Abelian Primitive Words

We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration problems and the relation to the theory of codes

A Note on Efficient Computation of All Abelian Periods in a String

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all `long' Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems

Identifying all abelian periods of a string in quadratic time and relevant problems

Abelian periodicity of strings has been studied extensively over the last years. In 2006 Constantinescu and Ilie defined the abelian period of a string and several algorithms for the computation of all abelian periods of a string were given. In contrast to the classical period of a word, its abelian version is more flexible, factors of the word are considered the same under any internal permutation of their letters. We show two O(|y|^2) algorithms for the computation of all abelian periods of a string y. The first one maps each letter to a suitable number such that each factor of the string can be identified by the unique sum of the numbers corresponding to its letters and hence abelian periods can be identified easily. The other one maps each letter to a prime number such that each factor of the string can be identified by the unique product of the numbers corresponding to its letters and so abelian periods can be identified easily. We also define weak abelian periods on strings and give an O(|y|log(|y|)) algorithm for their computation, together with some other algorithms for more basic problems.Comment: Accepted in the "International Journal of foundations of Computer Science

Two results on words

The study of combinatorial patterns of words has raised great interest since the early 20th century. In this master's thesis presentation we study two combinatorial patterns. The first pattern is “abelian k-th power free” and the second one is “representability of sets of words of equal length”. For the first pattern we study the context-freeness of non-abelian k-th powers. A word is a non-abelian k-th power if it cannot be factorized in the form w1w2...wk where the wi are permutations of w1 for 2 ≤ i ≤ k. We show that neither the language of non-abelian squares nor the language of non- abelian cubes is context-free. For the second pattern we study the representability of a set of words of fixed length. A set S of words of length n is representable if there exists some word w such that the set of length-n factors of w equals S. We will give lower and upper bounds for the number of such representable sets. Furthermore, we study a variation of the problem: we fix a length t, and try to evaluate the number of sets of words of length n such that there exists some word w of length t such that the set of length-n factors of w equals S. We give a closed-form formula in the case where n ≤ t < 2n. In particular, we give a characterization on two distinct words having the same subset of length-n factors

A Study of Pseudo-Periodic and Pseudo-Bordered Words for Functions Beyond Identity and Involution

Periodicity, primitivity and borderedness are some of the fundamental notions in combinatorics on words. Motivated by the Watson-Crick complementarity of DNA strands wherein a word (strand) over the DNA alphabet \{A, G, C, T\} and its Watson-Crick complement are informationally ``identical , these notions have been extended to consider pseudo-periodicity and pseudo-borderedness obtained by replacing the ``identity function with ``pseudo-identity functions (antimorphic involution in case of Watson-Crick complementarity). For a given alphabet $\Sigma$, an antimorphic involution $\theta$ is an antimorphism, i.e., $\theta(uv)=\theta(v) \theta(u)$ for all $u,v \in \Sigma^{*}$ and an involution, i.e., $\theta(\theta(u))=u$ for all $u \in \Sigma^{*}$. In this thesis, we continue the study of pseudo-periodic and pseudo-bordered words for pseudo-identity functions including involutions. To start with, we propose a binary word operation, $\theta$-catenation, that generates $\theta$-powers (pseudo-powers) of a word for any morphic or antimorphic involution $\theta$. We investigate various properties of this operation including closure properties of various classes of languages under it, and its connection with the previously defined notion of $\theta$-primitive words. A non-empty word $u$ is said to be $\theta$-bordered if there exists a non-empty word $v$ which is a prefix of $u$ while $\theta(v)$ is a suffix of $u$. We investigate the properties of $\theta$-bordered (pseudo-bordered) and $\theta$-unbordered (pseudo-unbordered) words for pseudo-identity functions $\theta$ with the property that $\theta$ is either a morphism or an antimorphism with $\theta^{n}=I$, for a given $n \geq 2$, or $\theta$ is a literal morphism or an antimorphism. Lastly, we initiate a new line of study by exploring the disjunctivity properties of sets of pseudo-bordered and pseudo-unbordered words and some other related languages for various pseudo-identity functions. In particular, we consider such properties for morphic involutions $\theta$ and prove that, for any $i \geq 2$, the set of all words with exactly $i$ $\theta$-borders is disjunctive (under certain conditions)

