Parikh matrices and M-ambiguity sequence

Acta Informatica

Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words. Given a word, the Parikh matrix of that word elegantly computes the number of occurrences of certain predefined subwords in that word. In this paper, we compute the Parikh matrix of any word raised to an arbitrary power. Furthermore, we propose canonical decompositions of both Parikh matrices and words into normal forms. Finally, given a Parikh matrix, the relation between its normal form and the normal forms of words in the corresponding M-equivalence class is established.

2014

Word recognition using confusion matrices for computing similarities among phonemes Abstract. A word is a finite sequence of symbols. Parikh matrix of a word, introduced by Mateescu et al , has become an effective tool in the study of certain numerical properties of words based on subwords. There have been several investigations on various properties of Parikh matrices such as M-ambiguity, M-equivalence, subword equalities and inequalities, commutativity and so on. Recently, Parikh matrices of words that are images under certain morphisms have been studied for their properties. On the other hand, Parikh matrices of words involving a certain ratio property called weak-ratio property have been investigated by . Here we consider two special morphisms called Fibonacci and Tribonacci morphisms and obtain properties of Parikh matrices of images of binary words under these morphisms, utilizing the notion of weak-ratio property.

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.

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.

ArXiv, 2018

We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as core prints of the word. The length of the path to attain a core print of a general word is scrutinized. This paper also studies the class of square-free ternary words with respect to the Parikh matrix mapping, which is an extension of the classical Parikh mapping. It is shown that there are only finitely many matrix-equivalence classes of ternary words such that all words in each class are square-free. Finally, we employ square-free morphisms to generate infinitely many pairs of square-free ternary words that share the same Parikh matrix.

International Journal of Foundations of Computer Science, 2016

Parikh matrices have been a powerful tool in arithmetizing words by numerical quantities. However, the dependence on the ordering of the alphabet is inherited by Parikh matrices. Strong M -equivalence is proposed as a canonical alternative to M -equivalence to get rid of this undesirable property. Some characterization of strong M -equivalence for a restricted class of words is obtained. Finally, the existential counterpart of strong M -equivalence is introduced as well.

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.

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.

A word is a sequence of symbols. A scattered subword or simply a subword of the word is a subsequence of. Parikh matrix ()is an ingenius tool introduced by Mateescu et al (2001) to count certain subwords in a word. Various properties of Parikh matrices have been established. Two words and are said to be M-ambiguous or amiable if their Parikh matrices () and ()are the same. On the other hand a morphism is a mapping on words whose images ()are also words with the property that, () = () ()for given words and. Istrailmorphism (Istrail, 1977) is a specific kind of morphism on a set { , , }of three symbols. Using this morphism, M-ambiguity or amiability of words based on Parikh matrices is investigated by Atanasiu (2010). Parikh matrices of words that involve certain ratio-property are investigated by Subramanian et al (2009). Here we consider this kind of ratio-property in the context of Istrailmorphism and obtain certain properties of morphic images of words under Istrailmorphism. Using these properties,conditions are obtained for product of Parikh matrices of such morphic images under Istrailmorphismto commute.

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.