Alphabet Of Cardinality Research Articles

On k-ary n-cubes and isometric words

The k-ary n-cubes are a generalization of the hypercubes to alphabets of cardinality k, with k≥2. More precisely, a k-ary n-cube is a graph with kn vertices associated to the k-ary words of length n. Given a k-ary word f, the k-ary n-cube avoiding f is the subgraph obtained deleting those vertices which contain f as a factor. When such a subgraph is isometric to the cube, for any n≥1, the word f is said isometric. A binary word f is isometric if and only if it is Ham-isometric, i.e., for any pair of f-free binary words u and v, u can be transformed in v by complementing the bits on which they differ and generating only f-free words. The case of a k-ary alphabet, with k≥2, is here investigated. From k≥4, the isometricity in terms of cubes is no longer captured by the Ham-isometricity, but by the Lee-isometricity. Then, Ham-isometric and Lee-isometric k-ary words are characterized in terms of their overlaps with errors. The minimal length of two words which witness the non-isometricity of a word f is called its index. The index of f is bounded in terms of its length and the bounds are shown tight by examples.

Weak stuffle algebras

Motivated by $q$-shuffle products determined by Singer from $q$-analogues of multiple zeta values, we build in this article a generalisation of the shuffle and stuffle products in terms of weak shuffle and stuffle products. Then, we characterise weak shuffle products and give as examples the case of an alphabet of cardinality two or three. We focus on a comparison between algebraic structures respected in the classical case and in the weak case. As in the classical case, each weak shuffle product can be equipped with a dendriform structure. However, they have another behaviour towards the quadri-algebra and the Hopf algebra structure. We give some relations satisfied by weak stuffle products.

The Relation Between k-Circularity and Circularity of Codes.

A code X is k-circular if any concatenation of at most k words from X, when read on a circle, admits exactly one partition into words from X. It is circular if it is k-circular for every integer k. While it is not a priori clear from the definition, there exists, for every pair (n,ell ), an integer k such that every k-circular ell -letter code over an alphabet of cardinality n is circular, and we determine the least such integer k for all values of n and ell . The k-circular codes may represent an important evolutionary step between the circular codes, such as the comma-free codes, and the genetic code.

Asymptotic bounds for spherical codes

The set of all error-correcting codes over a fixed finite alphabet of cardinality determines the set of code points in the unit square with coordinates := (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximising simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in inventing explicit constructions of “good codes” and comparing new classes of codes with earlier ones. A less classical approach studies the geometry of the whole set of code points (with fixed), at first independently of its computability properties, and only afterwards turning to problems of computability, analogies with statistical physics, and so on. The main purpose of this article consists in extending this latter strategy to the domain of spherical codes.

Limited automata and unary languages

Limited automata are one-tape Turing machines that can rewrite the contents of tape cells only in the first d visits, for a fixed d. When d=1 these models characterize regular languages.We show an exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata. Despite this gap, there are unary regular languages for which d-limited automata cannot be significantly smaller than finite automata, for any arbitrarily large d.We also prove that from each unary context-free grammar G it is possible to obtain an equivalent 1-limited automaton whose description has a size that is polynomial in the size of G.For alphabets of cardinality at least 2, for each grammar G generating a context-free language L, it is possible to obtain a 1-limited automaton whose description has polynomial size in that of G and whose accepted language L′ is Parikh-equivalent to L.

Fast computation of abelian runs

Given a word w and a Parikh vector P, an abelian run of period P in w is a maximal occurrence of a substring of w having abelian period P. Our main result is an online algorithm that, given a word w of length n over an alphabet of cardinality σ and a Parikh vector P, returns all the abelian runs of period P in w in time O(n) and space O(σ+p), where p is the norm of P, i.e., the sum of its components. We also present an online algorithm that computes all the abelian runs with periods of norm p in w in time O(np), for any given norm p. Finally, we give an O(n2)-time offline randomized algorithm for computing all the abelian runs of w. Its deterministic counterpart runs in O(n2log⁡σ) time.

A Deterministic Algorithm for Alpha-Numeric Sequence Comparison with Application to Protein Sequence Detection

This paper is an extension of a deterministic algorithm, [1, 2], that was initially designed to measure the rate of similarity between DNA sequences, and any sequences made up with symbols of alphabets of cardinality 4. Here, a modified and extended version to handle sequences of symbols from alphabets of cardinality > 4 is presented. This extension opens up its application area. As a test ground, we search for peptides within a protein database. Computational results on real data and a comparison with BLAST will be discussed.

