Finite Word Research Articles

An upper bound of the number of distinct powers in binary words

Let u be a finite word and r be a positive integer, a power is a word of the form uu...u︸rtimes. Particularly, a square is a word of the form uu. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is bounded by its length. This conjecture was proved recently by Brlek and Li. Besides, a stronger upper bound for binary words was conjectured by Jonoska, Manea and Seki by stating that, for a word of length n over the alphabet {a,b}, if k is the least of the number of a's and the number of b's and k≥2, then the number of distinct squares is upper bounded by 2k−12k+2n. In this article, we prove this conjecture by giving a stronger statement on the number of distinct powers in a binary word.

BFT—Low-Latency Bit-Slice Design of Discrete Fourier Transform

Structures for the evaluation of fast Fourier transforms are important components in several signal-processing applications and communication systems. Their capabilities play a key role in the performance enhancement of the whole system in which they are embedded. In this paper, a novel implementation of the discrete Fourier transform is proposed, based on a bit-slice approach and on the exploitation of the input sequence finite word length. Input samples of the sequence to be transformed are split into binary sequences and each one is Fourier transformed using only complex sums. An FPGA-based solution characterized by low latency and low power consumption is designed. Simulations have been carried out, first in the Matlab environment, then emulated in Quartus IDE with Intel. The hardware implementation of the conceived system and the test for the functional accuracy verification have been performed, adopting the DE2-115 development board from Terasic, which is equipped with the Cyclone IV EP4CE115F29C7 FPGA by Intel.

Modeling and Simulation of a Micro strip Antenna in Annular Geometry

In the last decade, the study of multidimensional systems for signal filtering purpose has attracted the attention of many research teams, which are motivated by various interesting real world applications such Image processing, video signal filtering processing, video signal filtering Computational tomography, distributed parameters systems. The representation of the state space realization for two-dimensional (2D) filters by a transfer function, with minimal delays is a crucial problem; the minimal realization can avoidmathematical complexities as well as technical benefit to hardware requirement. Unfortunately, it is not possible to find the minimal state space realization for every 2D system other than the 1D system, but we can find it for a specific cases of all-pole or the all-zero. Lattice filter is classified as an important structure in digital signal processing. In fact, it minimizes the computation complexity, reduces the finite word length and requires less hardware compared to the direct form digital filter with similar design specifications. The authors in two delay units z1-1 , z2-1 are used. In this paper we propose a new lattice-ladder structure of 2-D digital filter, composed of a delay units z1-1 and a basic lattice section z1 and z2-1 - Using matrix representation of the basic lattice sections and we derived transfer functions of the proposed 2-D digital ladder-lattice filters. The proposed structure is composed by a minimal number of delays and a minimal number of basic lattice sections. Furthermore, we have considered the description of the proposed lattice structure by using a state-space n-D Roesser form description presented in below and the implementations suggested by the authors in literatures. Besides, the state space equations for 2-D digital filter are presented, where the dimension of the matrices of the proposed lattice structure verify the minimal state space realization. The remainder of the paper is organized as follows; first is devoted to the proposed -D lattice-ladder structure and realization and in the second is dedicated to the state space realization. As conclusion this work has allowed us to approach a field that has experienced a great growth in recent years which is constantly evolving in order to improve, understanding and control the complex systems and processes that surround us.

Search Algorithms for the Combinatorial Generation of Bordered Box Repetition-Free Words

Four search algorithms to search for the longest box repetition-free word over a given alphabet are presented, giving an empirical result on the upper bound of the length of these words. A box repetition-free word is a finite word w where any given factor of the form asa, with a ∈ Σ and s ∈ Σ∗, occurs at most once. A maximal box repetition-free word is a box repetition-free word which cannot be extended by appending to it any a ∈ Σ without it ceasing to be repetition free.

