Constants and label-equivalence: A decision procedure for reflexive regular splicing languages

A structural characterization of reflexive splicing languages has been recently given in [P. Bonizzoni, C. De Felice, R. Zizza, The structure of reflexive regular splicing languages via Schützenberger constants, Theoretical Computer Science 334 (2005) 71–98] and [P. Bonizzoni, G. Mauri, Regular splicing languages and subclasses, Theoretical Computer Science 340 (2005) 349–363] showing surprising connections between long standing notions in formal language theory, the syntactic monoid and Schützenberger constant and the splicing operation. In this paper, we provide a procedure to decide whether a regular language is a reflexive splicing language, based on the above-mentioned characterization that is given in terms of a finite set of constants for the language. The procedure relies on the notion of label-equivalence that induces a finite refinement of the syntactic monoid of a regular language L. A finite set of representatives for label-equivalent classes of constant words in L is defined and it is proved that such a finite set provides the splice sites of splicing rules generating language L.