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Parikh Matrices for Powers of Words [article]

Adrian Atanasiu, Ghajendran Poovanandran, Wen Chean Teh
2018 arXiv   pre-print
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.  ...  Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words.  ...  Acknowledgement The second and third authors gratefully acknowledge support for this research by a Research University Grant No. 1011/PMATHS/8011019 of Universiti Sains Malaysia.  ... 
arXiv:1808.04102v1 fatcat:fr5zjvtqx5an7dktebx7l7mbxu

Algebraic Properties of Parikh Matrices of Binary Picture Arrays

Somnath Bera, Sastha Sriram, Atulya K. Nagar, Linqiang Pan, K. G. Subramanian
2020 Journal of Mathematics  
Here, we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness, and a restricted shuffle operator extending the corresponding notions studied  ...  We also obtain properties of Parikh matrices of arrays formed by certain geometric operations.  ...  Pan were partially supported by the National Natural Science Foundation of China (61772214).  ... 
doi:10.1155/2020/3236405 fatcat:ip5dtrlmzfd5xhemdugs3y554u

Parikh matrix on the Context-Free Grammar for Natural Languages

Amrita Bhattacharjee
2016 International Journal Of Engineering And Computer Science  
In this paper Parikh Matrices over context-free languages are investigated. Context-free grammars for Natural languages are a developing area of investigation.  ...  As a case study in this paper Bengali letters, Bengali words and Bengali sentences are studied by using Parikh matrix.  ...  The combination of Parikh matrix and this extension give a more powerful tool for the study of algebraic properties of words.  ... 
doi:10.18535/ijecs/v5i11.13 fatcat:sex2spbwsvedxh4twnge43nzwy

Connections between subwords and certain matrix mappings

Arto Salomaa
2005 Theoretical Computer Science  
Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of the theory of words.  ...  This paper continues the investigation of Parikh matrices and subword occurrences.  ...  Acknowledgements The author is grateful to the referee for useful suggestions.  ... 
doi:10.1016/j.tcs.2005.03.024 fatcat:uh25jsbjqfhivekwpssmyqg7ui

Reducing the Ambiguity of Parikh Matrices [chapter]

Jeffery Dick, Laura K. Hutchinson, Robert Mercaş, Daniel Reidenbach
2020 Lecture Notes in Computer Science  
More specifically, for any word, we study the Parikh matrix of its Lyndon conjugate as well as that of its projection to a smaller alphabet.  ...  This work looks at how considering the Parikh matrices of various transformations of a given word can decrease that ambiguity.  ...  From Corollary 1, we learn that P-Parikh matrices cannot reduce the ambiguity of a Parikh matrix that describes words in a binary alphabet, but are very powerful when it comes to reducing the ambiguity  ... 
doi:10.1007/978-3-030-40608-0_28 fatcat:btqhbzqvtngt3aftjkru2mzche

A sharpening of the Parikh mapping

Alexandru Mateescu, Arto Salomaa, Kai Salomaa, Sheng Yu
2001 RAIRO - Theoretical Informatics and Applications  
We characterize the matrix products and establish also an interesting interconnection between mirror images of words and inverses of matrices.  ...  In this paper we introduce a sharpening of the Parikh mapping and investigate its basic properties. The new mapping is based on square matrices of a certain form.  ...  For instance, in view of Remark 2.3, we might look for a "natural" family of sets of matrices such that the set of Parikh matrices resulting from a context-free language is in this family.  ... 
doi:10.1051/ita:2001131 fatcat:k6xqvcmhxjgstlozmp5a2xta5m

Independence of certain quantities indicating subword occurrences

Arto Salomaa
2006 Theoretical Computer Science  
Parikh matrices recently introduced have these quantities as their entries.  ...  The numerical quantity investigated in this paper is |w| u , the number of occurrences of a word u as a (scattered) subword of a word w.  ...  Acknowledgements The useful suggestions of two referees are gratefully acknowledged. In particular, the second referee sharpened the proof of Lemma 3.  ... 
doi:10.1016/j.tcs.2006.06.025 fatcat:p3cckqjtxjhijgsuuh6h577asm