Using the past for resolving the future

Nondeterminism models an ability to see the future: An automaton with an infinite look ahead can successfully resolve its nondeterministic choices. An automaton ishistory deterministic(HD) if it can successfully resolve its nondeterministic choices in a way that only depends on the past. Formally, an HD automaton has a strategy that maps each finite word to the transition to be taken after the word is read and following this strategy results in accepting all the words in the language of the automaton. Beyond being theoretically interesting and intriguing, HD automata can replace deterministic automata in several applications, most notably reactive synthesis, and they attract a lot of interest in the research community. The survey describes the development of HD ω-regular automata, relates history determinism to other types of bounded nondeterminism, studies the determinization of HD automata and their succinctness with respect to deterministic ones, and discusses variants, extensions, and open problems around HD automata.

A unified approach for digital realization of fractional order operator in delta domain

The fractional order operator (s?a,0 < a <1) plays the pivotal role for the realization of fractional orders systems (FOS). For the realization of the FOS, fractional order operator (FOO) needs to be realized either in discrete or continuous time domain. Discrete time rational approximation of FOO in the z -domain fails to provide meaningful information at fast sampling interval. Moreover, z domain rational transfer function becomes highly sensitive with respect to its coefficients variation resulting to the poor finite word length effects for digital realization. In the other hand delta operator parameterized system allows to develop unification of continuous and discrete time formulations leading to the development of a unified framework for digital realization at fast sampling interval. The discrete time approximation of the FOO in delta domain is found to be robust to its coefficient variation in comparison to the shift operator based discretization of FOO. In this paper, discrete d -operator parameterization is proposed for the digital realization using direct discretization of FOO. As a result, superior finite word length effect is observed for the realization of the FOO in discrete delta domain. Fractional order operator with different orders (a ) are considered for the realization purpose using the proposed method and the results obtained using MATLare presented for validation.

Palindromic length and reduction of powers

The palindromic length of a finite word v is the minimum number of palindromes whose concatenation is equal to v. It was conjectured in 2013 that for every aperiodic infinite word x, the palindromic length of its factors is not bounded.Let x be an infinite aperiodic word with a bounded palindromic length and containing infinitely many occurrences of the fifth power u5 of some word u. In this paper, we prove that we can erase from x extra powers of u and of its reversal uR to get another infinite aperiodic word of bounded palindromic length which does not contain u5 nor (uR)5.This result is not sufficient to prove that x does not exist but may give a hint for this proof.

Palindromic factorization of rich words

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. For every finite rich word $w$ there are distinct nonempty palindromes $w_1, w_2,\dots,w_p$ such that $w=w_pw_{p-1}\cdots w_1$ and $w_i$ is the longest palindromic suffix of $w_pw_{p-1}\cdots w_i$, where $1\leq i\leq p$. This palindromic factorization is called \emph{UPS-factorization}. Let $luf(w)=p$ be \emph{the length of UPS-factorization} of $w$. In 2017, it was proved that there is a constant $c$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then $luf(w)\leq c\frac{n}{\ln{n}}$. We improve this result as follows: There are constants $\mu, \pi$ such that if $w$ is a finite rich word and $n=\vert w\vert$ then \[luf(w)\leq \mu\frac{n}{e^{\pi\sqrt{\ln{n}}}}\mbox{.}\] The constants $c,\mu,\pi$ depend on the size of the alphabet.

On the number of k-powers in a finite word

This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number of distinct squares in a finite word. By counting the number of (right-)special factors, we give an upper bound for the number of k-powers in a finite word for any integer k≥3. By k-power, we mean a word of the form uu...u︸ktimes.

The critical exponent functions

The critical exponent of a finite or infinite word w over a given alphabet is the supremum of the reals α for which w contains an α-power. We study the maps associating to every real in the unit interval the inverse of the critical exponent of its base-n expansion. We strengthen a combinatorial result by J.D. Currie and N. Rampersad to show that these maps are left- or right-Darboux at every point, and use dynamical methods to show that they have infinitely many nontrivial fixed points and infinite topological entropy. Moreover, we show that our model-case map is topologically mixing.

