A New Operator over Parikh Languages

Abstract

The characterization of M-equivalence for the Parikh matrices is a decade old open problem. This paper studies Parikh matrices and M-equivalence in relation to the s-shuffle operator for the binary alphabet. We also study the distance between images under the s-shuffle operator in a graph associated to the corresponding class of M-equivalent words.

Properties of Parikh Matrices of Binary Words Obtained by an Extension of a Restricted Shuffle Operator

“…In [3], a restricted s-shuffle operator on two binary words, denoted as SShuf, which is initially due to Pȃun [10], is utilized for deriving certain properties of Parikh matrices and in particular, the M −equivalence of words over a binary alphabet. Here we introduce an extension of the s−shuffle operator as follows:…”

“…The concept of Parikh matrix mapping, initiated by Mateescu et al [8], opened up a series of investigations on different problems on words and Parikh matrices (See, for example, [1,2,4,5,[12][13][14][15]). The well-known notion of Parikh vector [9] of a word which gives the number of each of the different symbols of the alphabet that define the word, has been of great significance, especially in formal language theory [11].…”

“…In the case of binary alphabet, very elegant characterizations for M −equivalence of words are known [5]. Very recently, Atanasiu and Teh [3] derived a number of properties, especially on M −equivalence of words, by developing an interesting connection between the Parikh matrix and a restricted shuffle operator on words, named SShuf f le in [3], initially considered in [10].…”

“…Here we introduce a special type of shuffle operator on binary words which we call as S m,n and which is an extension of SShuf f le in [3]. The shuffle operator S m,n , m, n ≥ 1, in operating on a pair of words (u, v) forms a word S m,n (u, v) by concatenating alternately consecutive factors of u and v, with the factors having length m for the former and length n for the latter.…”

Strong (2 ⋅ t) and Strong (3 ⋅ t) Transformations for Strong M-Equivalence

“…However, not every word is uniquely determined by its Parikh matrix. The injectivity problem, which asks for a characterization of words sharing the same Parikh matrix, has received extensive interest (for example, see [1][2][3][4][5][6][7][14][15][16][17][18][19][20][21][22]). This, together with the fact that Parikh matrices are dependent on the ordering of the alphabet, led to the introduction of strong M-equivalence in [18].…”