RECOGNIZABILITY AND REPRESENTABILITY OF FORMAL TREE POWER SERIES
Abstract
WE PRESENT A THEORY OF RECOGNIZABLE FORMAL TREE POWER SERIES USING THE NOTION OF SYNTACTIC IK-Σ-ALGEBRA OF A TREE SERIES. THE RECOGNIZABILITY OF SUCH A SERIESIS RELATED WITH THE EXISTENCE OF A REALIZATION FOR THIS SERIES. WE PROVE THAT A TREE SERIES IS RECOGNIZABLE IF AND ONLY IF ITS SYNTACTIC ΙΚ-Σ-ALGEBRA IS OF FINITE TYPE. NEXT WE CHARACTERIZE RECOGNIZABLE FORMAL SERIES ON TREES BY MEANS OF MATRIX REPRESENTATIONS. FURTHERMORE, WE PROVE THAT THE IMAGE OF SUCH A REPRESENTATION WITH MINIMUM DIMENSION IS ISOMORPHIC TO SYNTACTIC ΙΚ-Σ-ALGEBRA OF THE INDUCED SERIES. FINALLY, WE SHOW THAT TWO MINIMAL MATRIX REPRESENTATIONS OF THESAME SERIES, ARE "SIMILAR". ALSO, USING REGULAR TREE GRAMMARS WEIGHTED OVER A SEMIRING, WE ESTABLISH KLEENE'S THEOREM IN THE CONTEST OF FORMAL TREE POWER SERIES.
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