Unbordered Words Research Articles

Ranking and unranking bordered and unbordered words

A border of a word w is a word that is both a non-empty proper prefix and suffix of w. If w has a border, then it is said to be bordered; otherwise, it is said to be unbordered. The main results of this paper are the first algorithms to rank and unrank length-n bordered and unbordered words over a k-letter alphabet. We show that, under the unit-cost RAM model, ranking bordered and unbordered words can be done in O(kn3) time using O(n) space, and unranking them can be done in O(n4klog⁡k) time using O(n) space.

Borders, palindrome prefixes, and square prefixes

We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix. We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square prefix, thus proving a 2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.

ON THE NUMBER OF ABELIAN BORDERED WORDS (WITH AN EXAMPLE OF AUTOMATIC THEOREM-PROVING)

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in ℤ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue–Morse word using some kind of automatic theorem-proving provided by a logical characterization of the k-automatic sequences.

Linear computation of unbordered conjugate on unordered alphabet

We present an algorithm that, given a word w of length n on an unordered alphabet, computes one of its unbordered conjugates. If such a conjugate does not exist, the algorithm computes one of its conjugates that is a power of an unbordered word. The time complexity of the algorithm is O(n): the number of comparisons between letters of w is bounded by 4n.

Automaticity of primitive words and irreducible polynomials

Automata, Logic and Semantics If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

On the number of frames in binary words

A frame is a square u u , where u is an unbordered word. Let F ( n ) denote the maximum number of distinct frames in a binary word of length n . We count this number for small values of n and show that F ( n ) is at most ⌊ n / 2 ⌋ + 8 for all n and greater than 7 n / 30 − ϵ for any positive ϵ and infinitely many n . We also show that Fibonacci words, which are known to contain plenty of distinct squares, have only a few frames. Moreover, by modifying the Thue–Morse word, we prove that the minimum number of occurrences of frames in a word of length n is ⌈ n / 2 ⌉ − 2 .

Inverse star, borders, and palstars

A language L is closed if L = L ⁎ . We consider an operation on closed languages, L − ⁎ , that is an inverse to Kleene closure. It is known that if L is closed and regular, then L − ⁎ is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free.

A note of involutively bordered words

In this paper we study some properties of involutively bordered words. When θ is a morphic involution, the relationship between the θ - d -descent of words in Dθ (i) and Dθ (k), k < i is derived. We provide methods to construct words in for k ⩾ 1 . We also obtain that the annohilator αub (u) of unbordered word u is a disjunctive post-plus language.

Detecting palindromes, patterns and borders in regular languages

Given a language L and a non-deterministic finite automaton M, we consider whether we can determine efficiently (in the size of M) if M accepts at least one word in L, or infinitely many words. Given that M accepts at least one word in L, we consider how long a shortest word can be. The languages L that we examine include the palindromes, the non-palindromes, the k-powers, the non- k-powers, the powers, the non-powers (also called primitive words), the words matching a general pattern, the bordered words, and the unbordered words.

WATSON-CRICK BORDERED WORDS AND THEIR SYNTACTIC MONOID

DNA strands that, mathematically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is Watson-Crick complementary to T and G to C, DNA single strands that are Watson-Crick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper studies a mathematical formalization of a particular case of hairpins, the Watson-Crick bordered words. A Watson-Crick bordered word is a word with the property that it has a prefix that is Watson-Crick complementary to its suffix. We namely study algebraic properties of Watson-Crick bordered and unbordered words. We also give a complete characterization of the syntactic monoid of the language consisting of all Watson-Crick bordered words over a given alphabet. Our results hold for the more general case where the Watson-Crick complement function is replaced by an arbitrary antimorphic involution.

On pseudoknot-bordered words and their properties

We study a generalization of the classical notions of bordered and unbordered words, motivated by biomolecular computing. DNA strands can be viewed as finite strings over the alphabet { A, G, C, T}, and are used in biomolecular computing to encode information. Due to the fact that A is Watson–Crick complementary to T and G to C, DNA single strands that are Watson–Crick complementary can bind to each other or to themselves forming so-called secondary structures. Most of these secondary structures are undesirable for biomolecular computational purposes since the strands they involve cannot further interact with other strands. This paper studies pseudoknot-bordered words, a mathematical formalization of pseudoknot-like inter- and intra-molecular structures. In this context, pseudoknot-unbordered words model DNA or RNA strands that will be free of such secondary structures. We obtain several properties of pseudoknot-bordered and -unbordered words. We also address following problem: Given a pseudoknot-unbordered word u, does { u } + consist of pseudoknot-unbordered words only? We show that this is not generally true. We find that a sufficient condition for { u } + to consist of pseudoknot-unbordered words only is that u be not primitive. All of our results hold for arbitrary antimorphic involutions, of which the DNA Watson–Crick complementarity function is a particular case.

