Position Automaton Research Articles

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 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 (Brüggemann–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.

Subset construction complexity for homogeneous automata, position automata and ZPC-structures

The aim of this paper is to investigate how subset construction performs on specific families of automata. A new upper bound on the number of states of the subset-automaton is established in the case of homogeneous automata. The complexity of the two basic steps of subset construction, i.e. the computation of deterministic transitions and the set equality tests, is examined depending on whether the nondeterministic automaton is an unrestricted one, an homogeneous one, a position one or a ZPC-structure, which is an implicit construction for a position automaton.