Some Alternative Ways to Find M-Ambiguous Binary Words Corresponding to a Parikh Matrix

Amrita Bhattacharjee, Bipul Syam Purkayastha
2014 International Journal on Computational Science & Applications  
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.  ...  The combination of Parikh matrix and this extension give a more powerful tool for the study of algebraic properties of words.  ... 
doi:10.5121/ijcsa.2014.4106 fatcat:zh4rr32znvgmfmyxvn4vvysxli

Subword histories and associated matrices

Arto Salomaa
2008 Theoretical Computer Science  
Simple polynomial formulas for computing the value of a subword history for arbitrary powers of a word are obtained.  ...  Reducing subword histories to linear ones, as well as the recently introduced Parikh matrices, will be important tools.  ...  Acknowledgements We want to thank the two referees for many useful suggestions. The important contribution of one of the referees was already acknowledged in the remark in Section 3.  ... 
doi:10.1016/j.tcs.2008.05.023 fatcat:x22w6preqzdjjoxcblbvp7w4s4

Extending Parikh matrices

Traian-Florin Şerbănuţă
2004 Theoretical Computer Science  
We introduce the notion of Parikh matrix induced by a word, a natural extension to the notion of Parikh matrix and prove a set of properties for this kind of matrices.  ...  We show that combining properties from both we obtain a more powerful tool for proving algebraic properties of words.  ...  general case of Parikh matrices induced by words.  ... 
doi:10.1016/s0304-3975(03)00396-7 fatcat:mv6munw3eva65gzbkop7qwucuq

On some decidability problems for HDOL systems with nonsingular Parikh matrices

Keijo Ruohonen
1979 Theoretical Computer Science  
The problem of decidability of ultimate equivalence, emptiness of intersection and finiteness of intersection, for HDOL sequences the generating morphisms of which have nonsingular Parikh matrices is investigated  ...  The Grst and the third of these decidability problems are shown to be closely connected with the equivalence problem for these sequences.  ...  For basic facts about (H)DOL systems and sequences the reader is referred to [8] . By IPI we denote the length of the word P and by [P] its Parikh vector.  ... 
doi:10.1016/0304-3975(79)90037-9 fatcat:bypis4wz6rairl4bcc5hzm4zfq

A note on equal powers of word morphisms

K. Ruohonen
1977 RAIRO Informatique théorique  
Acknowledgement : I would like thank an anonymous référée for his comments which helped to simplify an earlier version of this note.  ...  A NOTE ON EQUAL POWERS OF WORD MORPHISMS 211 REMARK 2 : A generalization of our methods to prove the above kind of results for endomorphisms (or even monomorphisms) with singular Parikh matrices does  ...  RESULTS We begin with a theorem on equal powers of integer matrices.  ... 
doi:10.1051/ita/1977110302071 fatcat:c4w253q76jadvhtgrlvcwfwlxm

Subword balance, position indices and power sums

Arto Salomaa
2010 Journal of computer and system sciences (Print)  
We establish some characterizations, connections with Parikh matrices, and connections with power sums. One particular emphasis concerns the effect of morphisms and iterated morphisms on words.  ...  In this paper, we investigate various ways of characterizing words, mainly over a binary alphabet, using information about the positions of occurrences of letters in words.  ...  Acknowledgment The detailed comments and suggestions of a referee significantly improved the paper.  ... 
doi:10.1016/j.jcss.2010.04.012 fatcat:c3sam2fgqnfjjllyosl5tkxxiy

Parikh Matrices and Strong M-Equivalence [article]

Wen Chean Teh
2015 arXiv   pre-print
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.  ...  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.  ...  Finally, the author gratefully acknowledges support for this research by a short term grant No. 304/PMATHS/6313077 of Universiti Sains Malaysia.  ... 
arXiv:1501.07354v2 fatcat:fnzbdx6qtvbmxlfwdh7vigk5ji

Page 8169 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Finally, some observations about the structure of Parikh matrices are made. Peter R. J.  ...  The authors establish a normal form (involving — and + only) for SHs and show the decidability of the equivalence problem for SHs; they use minors of Parikh matrices to prove some inequal- ities between  ... 
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