Properties of Abelian Primitive Languages

Properties of Abelian Primitive LanguagesLet the alphabet X be indexed as X={a0,a1,…,an}. For x∈X* and aj∈X, 0≤j≤n, let #j(x) denote the number of occurrences of aj. The Parikh vector of x, denoted by ψ(x)=(#0(x), #1(x), #2(x), …, #n(x)). An abelian k-power is a non-empty word of the form x1x2…xk, where ψ(x1)= ψ(x2)= …=ψ(xk). If a word is abelian 2-power, then it is called an abelian square. The avoidability concerned abelian squares are investigated in [1]. A non-empty word is called abelian primitive if it is not an abelian k-power for any k≥2. Let Ap be the set of all abelian primitive words over an alphabet X. This project is devoted to investigate the properties of Ap.A language L1⊆X* is said to pseudo-commute with L2 under f : X*→X* iff L1L2= f(L2)L1 [4]. Of special interest from the DNA computing perspective is the special case where f is an antimorphic involution modeling the Watson-Crick complementarity of DNA strands (see [2]). Two languages L1 and L2 over an alphabet X commute if L1L2=L2L1 [5,6]. If two languages L1 and L2 commute and L2 is singleton, then the word in L2 and all words in L1 have the same primitive root [5]. While the properties of abelian primitive words are greatly related to properties of cyclic permutation primitive words and the solutions of the equation L1L=LL1.[1] Keranen, Abelian Squares are Avoidable on 4 Letters, Springer LNCS 623 (1992), 41--52.[2] M. Daley and L. Kari, DNA Computing: Models and Implementations, Comments on Theoretical Biology, 7(3) (2002), 177—198.[3] R.C. Lyndon and M.P. Schutzenberger, The Equation aM=bNcP in a Free Group, Michigan Math. J., 9 (1962), 289--298.[4] L. Kari, K. mahalingam, and S. Seki, On the Pseudo-Commutativity of Words and Languages, (submitted), 2007.[5] H.J. Shyr, On Two Languages That Commute, Note on Semigroup 9, K. Marx. University, Economics, Budapest, (1983), 257—269.[6] S.S. Yu, Languages and Codes, Lecture Notes, Tsang Hai Book Publishing Co., Taichung, Taiwan 2005.交換質式言語的性質令字母集X為有序集合X={a0,a1,…,an}。對於x∈X*及aj∈X，0≤j≤n，令#j(x)表示字母aj在x中出現的次數。x的Parikh向量則定義為ψ(x)=(#0(x), #1(x), #2(x), …, #n(x))。一個交換k-次元是一個非空的字具有x1x2…xk的型式，其中ψ(x1)= ψ(x2)= …=ψ(xk)。如果一個字是交換2-次元，則稱之為一個交換平方字。在[1]中是研究關於交換平方字的可避免性。當一個非空的字不是任何k≥2的交換k-次元，則稱之為一個交換質式字。令Ap表示建構於字母集X的所有交換質式字的集合。我們將在此計畫中研究關於Ap的性質。給一個對映f : X*→X*，一個言語L1⊆X*稱為與另一個言語L2為pseudo- commute就是說它們滿足等式L1L2= f(L2)L1 [4]。其中，特別由DNA計算透視所導致得吸引人之處，是在於f為一個非同構次方用以塑造DNA双旋股的Watson-Crick互補模式的特殊狀況(參考[2])。當建構於字母集X的兩個言語L1與L2滿足L1L2=L2L1時，稱它們為可交換。當L1與L2為可交換且L2為含單一元素時，L2中的那個字及所有L1中的字皆有同一個質式根[5]。而交換質式字的性質與旋轉交換質式字及言語等式L1L=LL1的解之性質息息相關。[1] Keranen, Abelian Squares are Avoidable on 4 Letters, Springer LNCS 623 (1992), 41--52.[2] M. Daley and L. Kari, DNA Computing: Models and Implementations, Comments on Theoretical Biology, 7(3) (2002), 177—198.[3] R.C. Lyndon and M.P. Schutzenberger, The Equation aM=bNcP in a Free Group, Michigan Math. J., 9 (1962), 289--298.[4] L. Kari, K. mahalingam, and S. Seki, On the Pseudo-Commutativity of Words and Languages, (submitted), 2007.[5] H.J. Shyr, On Two Languages That Commute, Note on Semigroup 9, K. Marx. University, Economics, Budapest, (1983), 257—269.[6] S.S. Yu, Languages and Codes, Lecture Notes, Tsang Hai Book Publishing Co., Taichung, Taiwan 2005