Several extensions of the Parikh matrix L-morphism

Abstract

The Parikh matrix mapping is a morphism assigning to each word w over a k-letter alphabet a (k+1)×(k+1) upper triangular matrix with entries expressing the number of occurrences in w of some specific subwords. To tackle the problem of ambiguity of this mapping two new mappings have been proposed in literature, assigning to words matrices with polynomial entries (q-matrices). One is a more subtle, but still ambiguous morphism, the other is unambiguous but not a morphism. We show that the former mapping can be extended to match even a fairly general extension of the original Parikh matrix morphism. Then we introduce an unambiguous q-matrix morphism based on the same general Parikh matrix mapping. Finally, we consider the problem of incomplete information on word symbols and show that the general Parikh matrix mapping can be further extended to deal with counts of fuzzy subword occurrences.