Critical population and error threshold on the sharp peak landscape for the Wright–Fisher model

We pursue the task of developing a finite population counterpart to Eigen's model. We consider the classical Wright-Fisher model describing the evolution of a population of size $m$ of chromosomes of length $\ell$ over an alphabet of cardinality $\kappa$. The mutation probability per locus is $q$. The replication rate is $\sigma>1$ for the master sequence and $1$ for the other sequences. We study the equilibrium distribution of the process in the regime where $\ell\to+\infty$, $m\to+\infty$, $q\to0$, $\ell q\to a\in\,]0,+\infty[$, $\frac{m}{\ell}\to\alpha\in [0,+\infty]$. We obtain an equation $\alpha\psi(a)=\ln\kappa$ in the parameter space $(a,\alpha)$ separating the regime where the equilibrium population is totally random from the regime where a quasispecies is formed. We observe the existence of a critical population size necessary for a quasispecies to emerge, and we recover the finite population counterpart of the error threshold. The result is the twin brother of the corresponding result for the Moran model. The proof is more complex, and it relies on the Freidlin-Wentzell theory of random perturbations of dynamical systems.

Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods

This paper is devoted to subexponential estimates in Shirshov’s height theorem. A word W is n-divisible if it can be represented in the form W = W0W1⋯Wn, where W1 ≺ W2 ≺⋯≺ Wn. If an affine algebra A satisfies a polynomial identity of degree n, then A is spanned by non-n-divisible words of generators a1 ≺⋯≺ al. A. I. Shirshov proved that the set of non-n-divisible words over an alphabet of cardinality l has bounded height h over the set Y consisting of all words of degree ≤ n − 1. We show that h < Φ (n, l), where Φ(n, l) = 287l∙n12 log3 n+48. Let l, n, and d ≥ n be positive integers. Then all words over an alphabet of cardinality l whose length is greater than ψ(n, d, l) are either n-divisible or contain the dth power of a subword, where ψ(n, d, l) = 218l(nd)3 log3(nd)+13d2. In 1993, E. I. Zelmanov asked the following question in the Dniester Notebook: Suppose that F2,m is a 2-generated associative ring with the identity xm = 0. Is it true that the nilpotency degree of F2,m has exponential growth? We give the definitive answer to E. I. Zelmanov by this result. We show that the nilpotency degree of the l-generated associative algebra with the identity xd = 0 is smaller than ψ(d, d, l). This implies subexponential estimates on the nilpotency index of nil-algebras of arbitrary characteristic. Shirshov’s original estimate was just recursive; in 1982 a double exponent was obtained, and an exponential estimate was obtained in 1992. Our proof uses Latyshev’s idea of an application of the Dilworth theorem. We think that Shirshov’s height theorem is deeply connected to problems of modern combinatorics. In particular, this theorem is related to the Ramsey theory. We obtain lower and upper estimates of the number of periods of length 2, 3, n − 1 in some non-n-divisible word. These estimates differ only by a constant.

Fitness Function Distributions over Generalized Search Neighborhoods in the q-ary Hypercube

The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length-n strings from a finite alphabet of cardinality q. Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q-ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.

Robust Parent-Identifying Codes and Combinatorial Arrays

An n-word y=(y1,..., yn) over a finite alphabet of cardinality q is called a descendant of a set of t words x1,..., xt if every coordinate yi,i=1,..., n, is contained in the set {x1i,..., xti}. A code C={x1,..., xM} is said to have the t-IPP property if for any n -word y that is a descendant of at most t parents belonging to the code, it is possible to identify at least one of them. From earlier works, it is known that t-IPP codes of positive rate exist if and only if t ≤ q-1. We introduce a robust version of IPP codes which allows error-free identification of parents in the presence of a certain number of mutations, i.e., coordinates in y that can break away from the descent rule, taking arbitrary values from the alphabet or becoming completely unreadable. We show existence of robust t-IPP codes for all t ≤ q-1 and some positive proportion of such coordinates. We uncover a relation between the hash distance of codes and the IPP property and use it to find the exact proportion of mutant coordinates that permits identification of pirates with zero probability of error in the case of size-2 coalitions.

Subexponential estimates in Shirshov's theorem on height

Suppose that is a free -generated associative ring with the identity . In 1993 Zelmanov put the following question: is it true that the nilpotency degree of has exponential growth?We give the definitive answer to Zelmanov's question by showing that the nilpotency class of an -generated associative algebra with the identity is smaller than , where This result is a consequence of the following fact based on combinatorics of words. Let , and be positive integers. Then all words over an alphabet of cardinality whose length is not less than are either -divisible or contain ; a word is -divisible if it can be represented in the form so that are placed in lexicographically decreasing order. Our proof uses Dilworth's theorem (according to V.N. Latyshev's idea). We show that the set of not -divisible words over an alphabet of cardinality has height over the set of words of degree , where Bibliography: 40 titles.

