Parikh Matrix Mapping Research Articles

Reducing the ambiguity of Parikh matrices

The Parikh matrix mapping allows us to describe words using matrices. Whilst compact, this description comes with a level of ambiguity since a single matrix may describe multiple words. In this paper, we investigate how considering the Parikh matrices of various transformations of a given word can decrease that ambiguity. More specifically, for any word, we study the Parikh matrix of its projection to a smaller alphabet as well as that of its Lyndon conjugate. Our results demonstrate that ambiguity can often be reduced using these concepts, and we give conditions on when they succeed.

On M-unambiguity of Parikh matrices

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The Parikh matrix mapping was introduced by Mateescu <em>et al</em>. 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 </span><em>M</em><span>-ambiguous words are crucial in Serbanuta's findings about the number of </span><em>M</em><span>-unambiguous prints. We will show that these words are in fact strongly </span><em>M</em><span>-ambiguous, thus suggesting a possible extension of Serbanuta’s work to the context of strong </span><span>M</span><span>-equivalence. In addition, initial results pertaining to a related conjecture by Serbanuta will be presented.</span></p></div></div></div>

Parikh q-Matrices and q-Ambiguous Words

The Parikh matrix mapping plays an important role in the study of words through numerical properties. The Parikh [Formula: see text]-matrix mapping, introduced by Egecioglu and Ibarra (2004) as an extension of the Parikh matrix mapping, maps words to matrices with polynomial entries in [Formula: see text] A word [Formula: see text] over an ordered alphabet [Formula: see text] is said to be [Formula: see text]-ambiguous if there exists another word [Formula: see text] over [Formula: see text] such that both the words have same Parikh [Formula: see text]-matrix. Here we derive several properties of [Formula: see text]-ambiguous words, in particular, for a binary alphabet.

Elementary matrix equivalence and core transformation graphs for Parikh matrices

The introduction of the Parikh matrix mapping by Mateescu et al. in 2001 gave rise to the injectivity problem, that is, the characterization of words having the same Parikh matrix. Certain elementary rewriting rules have been successful in solving this problem for the binary alphabet, yet they do not suffice for the general case. This paper studies these elementary rewriting rules exclusively for the ternary alphabet. Certain labeled graphs, termed as core transformation graphs, are introduced and used to study the structural changes in ternary words upon applications of these elementary rewriting rules. For an arbitrary core transformation graph, all possibilities of the labels on its edges are exhausted and, for each label, the pairs of vertices forming the corresponding edges are characterized. Finally, we obtain results relating core transformation graphs to elementarily matrix equivalent words.

On strongly M-unambiguous prints and Şerbǎnuţǎ's conjecture for Parikh matrices

In the combinatorial study of words, the Parikh matrix mapping was introduced by Mateescu et al. in 2001 as a natural expansion of the classical Parikh mapping. Solving the general injectivity problem of Parikh matrices remains as one of the most sought after triumph among researchers in this area of study. In this paper, we tackle this problem by extending Şerbǎnuţǎ's work regarding prints and M-unambiguity to the context of strong M-equivalence. Consequently, we obtain results on the finiteness of strongly M-unambiguous prints for any finite alphabet. Finally, a related conjecture by Şerbǎnuţǎ is conclusively addressed.

Parikh Matrices and Parikh Rewriting Systems

Since the introduction of the Parikh matrix mapping, its injectivity problem is on top of the list of open problems in this topic. In 2010 Salomaa provided a solution for the ternary alphabet in terms of a Thue system with an additional feature called counter. This paper proposes the notion of a Pa rikh rewriting system as a generalization and systematization of Salomaa’s result. It will be shown that every Parikh rewriting system induces a Thue system without counters that serves as a feasible solution to the injectivity problem.

On Core Words and the Parikh Matrix Mapping

Core of a binary word, recently introduced, is a refined way to characterize binary words having the same Parikh matrices, as well as bridging the connection between binary words and partitions of natural numbers. This paper continues the work by generalizing to higher alphabet. The core of a word as well as the relatived version is the essential part of a word that captures the key information of the word from the perspective of its Parikh matrix. Various nice properties of the cores and some interesting results regarding the M-equivalence classes of ternary words are obtained.

TWO-DIMENSIONAL DIGITIZED PICTURE ARRAYS AND PARIKH MATRICES

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained.

Enriching Parikh matrix mappings

In this paper, we define the Super-Parikh (S-Parikh) matrix mapping as an extension of the Parikh matrix mapping introduced by Mateescu et al. Like the Parikh matrix, the extension revolves around a certain type of square matrices, but instead of non-negative integers, its matrix-mapped elements are non-negative rationals (fractions). We study the basic properties of the newly defined formalism and later on we investigate the injectivity of the mapping. Also, we begin a search for the reverse mapping – that is a method for obtaining a word, given the S-Parikh matrix.

Several extensions of the Parikh matrix L-morphism

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.

PARIKH MATRICES, AMIABILITY AND ISTRAIL MORPHISM

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.

PARIKH MATRIX MAPPING AND LANGUAGES

Restricting the Parikh Matrix mapping to a language rather than to an alphabet rises a set of problems that seem interesting to us. Moreover, amiability (or M -equivalence as it is named by other authors) is exploited in order to characterise certain types of languages. The paper also proposes a series of results establishing relations between classes of languages defined by Parikh matrix mappings and the Chomsky hierarchy. Finally, a result related to word composition concludes the paper, showing that for every two arbitrary words there exist two other words such that their composition (two by two) is amiable.

Parikh matrices and amiable words

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of words with the same Parikh matrix; these words are called “amiable”. The presented results extend the results obtained in [A. Atanasiu, Binary amiable words, Int. J. Found. Comput. Sci. 18 (2) (2007) 387–400] for the binary case. In particular it is shown that all the words having the same Parikh matrix can be obtained one from another by applying only two types of transformations. Moreover, the mirrors of two amiable words are also amiable (thus forming a symmetrical class of words).

BINARY AMIABLE WORDS

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of the binary words having the same Parikh matrix; these words are called "amiable" Some results concerning the conditions when the equivalence classes of amiable words have more than one element, a characterization theorem concerning a graph associated to an equivalence class of amiable words, and some basic properties of a rank distance defined on these classes are the main subjects considered here.