associated c-continuation. If states are viewed as positions, CE is isomorphic to PE. On the other hand, a partial derivative, as defined by Antimirov, is a class of c-derivatives for some equivalence relation, thus CE reduces to EE. Finally CE makes it possible to go from PE to EE, while this cannot be achieved directly (from the state graphs). These theoretical results lead to an O(|E|2) space and time algorithm to compute the equation automaton, where |E| is the size of the expression. This is the complexity of the most efficient constructions yielding the position automaton, while the size of the equation automaton is not greater and generally much smaller than the size of the position automaton.

NFA reduction algorithms by means of regular inequalities

We present different techniques for reducing the number of states and transitions in nondeterministic automata. These techniques are based on the two preorders over the set of states, related to the inclusion of left and right languages. Since their exact computation is NP-hard, we focus on polynomial approximations which enable a reduction of the NFA all the same. Our main algorithm relies on a first approximation, which can be easily implemented by means of matrix products with an O(mn3) time complexity, and optimized to an O(mn) time complexity, where m is the number of transitions and n is the number of states. This first algorithm appears to be more efficient than the known techniques based on equivalence relations as described by Lucian Ilie and Sheng Yu. Afterwards, we briefly describe some more accurate approximations and the exact (but exponential) calculation of these preorders by means of determinization.

Random generation of DFAs

This document gives a generalization on the alphabet size of the method that is described in Nicauds thesis for randomly generating complete DFAs. First, we recall some properties of m-ary trees and we give a bijection between the set of m-ary trees and the set R(m, n) of generalized tuples. We show that this bijection can be built on any total prefix order on Σ*, Then we give the relations that exist between the elements of R(m,n) and complete DFAs built on an alphabet of size greater than 2. We give algorithms that allow us to randomly generate accessible complete DFAs. Finally, we provide experimental results that show that most of the accessible complete DFAs built on an alphabet of size greater than 2 are minimal.

Partial derivatives of an extended regular expression

The notion of expression derivative due to Brzozowski leads to the construction of a deterministic automaton from an extended regular expression, whereas the notion of partial derivative due to Antimirov leads to the construction of a non-deterministic automaton from a simple regular expression. In this paper, we generalize Antimirov partial derivatives to regular expressions extended to complementation and intersection. For a simple regular expression with n symbols, Antimirov automaton has at most n+1 states. As far as an extended regular expression is concerned, we show that the number of states can be exponential.

FROM C-CONTINUATIONS TO NEW QUADRATIC ALGORITHMS FOR AUTOMATON SYNTHESIS

Two classical non-deterministic automata recognize the language denoted by a regular expression: the position automaton which deduces from the position sets defined by Glushkov and McNaughton–Yamada, and the equation automaton which can be computed via Mirkins prebases or Antimirovs partial derivatives. Let |E| be the size of the expression and ‖E‖ be its alphabetic width, i.e. the number of symbol occurrences. The number of states in the equation automaton is less than or equal to the number of states in the position automaton, which is equal to ‖E‖+1. On the other hand, the worst-case time complexity of Antimirov algorithm is O(‖E‖3· |E|2), while it is only O(‖E‖·|E|) for the most efficient implementations yielding the position automaton (Bruggemann–Klein, Chang and Paige, Champarnaud et al.). We present an O(|E|2) space and time algorithm to compute the equation automaton. It is based on the notion of canonical derivative which makes it possible to efficiently handle sets of word derivatives. By the way, canonical derivatives also lead to a new O(|E|2) space and time algorithm to construct the position automaton.

A New Quadratic Algorithm to Convert a Regular Expression into an Automaton

We present a new sequential algorithm to convert a regular expression into its Glushkov automaton. This conversion runs in quadratic time, so it has the same time complexity as the Bruggemann-Klein algorithm and the Chang and Paige one. It provides, however, a representation of the Glushkov automaton that needs only linear space.

An Efficient Computation of the Equation K-automaton of a Regular K-expression

The aim of this paper is to describe a quadratic algorithm to compute the equation K-automaton of a regular K-expression as defined by Lombardy and Sakarovitch. Our construction is based on an extension to regular K-expressions of the notion of c-continuation that we introduced to compute the equation automaton of a regular expression as a quotient of its position automaton.

New Finite Automaton Constructions Based on Canonical Derivatives

Two classical constructions to convert a regular expression into a finite non-deterministic automaton provide complementary advantages: the notion of position of a symbol in an expression, introduced by Glushkov and McNaugthon-Yamada, leads to an efficient computation of the position automaton (there exist quadratic space and time implementations w.r.t. the size of the expression), whereas the notion of derivative of an expression w.r.t. a word, due to Brzozowski, and generalized by Antimirov, yields a small automaton. The number of states of this automaton, called the equation automaton, is less than or equal to the number of states of the position automaton, and in practice it is generally much smaller. So far, algorithms to build the equation automaton, such as Mirkins or Antimirovs ones, have a high space and time complexity. The aim of this paper is to present new theoretical results allowing to compute the equation automaton in quadratic space and time, improving by a cubic factor Antimirovs construction. These results lay on the computation of a new kind of derivative, called canonical derivative, which makes it possible to connect the notion of continuation in a linear expression due to Berry and Sethi, and the notion of partial derivative of a regular expression due to Antimirov. A main interest of the notion of canonical derivative is that it leads to an efficient computation of the equation automaton via a specific reduction of the position automaton.

UNAVOIDABLE SETS OF CONSTANT LENGTH

A set of words X is called unavoidable on a given alphabet A if every infinite word on A has a factor in X. For k,q≥1, let c(k,q) be the number of conjugacy classes of words of length k on q letters. An unavoidable set of words of length k on q symbols has at least c(k,q) elements. We show that for any k,q≥1, there exists an unavoidable set of words of length k on q symbols having c(k,q) elements.

Computing the Equation Automaton of a Regular Expression in Space and Time

Let E be a regular expression the size of which is s. Mirkins prebases and Antimirovs partial derivatives lead to the construction of the same automaton, called the equation automaton of E. The number of states in this automaton is less than or equal to the number of states in the position automaton. On the other hand, it can be computed by Antimirovs algorithm with an O(s5) time complexity, whereas there exist O(s2) implementations for the position automaton. We present an O(s2) space and time algorithm to compute the equation automaton. It is based on the notion of canonical derivative which is related both to word and partial derivatives. This work is tightly connected to pattern matching area since the aim is, given a regular expression, to produce an as small as possible recognizer with the best space and time complexity.