Counting Subwords in Circular Words and Their Parikh Matrices

The word inference problem is to determine languages such that the information on the number of occurrences of those subwords in the language can uniquely identify a word. A considerable amount of work has been done on this problem, but the same cannot be said for circular words despite growing interests on the latter due to their applications—for example, in splicing systems. Meanwhile, Parikh matrices are useful tools and well established in the study of subword occurrences. In this work, we propose two ways of counting subword occurrences in circular words. We then extend the idea of Parikh matrices to the context of circular words and investigate this extension. Motivated by the word inference problem, we study ambiguity in the identification of a circular word by its Parikh matrix. Accordingly, two rewriting rules are developed to generate ternary circular words which share the same Parikh matrix.

[1]  Rani Siromoney,et al.  Circular DNA and Splicing Systems , 1992, ICPIA.

[2]  Wen Chean Teh,et al.  On strongly M-unambiguous prints and Şerbǎnuţǎ's conjecture for Parikh matrices , 2018, Theor. Comput. Sci..

[3]  Miroslav Dudík,et al.  Reconstruction from subsequences , 2003, J. Comb. Theory, Ser. A.

[4]  James D. Currie,et al.  Circular Words Avoiding Patterns , 2002, Developments in Language Theory.

[5]  Wen Chean Teh,et al.  On a conjecture about Parikh matrices , 2016, Theor. Comput. Sci..

[6]  Virgil Nicolae Serbanuta.  On Parikh Matrices, Ambiguity, and PRINTS , 2009, Int. J. Found. Comput. Sci..

[7]  Aleksi Saarela.  Separating many words by counting occurrences of factors , 2019, DLT.

[8]  Arto Salomaa.  Independence of certain quantities indicating subword occurrences , 2006, Theor. Comput. Sci..

[9]  Paola Bonizzoni,et al.  A characterization of (regular) circular languages generated by monotone complete splicing systems , 2010, Theor. Comput. Sci..

[10]  Our Correspondent in Molecular Biology.  Circular DNA , 1967, Nature.

[11]  Ion Petre,et al.  Parikh matrices and amiable words , 2008, Theor. Comput. Sci..

[12]  Paul K. Stockmeyer,et al.  Reconstruction of sequences , 1991, Discret. Math..

[13]  Arto Salomaa,et al.  Subword histories and Parikh matrices , 2004, J. Comput. Syst. Sci..

[14]  Alexandru Mateescu.  Algebraic Aspects of Parikh Matrices , 2004, Theory Is Forever.

[15]  Wen Chean Teh,et al.  Parikh word representability of bipartite permutation graphs , 2020, Discret. Appl. Math..

[16]  Wen Chean Teh,et al.  Parikh Determinants , 2019, WORDS.

[17]  D. Sean Fitzpatrick.  There are binary circular cube-free words of length n contained within the Thue-Morse word for all positive integers n , 2005, Ars Comb..

[18]  Traian-Florin Serbanuta,et al.  Injectivity of the Parikh Matrix Mappings Revisited , 2006, Fundam. Informaticae.

[19]  Wen Chean Teh,et al.  Parikh matrices for powers of words , 2018, Acta Informatica.

[20]  Arto Salomaa.  Criteria for the matrix equivalence of words , 2010, Theor. Comput. Sci..

[21]  K. G. Subramanian,et al.  Product of Parikh Matrices and Commutativity , 2012, Int. J. Found. Comput. Sci..

[22]  K. G. Subramanian,et al.  On Parikh Matrices , 2009, Int. J. Found. Comput. Sci..

[23]  Michel Rigo,et al.  Relations on words , 2016, ArXiv.

[24]  Florin Manea,et al.  k-Spectra of Weakly-c-Balanced Words , 2019, DLT.

[25]  Arto Salomaa,et al.  A sharpening of the Parikh mapping , 2001, RAIRO Theor. Informatics Appl..

[26]  K. G. Subramanian,et al.  Two-Dimensional Digitized Picture Arrays and Parikh Matrices , 2013, Int. J. Found. Comput. Sci..

[27]  Jamie Simpson.  Palindromes in circular words , 2014, Theor. Comput. Sci..

[28]  Wen Chean Teh,et al.  Elementary matrix equivalence and core transformation graphs for Parikh matrices , 2018, Discret. Appl. Math..

[29]  J An,et al.  Characterization of a Word by Its Subwords , .

[30]  Paola Bonizzoni,et al.  On the power of circular splicing , 2005, Discret. Appl. Math..

[31]  Arseny M. Shur.  On Ternary Square-free Circular Words , 2010, Electron. J. Comb..

[32]  Robert Mercas,et al.  Reducing the Ambiguity of Parikh Matrices , 2020, LATA.

[33]  W. C. Teh,et al.  M-ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited , 2019, Bulletin of the Malaysian Mathematical Sciences Society.