Journal ArticleDOI
From c-continuations to new quadratic algorithms for automaton synthesis
TLDR
An O(|E|2) space and time algorithm to compute the equation automaton is presented, based on the notion of canonical derivative which makes it possible to efficiently handle sets of word derivatives.Abstract:
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 Mirkin's prebases or Antimirov's 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.read more
Citations
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Journal ArticleDOI
Canonical derivatives, partial derivatives and finite automaton constructions
TL;DR: The 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.
Journal Article
From Mirkin's Prebases to Antimirov's Word Partial Derivatives
TL;DR: A proof of the fact that two notions related to regular expressions, the prebases due to Mirkin and the partial derivatives introduced by Antimirov lead to the construction of identical nondeterministic automata recognizing the language of a given regular expression is given.
Proceedings Article
An efficient computation of the equation K-automaton of a regular K-expression
TL;DR: An extension to regular K-expressions of the notion of c-continuation that is introduced to compute the equation automaton of a regular expression as a quotient of its position automaton is described.
Book ChapterDOI
New Finite Automaton Constructions Based on Canonical Derivatives
TL;DR: New theoretical results are presented allowing to compute the equation automaton in quadratic space and time, improving by a cubic factor Antimirov's construction, 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.
Journal ArticleDOI
An Efficient Computation of the Equation K-automaton of a Regular K-expression
TL;DR: In this article, a quadratic algorithm for computing the equation K-automaton of a regular K-expression as defined by Lombardy and Sakarovitch is described. But this algorithm is based on an extension to regular Kexpressions of the notion of c-continuation that was introduced to compute the equation automaton as a quotient of its position automaton.
References
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Journal ArticleDOI
Three partition refinement algorithms
Robert Paige,Robert E. Tarjan +1 more
TL;DR: This work presents improved partition refinement algorithms for three problems: lexicographic sorting, relational coarsest partition, and double lexical ordering that uses a new, efficient method for unmerging two sorted sets.