The topography of zeros and poles of a third order IIR digital filter with finite word length in the z-plane

Earlier it was found that the zeros and poles of IIR digital filters with finite word length are elements of the set of algebraic numbers. Therefore, not every point of the unit circle of the z-plane can be a zero and / or a pole of such digital filters. The position of admissible positions for zeros and poles depends on the degree of algebraic numbers and the length of the fractional part of the coefficients of the equivalent canonical structure of the corresponding order. The formation of the corresponding configurations is considered as a z-plane discretization due to the quantization of the filter coefficients. The z-plane discretization for second-order filters has been well studied. The geometric locus of the corresponding algebraic numbers is a system of concentric circles, that is, plane algebraic curves of the second degree. This paper presents the results of studying the geometrical place of the third order IIR filter zeros and poles.

On Morphisms Preserving Palindromic Richness

It is known that each word of length n contains at most n + 1 distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric d-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow us to construct new rich words from already known rich words. We focus on morphisms in Class Pret. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism ϕ in the class there exists a palindrome w such that ϕ(a)w is a first complete return word to w for each letter a. We characterize Pret morphisms which preserve richness over a binary alphabet. We also study marked Pret morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class Pret and that it preserves richness.

The Big-O Problem

Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the problem is undecidable, even for the instantiation of weighted automata as labelled Markov chains. Moreover, even when it is known that one weighted automaton is big-O of another, the problem of finding or approximating the associated constant is also undecidable. Our positive results show that the big-O problem is polynomial-time solvable for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e., when the alphabet is a single character) and decidable, subject to Schanuel's conjecture, when the language is bounded (i.e., a subset of $w_1^*\dots w_m^*$ for some finite words $w_1,\dots,w_m$) or when the automaton has finite ambiguity. On labelled Markov chains, the problem can be restated as a ratio total variation distance, which, instead of finding the maximum difference between the probabilities of any two events, finds the maximum ratio between the probabilities of any two events. The problem is related to $\varepsilon$-differential privacy, for which the optimal constant of the big-O notation is exactly $\exp(\varepsilon)$.

HV-Palindromes in Two-Dimensional Words

A two-dimensional (2D) word is a 2D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are 1D palindromes. We characterize such words and study some of their combinatorial and structural properties. We also find the number of possible palindromic conjugates of a 2D word. We investigate an upper bound on the number of distinct non-empty HV-palindromic subwords in any finite 2D word, thus, proving the conjecture given by Anisiu et al. We also identify the minimum number of HV-palindromic subwords in an infinite 2D word over a finite alphabet.

Maximal escape rate for shifts

<p style='text-indent:20px;'>We consider the shift transformation on the space of infinite sequences over a finite alphabet endowed with the invariant product measure, and examine the presence of a <i>hole</i> on the space. The holes we study are specified by the sequences that do not contain a given finite word as initial sub-string. The measure of the set of sequences that do not fall into the hole in the first <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> iterates of the shift is known to decay exponentially with <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and its exponential rate is called <i>escape rate</i>. In this paper we provide a complete characterization of the holes with maximal escape rate. In particular we show that, contrary to the case of equiprobable symbols, ordering the holes by their escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. Finally, we adapt our technique to the case of shifts endowed with Markov measures, where preliminary results show that a more intricate situation is to be expected.</p>

Exact Analysis of the Finite Precision Error Generated in Important Chaotic Maps and Complete Numerical Remedy of These Schemes

A first aim of the present work is the determination of the actual sources of the “finite precision error” generation and accumulation in two important algorithms: Bernoulli’s map and the folded Baker’s map. These two computational schemes attract the attention of a growing number of researchers, in connection with a wide range of applications. However, both Bernoulli’s and Baker’s maps, when implemented in a contemporary computing machine, suffer from a very serious numerical error due to the finite word length. This error, causally, causes a failure of these two algorithms after a relatively very small number of iterations. In the present manuscript, novel methods for eliminating this numerical error are presented. In fact, the introduced approach succeeds in executing the Bernoulli’s map and the folded Baker’s map in a computing machine for many hundreds of thousands of iterations, offering results practically free of finite precision error. These successful techniques are based on the determination and understanding of the substantial sources of finite precision (round-off) error, which is generated and accumulated in these two important chaotic maps.