Unbordered partial words

An unbordered word is a string over a finite alphabet such that none of its proper prefixes is one of its suffixes. In this paper, we extend the results on unbordered words to unbordered partial words. Partial words are strings that may have a number of “do not know” symbols. We extend a result of Ehrenfeucht and Silberger which states that if a word u can be written as a concatenation of nonempty prefixes of a word v , then u can be written as a unique concatenation of nonempty unbordered prefixes of v . We study the properties of the longest unbordered prefix of a partial word, investigate the relationship between the minimal weak period of a partial word and the maximal length of its unbordered factors, and also investigate some of the properties of an unbordered partial word and how they relate to its critical factorizations (if any).

INVOLUTIVELY BORDERED WORDS

In this paper we study a generalization of the classical notions of bordered and unbordered words, motivated by DNA computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in DNA computing to encode information. Due to the fact that A is Watson-Crick complementary to T and G to C, DNA single strands that are Watson-Crick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper studies a mathematical formalization of a particular case of hairpins, the Watson-Crick bordered words. A Watson-Crick bordered word is a word with the property that it has a prefix that is Watson-Crick complementary to its suffix. More generally, we investigate the notion of θ-bordered words, where θ is a morphic or antimorphic involution. We show that the set of all θ-bordered words is regular, when θ is an antimorphic involution and the set of all θ-bordered words is context-sensitive when θ is a morphic involution. We study the properties of θ-bordered and θ-unbordered words and also the relation between θ-bordered and θ-unbordered words and certain type of involution codes.

Periodicity and unbordered words

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this article. Consider a finite word w of length n . We call a word bordered if it has a proper prefix, which is also a suffix of that word. Let μ( w ) denote the maximum length of all unbordered factors of w , and let ∂( w ) denote the period of w . Clearly, μ( w ) ≤ ∂( w ). We establish that μ( w ) = ∂( w ), if w has an unbordered prefix of length μ( w ) and n ≥ 2μ( w ) − 1. This bound is tight and solves the stronger version of an old conjecture by Duval [1983]. It follows from this result that, in general, n ≥ 3μ( w ) − 3 implies μ( w ) = ∂( w ), which gives an improved bound for the question raised by Ehrenfeucht and Silberger in 1979.

Counting bordered and primitive words with a fixed weight

A word w is primitive if it is not a proper power of another word, and w is unbordered if it has no prefix that is also a suffix of w . We study the number of primitive and unbordered words w with a fixed weight, that is, words for which the Parikh vector of w is a fixed vector. Moreover, we estimate the number of words that have a unique border.

A proof of the extended Duval's conjecture

We give a short and elementary proof of the following stronger version of Duval's conjecture: let u be an unbordered word, and v a word of length | u | - 1 , such that v is not a prefix of u. Then u v contains an unbordered word of length at least | u | + 1 .

MINIMAL DUVAL EXTENSIONS

A word v=wu is a (nontrivial) Duval extension of the unbordered word w, if (u is not a prefix of v and) w is an unbordered factor of v of maximum length. After a short survey of the research topic related to Duval extensions, we show that, if wu is a minimal Duval extension, then u is a factor of w. We also show that finite, unbordered factors of Sturmian words are Lyndon words.

A Note on a Conjecture of Duval and Sturmian Words

We prove a long standing conjecture of Duval in the special case of Sturmian words. Mathematics Subject Classication. ??????????????. Let U be a nonempty word on a nite alphabet A: A nonempty word B dierent from U is called a border of U if B is both a prex and sux of U: We say U is bordered if U admits a border, otherwise U is said to be unbordered. For example, U = 011001011 is bordered by the factor 011; while 00010001001 is unbordered. An integer 1 k n is a period of a word U = U1 :::U n if and only if for all 1 i n k we have Ui = Ui+k. It is easy to see that k is a period of U if and only if the prex B of U of length n k is a border of U or is empty. Let (U) denote the smallest period of U.T henU is unbordered if and only if (U )i s equal to the length of U; that is (U )= jUj = n. We further denote by (U )t he length of the longest unbordered factor of U. Clearly U is unbordered if and only if (U )= jUj. In general, for any word U we have (U) (U )( cf. Prop. 2.2 of (3)). An interesting question is to ask for which words U does equality hold. In (3) Duval shows that (U )= (U) whenever jU j2(U) 2( cf. Cor. 4.2 in (3)). These notions extend directly to innite words. For an innite word !; if (!) is nite, then ! is periodic of period (!). (cf. (2) and (4)). In (3) Duval conjectured that: Conjecture 1 (Duval 1981). Let U be an unbordered word on an alphabet A and let W be a word of length 2jUj beginning in U and with the property that each factor of W of length greater than jUj is bordered. Then W has period jUj; i.e., W = UU: