Generalizations of Parikh mappings

Journal of Computer and System Sciences, 2004

Parikh matrices recently introduced give much more information about a word than just the number of occurrences of each letter. In this paper we introduce the closely related notion of a subword history and obtain a sequence of general results: elimination of products, decidability of equivalence, and normal form. We also investigate overall methods for proving the validity of such results. A general inequality of ''Cauchy type'' for subword occurrences is established. r

A new way to transform amiable words over a ternary ordered alphabet is defined and analysed: under some conditions, several subwords of a word can be modified in the same step, obtaining another amiable word. This transformation cannot be always decomposed in sequential transformations as defined in [1], and gives almost the same final results as in and , but without any output from the set of amiable words defined by a Parikh matrix. The class of amiable languages characterized by the number of subwords modified in the same step is defined and studied partially in the second part of the paper.

2018

The Parikh matrix of a word $w$ over an alphabet $\{a_1, \cdots , a_k \}$ with an ordering $a_1 < a_2 < \cdots a_k,$ gives the number of occurrences of each factor of the word $a_1 \cdots a_k$ as a (scattered) subword of the word $w.$ Two words $u,v$ are said to be $M-$equivalent, if the Parikh matrices of $u$ and $v$ are the same. On the other hand properties of image words under different morphisms have been studied in the context of subwords and Parikh matrices. Here an extension to three letters, introduced by S$\acute{e}\acute{e}$bold (2003), of the well-known Thue morphism on two letters, is considered and properties of Parikh matrices of morphic images of words are investigated. The significance of the contribution is that various classes of binary words are obtained whose images are $M-$equivalent under this extended morphism.

In this paper we introduce an extension of the Parikh mapping and investigate some of its basic properties. The extension is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix product gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting interconnection between mirror images of words and inverses of matrices.

International Journal on Computational Science & Applications, 2014

Parikh matrix of a word gives numerical information of the word in terms of its subwords. In this Paper an algorithm for finding Parikh matrix of a binary word is introduced. With the help of this algorithm Parikh matrix of a binary word, however large it may be can be found out. M-ambiguous words are the problem of Parikh matrix. In this paper an algorithm is shown to find the M-ambiguous words of a binary ordered word instantly. We have introduced a system to represent binary words in a two dimensional field. We see that there are some relations among the representations of M-ambiguous words in the two dimensional field. We have also introduced a set of equations which will help us to calculate the M-ambiguous words.

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of words having the same Parikh matrix; these words are called "amiable" or "M - equivalent". The presented paper uses the results obtained in [3] for the binary case. The aim is to distinguish the amiable words by using a morphism that provides additional information about them. The morphism proposed here is the Istrail morphism.

International Journal of Foundations of Computer Science, 2018

We introduce an extension of the restricted shuffle operator on binary words considered by Atanasiu and Teh (2016). We then derive properties on Parikh matrix equivalence of words over a binary alphabet based on this extended shuffle operator and a weak-ratio property of words. We also examine the recently introduced concept of core Parikh matrix equivalence of binary words in the context of the restricted shuffle operator.

In this paper we investigate the injectivity of the Parikh matrix mapping. This research is done mainly on the binary alphabet. We identify a family of binary words, refered to as "palindromicly amicable", such that two such words are palindromicly amicable if and only if they have the same image by the Parikh matrix mapping. Some other related problems are discussed, too.

International Journal of Foundations of Computer Science, 2010

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of words having the same Parikh matrix; these words are called "amiable" or "M - equivalent". The presented paper uses the results obtained in [3] for the binary case. The aim is to distinguish the amiable words by using a morphism that provides additional information about them. The morphism proposed here is the Istrail morphism.

Journal of Computer and System Sciences, 2013

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.