Parikh Motivated Study on Repetitions in Words

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.

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.

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 Computer Science, 2018

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 Şerbǎnu¸tǎ'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 Şerbǎnu¸tǎ is conclusively addressed.

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.

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.

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.

The combinatorics of squares in a word depends on how the equivalence of halves of the square is defined. We consider Abelian squares, parameterized squares and order-preserving squares. The word uv is an Abelian (parameterized, order-preserving) square if u and v are equivalent in the Abelian (parameterized, order-preserving) sense. The maximum number of ordinary squares is known to be asymptotically linear, but the exact bound is still investigated. We present several results on the maximum number of distinct squares for nonstandard subword equivalence relations. Let SQ Abel (n, k) and SQ Abel (n, k) denote the maximum number of Abelian squares in a word of length n over an alphabet of size k, which are distinct as words and which are nonequivalent in the Abelian sense, respectively. We prove that SQ Abel (n, 2) = Θ(n 2 ) and SQ Abel (n, 2) = Ω(n 1.5 / log n). We also give linear bounds for parameterized and order-preserving squares for small alphabets: SQ param (n, 2) = Θ(n) and SQ op (n, O(1)) = Θ(n). As a side result we construct infinite words over the smallest alphabet which avoid nontrivial order-preserving squares and nontrivial parameterized cubes (nontrivial parameterized squares cannot be avoided in an infinite word).

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.

Fundamenta Informaticae, 2016

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 Parikh 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.