A Catalog of Self-Affine Hierarchical Entropy Functions

For fixed k ≥ 2 and fixed data alphabet of cardinality m, the hierarchical type class of a data string of length n = kj for some j ≥ 1 is formed by permuting the string in all possible ways under permutations arising from the isomorphisms of the unique finite rooted tree of depth j which has n leaves and k children for each non-leaf vertex. Suppose the data strings in a hierarchical type class are losslessly encoded via binary codewords of minimal length. A hierarchical entropy function is a function on the set of m-dimensional probability distributions which describes the asymptotic compression rate performance of this lossless encoding scheme as the data length n is allowed to grow without bound. We determine infinitely many hierarchical entropy functions which are each self-affine. For each such function, an explicit iterated function system is found such that the graph of the function is the attractor of the system.

Pseudogeometries with clusters and an example of a recursive [4, 2, 3]42-code

In 1998, E. Couselo, S. Gonzalez, V. Markov, and A. Nechaev defined the recursive codes and obtained some results that allowed one to conjecture the existence of recursive MDS-codes of dimension 2 and length 4 over any finite alphabet of cardinality q ∉ {2, 6}. This conjecture remained open only for q ∈ {14, 18, 26, 42}. It is shown in this paper that there exist such codes for q = 42. We used a new construction, that of pseudogeometry with clusters.

The number of S-precomplete classes in the functional system P κ τ

The problem on the number of precomplete classes in the functional system Pκτ is considered, elements of Pκτ are deterministic S-functions defined on words of length τ composed from letters of an alphabet of cardinality κ. An asymptotics for the number of S-precomplete classes in Pκτ is obtained for arbitrary fixed κ and τ tending to infinity.

On perfect p-ary codes of length p + 1

Let p be a prime number and assume p ? 5. We will use a result of L. Redei to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p ? 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.

Universal quantum computation with unlabelled qubits

We show that an nth root of the Walsh–Hadamard transform (obtained from the Hadamard gate and a cyclic permutation of the qubits), together with two diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit) and a non-local phase-flip (for a fixed but arbitrary coefficient), can do universal quantum computation on n qubits. A quantum computation, making use of n qubits and based on these operations, is then a word of variable length, but whose letters are always taken from an alphabet of cardinality three. Therefore, in contrast with other universal sets, no choice of qubit lines is needed for the application of the operations described here. A quantum algorithm based on this set can be interpreted as a discrete diffusion of a quantum particle on a de Bruijn graph, corrected on-the-fly by auxiliary modifications of the phases associated with the arcs.

On average sequence complexity

In this paper we study the average behavior of the number of distinct substrings in a text of size n over an alphabet of cardinality k. This quantity is called the complexity index and it captures the “richness of the language” used in a sequence. For example, sequences with low complexity index contain a large number of repeated substrings and they eventually become periodic (e.g., tandem repeats in a DNA sequence). In order to identify unusually low- or high-complexity strings one needs to determine how far are the complexities of the strings under study from the average or maximum string complexity. While the maximum string complexity was studied quite extensively in the past, to the best of our knowledge there are no results concerning the average complexity. We first prove that for a sequence generated by a mixing model (which includes Markov sources) the average complexity is asymptotically equal to n 2 / 2 which coincides with the maximum string complexity. However, for a memoryless source we establish a more precise result, namely the average string complexity is n 2 / 2 - n log k n + 1 + ( 1 - γ ) / ln k + φ k ( log k n ) + o ( 1 ) n where γ ≈ 0.577 and φ k ( x ) is a periodic function with a small amplitude for small alphabet size.

Asymptotic Properties of the Factors of Words Over a Finite Alphabet

Let A be an alphabet of cardinality m, k$_n$ be a sequence of positive integers and A* (||=k$_n$). In this paper it is shown that if lim sup$_{n}$k$_n$/ln n<1/ln m, then almost all words of length ...

Isomorphisms of the De Bruijn digraph and free‐space optical networks

AbstractThe de Bruijn digraph B(d, D) has degree d, diameter D, dD vertices, and dD+1 arcs. It is usually defined by words of size D on an alphabet of cardinality d, through a cyclic left‐shift permutation on the words, after which the rightmost symbol is changed. In this paper, we show that any digraph defined on words of a given size, through an arbitrary permutation on the alphabet and an arbitrary permutation on the word indices, is isomorphic to the de Bruijn digraph, provided that this latter permutation is cyclic. We use this result to improve from O(dD+1) to $\Theta(\sqrt{d^{D+1}})$ the number of lenses required for the implementation of B(d, D) by the Optical Transpose Interconnection System proposed by Marsden et al. [Opt Lett 18 (1993), 1083–1085]. © 2002 Wiley Periodicals, Inc.