Parikh matrices and 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.

Theoretical Computer Science, 2005

Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of the theory of words. In particular, many inequalities between (scattered) subword occurrences have been obtained as consequences of the properties of the matrices. This paper continues the investigation of Parikh matrices and subword occurrences. In particular, we study certain inequalities, as well as information about subword occurrences sufficient to determine the whole word uniquely. Some algebraic considerations, facts about forbidden subwords, as well as some open problems are also included.

Journal of Physics: Conference Series

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked new investigations in the theory of formal languages by various researchers, among whom is Şerbǎnuţǎ. A decade-old conjecture by Şerbǎnuţǎ on M-ambiguous words was recently disproved and this led to new possibilities in the study of such words. In this paper, we will show that by selectively duplicating letters in a word, the M-ambiguity of the resulting words can be continuously altered both finitely and infinitely many times. The changes in the M-ambiguity of those words are then presented in sequences. Finally, a study on the periodicity of such sequences is posed as our future work.

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 aim is to reduce the number of amiable words using a morphism which provides additional information about them.

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.

Indonesian Journal of Combinatorics

The Parikh matrix mapping was introduced by Mateescu et al. in 2001 as a canonical generalization of the classical Parikh mapping. The injectivity problem of Parikh matrices, even for ternary case, has withstanded numerous attempts over a decade by various researchers, among whom is Serbanuta. Certain M-ambiguous words are crucial in Serbanuta's findings about the number of M-unambiguous prints. We will show that these words are in fact strongly M-ambiguous, thus suggesting a possible extension of Serbanuta’s work to the context of strong M-equivalence. In addition, initial results pertaining to a related conjecture by Serbanuta will be presented.

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.

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

Theoretical Informatics and Applications, 2009

Parikh matrices have become a useful tool for investigation of subword structure of words. Several generalizations of this concept have been considered. Based on the concept of formal power series, we describe a general framework covering most of these generalizations. In addition, we provide a new characterization of binary amiable wordswords having a common Parikh matrix.

Bulletin of the Malaysian Mathematical Sciences Society

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Şerbǎnuţǎ. Recently, a decade-old conjecture by Şerbǎnuţǎ on the M-ambiguity of words was disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of those words are then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word results in an M-ambiguity sequence that is eventually periodic.