Algorithms for Glushkov K-graphs

TL;DR

This paper extends the characterization of Glushkov automata to weighted graphs over factorial semirings, providing algorithms under certain restrictions and exploring open questions for general cases.

Contribution

It generalizes the characterization of Glushkov graphs to weighted automata with weights in factorial semirings, under the star normal form restriction.

Findings

Characterization of weighted Glushkov graphs over factorial semirings

Algorithms developed under star normal form restriction

Open problem for general semirings and equivalence of K-expressions

Abstract

The automata arising from the well known conversion of regular expression to non deterministic automata have rather particular transition graphs. We refer to them as the Glushkov graphs, to honour his nice expression-to-automaton algorithmic short cut (On a synthesis algorithm for abstract automata, Ukr. Matem. Zhurnal, 12(2):147-156, 1960, In Russian). The Glushkov graphs have been characterized (P. Caron and D. Ziadi, Characterization of Glushkov automata. Theoret. Comput. Sci., 233(1-2):75-90, 2000) in terms of simple graph theoretical properties and certain reduction rules. We show how to carry, under certain restrictions, this characterization over to the weighted Glushkov graphs. With the weights in a semiring K, they are defined as the transition Glushkov K-graphs of the Weighted Finite Automata (WFA) obtained by the generalized Glushkov construction (P. Caron and M. Flouret, Glushkov construction for series: the non commutative case, Internat. J. Comput. Math., 80(4):457-472, 2003) from the K-expressions. It works provided that the semiring K is factorial and the K-expressions are in the so called star normal form (SNF) of Bruggeman-Klein (Regular expressions into finite automata, Theoret. Comput. Sci., 120(2):197-213, 1993) The restriction to the factorial semiring ensures to obtain algorithms. The restriction to the SNF would not be necessary if every K-expressions were equivalent to some with the same litteral length, as it is the case for the boolean semiring B but remains an open question for a general K.