Universal Rewriting Rules for the Parikh Matrix Injectivity Problem

The injectivity problem of the Parikh matrix is closely related to the characterization of the M-equivalence. Current studies provide partial results such as a necessary condition of the M-equivalence or characterization of the M-equivalence over a binary or ternary alphabet. While these studies give rise to rewriting rules that construct M-equivalent strings of a given string over a binary or ternary alphabet, it has been open to designing general rewriting rules for M-equivalence independent of the alphabet size. We propose rewriting rules using exponent-strings, which are an extension of strings, as an intermediate representation. We introduce a special normal form for an exponent-string and prove that any string can be rewritten into an M-equivalent exponent-string in this special form. Then, we show that our rewriting rules characterize M-equivalence over an ordered alphabet of an arbitrary size.