Counting Subwords in Circular Words and Their Parikh Matrices

Natural Computing, 2010

Circular splicing has been introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we focus on the relationship between regular circular languages and languages generated by finite circular splicing systems. We survey the known results towards a characterization of the intersection between these two classes and provide new contributions on the open problem of finding this characterization. First, we exhibit a non-regular circular language generated by a circular simple system thus disproving a known result in this area. Then we give new results related to a restrictive class of circular splicing systems, the marked systems. Precisely, we review in a graph theoretical setting the recently obtained characterization of marked systems generating regular circular languages. In particular, we define a slight variant of marked systems, that is the g-marked systems, and we introduce the graph associated with a g-marked system. We show that a g-marked system generates a regular circular language if and only if its associated graph is a cograph. Furthermore, we prove that the class of g-marked systems generating regular circular languages is closed under a complement operation applied to systems. We also prove that marked systems with self-splicing generate only regular circular languages.

International Journal on Computational Science & Applications, 2014

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. In this paper an algorithm is shown to find the M-ambiguous words of a binary ordered word instantly. We have introduced a system to represent binary words in a two dimensional field. We see that there are some relations among the representations of M-ambiguous words in the two dimensional field. We have also introduced a set of equations which will help us to calculate the M-ambiguous words.

Discrete Applied Mathematics, 2005

Splicing systems are generative devices of formal languages, introduced by Head in 1987 to model biological phenomena on linear and circular DNA molecules. Via automata properties we show that it is decidable whether a regular language L on a one-letter alphabet is generated by a finite (Paun) circular splicing system: L has this property if and only if there is a unique final state q n on the closed path in the transition diagram of the minimal finite state automaton A recognizing L and q n is idempotent (i.e., (q n , a n ) = q n ). This result is obtained by an already known characterization of the unary languages L generated by a finite (Paun) circular splicing system and, in turn, allows us to simplify the description of the structure of L. This description is here extended to the larger class of the uniform languages, i.e., the circularizations of languages with the form A J = j ∈J A j , J being a subset of the set N of the nonnegative integers. Finally, we exhibit a regular circular language, namely ∼ ((A 2 ) * ∪ (A 3 ) * ), that cannot be generated by any finite circular splicing system.

Discrete Mathematics, 2016

The conjugacy relation defines a partition of words into equivalence classes. We call these classes circular words. Periodic properties of circular words are investigated in this article. The Periodicity Theorem of Fine and Wilf does not hold for weak periods of circular words; instead we give a strict upper bound on the length of a non-unary circular word that has two given relatively prime weak periods. Weak periods also lead to a way of representing circular words in a more compact form. We investigate in which cases are these representations unique or minimal. We will also analyze weak periods of circular Thue-Morse, Fibonacci and Christoffel words.

Theoretical Computer Science, 2006

When words are characterized in terms of numerical quantities, awkward considerations due to the noncommutativity of words are avoided. 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. Parikh matrices recently introduced have these quantities as their entries. According to the main result in this paper, no entry in a Parikh matrix, no matter how high the dimension, can be computed in terms of the other entries. Consequences concerning various inference problems between numbers |w| u themselves, as well as of the word w from these numbers, are obtained.

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, 2013

Trapezoidal words are words having at most n+1 distinct factors of length n for every n ≥ 0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, semicentral words, and show that they are characterized by the property that they can be written as uxyu, for a central word u and two different letters x, y. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.

2014

Word recognition using confusion matrices for computing similarities among phonemes Abstract. A word is a finite sequence of symbols. Parikh matrix of a word, introduced by Mateescu et al , has become an effective tool in the study of certain numerical properties of words based on subwords. There have been several investigations on various properties of Parikh matrices such as M-ambiguity, M-equivalence, subword equalities and inequalities, commutativity and so on. Recently, Parikh matrices of words that are images under certain morphisms have been studied for their properties. On the other hand, Parikh matrices of words involving a certain ratio property called weak-ratio property have been investigated by . Here we consider two special morphisms called Fibonacci and Tribonacci morphisms and obtain properties of Parikh matrices of images of binary words under these morphisms, utilizing the notion of weak-ratio property.

RAIRO - Theoretical Informatics and Applications, 2004

Circular splicing has been very recently introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we restrict our study to the relationship between regular circular languages and languages generated by finite circular splicing systems and provide some results towards a characterization of the intersection between these two classes. We consider the class of languages X * , called here star languages, which are closed under conjugacy relation and with X being a regular language. Using automata theory and combinatorial techniques on words, we show that for a subclass of star languages the corresponding circular languages are (Paun) circular splicing languages. For example, star languages belong to this subclass when X * is a free monoid or X is a finite set. We also prove that each (Paun) circular splicing language L over a one-letter alphabet has the form L = X + ∪ Y , with X, Y finite sets satisfying particular hypotheses. Cyclic and weak cyclic languages, which will be introduced in this paper, show that this result does not hold when we increase the size of alphabets, even if we restrict ourselves to regular languages.

Discrete Mathematics and Theoretical …, 2003

We find generating functions for the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.