On p/q-recognisable sets

Let p/q be a rational number. Numeration in base p/q is defined by a function that evaluates each finite word over A_p={0,1,...,p-1} to some rational number. We let N_p/q denote the image of this evaluation function. In particular, N_p/q contains all nonnegative integers and the literature on base p/q usually focuses on the set of words that are evaluated to nonnegative integers; it is a rather chaotic language which is not context-free. On the contrary, we study here the subsets of (N_p/q)^d that are p/q-recognisable, i.e. realised by finite automata over (A_p)^d. First, we give a characterisation of these sets as those definable in a first-order logic, similar to the one given by the B\"uchi-Bruy\`ere Theorem for integer bases numeration systems. Second, we show that the natural order relation and the modulo-q operator are not p/q-recognisable.

Enumeration of two dimensional palindromes

We investigate the maximum number of distinct palindromic sub-words in a two-dimensional finite word. We give tight bounds for the maximum number of distinct palindromic sub-words in two-row periodic/aperiodic words. Using this, we give upper bounds for the number of distinct palindromic sub-words for 2D words of larger sizes. Lastly, we propose a better bound for any finite 2D word.

On cogrowth function of algebras and its logarithmical gap

Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a subword-minimal $I$-reducible word. A {\em cogrowth} function is number of obstructions of length $\le n$. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.

Determinization and Minimization of Automata for Nested Words Revisited

We consider the problem of determinizing and minimizing automata for nested words in practice. For this we compile the nested regular expressions (NREs) from the usual XPath benchmark to nested word automata (NWAs). The determinization of these NWAs, however, fails to produce reasonably small automata. In the best case, huge deterministic NWAs are produced after few hours, even for relatively small NREs of the benchmark. We propose a different approach to the determinization of automata for nested words. For this, we introduce stepwise hedge automata (SHAs) that generalize naturally on both (stepwise) tree automata and on finite word automata. We then show how to determinize SHAs, yielding reasonably small deterministic automata for the NREs from the XPath benchmark. The size of deterministic SHAs automata can be reduced further by a novel minimization algorithm for a subclass of SHAs. In order to understand why the new approach to determinization and minimization works so nicely, we investigate the relationship between NWAs and SHAs further. Clearly, deterministic SHAs can be compiled to deterministic NWAs in linear time, and conversely NWAs can be compiled to nondeterministic SHAs in polynomial time. Therefore, we can use SHAs as intermediates for determinizing NWAs, while avoiding the huge size increase with the usual determinization algorithm for NWAs. Notably, the NWAs obtained from the SHAs perform bottom-up and left-to-right computations only, but no top-down computations. This NWA behavior can be distinguished syntactically by the (weak) single-entry property, suggesting a close relationship between SHAs and single-entry NWAs. In particular, it turns out that the usual determinization algorithm for NWAs behaves well for single-entry NWAs, while it quickly explodes without the single-entry property. Furthermore, it is known that the class of deterministic multi-module single-entry NWAs enjoys unique minimization. The subclass of deterministic SHAs to which our novel minimization algorithm applies is different though, in that we do not impose multiple modules. As further optimizations for reducing the sizes of the constructed SHAs, we propose schema-based cleaning and symbolic representations based on apply-else rules that can be maintained by determinization. We implemented the optimizations and report the experimental results for the automata constructed for the XPathMark benchmark.