Weak Separation Problem for Tree Languages

2020 ◽  
Vol 31 (05) ◽  
pp. 583-593
Author(s):  
Saeid Alirezazadeh ◽  
Khadijeh Alibabaei
Keyword(s):  
Computer Science ◽  
Natural Extension ◽  
Classical Problem ◽  
Finite Alphabet ◽  
Direct Products ◽  
Separation Problem ◽  
Finite Algebras ◽  
Tree Languages ◽  
Weak Separation ◽  
Recognizable Language

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. By looking at the syntactic congruence for monoids and as the natural extension in the case of forest algebras, we could define a version of syntactic congruence of a subset of the free forest algebra, not just a forest language. Let [Formula: see text] be a finite alphabet and [Formula: see text] be a pseudovariety of finite forest algebras. A language [Formula: see text] is [Formula: see text]-recognizable if its syntactic forest algebra belongs to [Formula: see text]. Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. Suppose that a forest language [Formula: see text] and a forest [Formula: see text] are given. We want to find if there exists any proof for that [Formula: see text] does not belong to [Formula: see text] just by using [Formula: see text]-recognizable languages, i.e. given such [Formula: see text] and [Formula: see text], if there exists a [Formula: see text]-recognizable language [Formula: see text] which contains [Formula: see text] and does not contain [Formula: see text]. In this paper, we present how one can use profinite forest algebra to separate a forest language and a forest term and also to separate two forest languages.

On Pseudovarieties of Forest Algebras

2016 ◽  
Vol 27 (08) ◽  
pp. 909-941 ◽  
Author(s):  
Saeid Alirezazadeh
Keyword(s):  
Natural Extension ◽  
Direct Products ◽  
Property A ◽  
Residually Finite ◽  
Finite Algebras

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. In the study of forest algebras one of the main difficulties is how to handle the faithfulness property. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. We tried to adapt in this context some of the results in the theory of semigroups, specially the studies on relatively free profinite semigroups, which are an important tool in the theory of pseudovarieties of semigroups. We define a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language. This new version is the natural extension of the syntactic congruence for monoids in the case of forest algebras and is used in the proof of an analog of Hunter’s Lemma. We show that under a certain assumption the two versions of syntactic congruences coincide. We adapt some results of Almeida on metric semigroups to the context of forest algebras. We show that the analog of Hunter’s Lemma holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite. We show an analog of Reiterman’s Theorem, which is based on a study of the structure profinite forest algebras.

scholarly journals Fixed Points Results in Algebras of Split Quaternion and Octonion

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 405 ◽  
Author(s):  
Mobeen Munir ◽  
Asim Naseem ◽  
Akhtar Rasool ◽  
Muhammad Saleem ◽  
Shin Kang
Keyword(s):  
Game Theory ◽  
Computer Science ◽  
Fixed Points ◽  
Quadratic Polynomial ◽  
Split Quaternion ◽  
Finite Algebras ◽  
Prime Fields

Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields Z p. Some characterizations of fixed points in terms of the coefficients of these polynomials are also given. Particularly, cardinalities of these fixed points have been determined depending upon the characteristics of the underlying field.

Partial clones

2020 ◽  
Vol 13 (08) ◽  
pp. 2050161
Author(s):  
Klaus Denecke
Keyword(s):  
Computer Science ◽  
Natural Numbers ◽  
Tree Languages ◽  
Associative Law ◽  
The Many ◽  
Composition Of Operations

A set [Formula: see text] of operations defined on a nonempty set [Formula: see text] is said to be a clone if [Formula: see text] is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the [Formula: see text]-ary operations defined on set [Formula: see text] for all natural numbers [Formula: see text] and the operations are the so-called superposition operations [Formula: see text] for natural numbers [Formula: see text] and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set [Formula: see text] and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

Data Analyzing via Probabilistic Modeling

2021 ◽  
pp. 25-51
Author(s):  
Dariusz Jacek Jakóbczak
Keyword(s):  
Artificial Intelligence ◽  
Image Processing ◽  
Computer Vision ◽  
Object Recognition ◽  
Computer Science ◽  
Classical Problem ◽  
Contour Reconstruction ◽  
Curve Interpolation ◽  
Characteristic Features ◽  
Unknown Object

Object recognition is one of the topics of artificial intelligence, computer vision, image processing, and machine vision. The classical problem in these areas of computer science is that of determining object via characteristic features. An important feature of the object is its contour. Accurate reconstruction of contour points leads to possibility to compare the unknown object with models of specified objects. The key information about the object is the set of contour points which are treated as interpolation nodes. Classical interpolations (Lagrange or Newton polynomials) are useless for precise reconstruction of the contour. The chapter is dealing with proposed method of contour reconstruction via curves interpolation. First stage consists in computing the contour points of the object to be recognized. Then one can compare models of known objects, given by the sets of contour points, with coordinates of interpolated points of unknown object. Contour points reconstruction and curve interpolation are possible using a new method of Hurwitz-Radon matrices.

Light Diacritic Restoration to Disambiguate Homographs in Modern Arabic Texts

2022 ◽  
Vol 21 (3) ◽  
pp. 1-14
Author(s):  
Aqil M. Azmi ◽  
Rehab M. Alnefaie ◽  
Hatim A. Aboalsamh
Keyword(s):  
Computer Science ◽  
Large Class ◽  
Statistical Approach ◽  
Reading Speed ◽  
Classical Problem ◽  
Modern Arabic

Diacritic restoration (also known as diacritization or vowelization) is the process of inserting the correct diacritical markings into a text. Modern Arabic is typically written without diacritics, e.g., newspapers. This lack of diacritical markings often causes ambiguity, and though natives are adept at resolving, there are times they may fail. Diacritic restoration is a classical problem in computer science. Still, as most of the works tackle the full (heavy) diacritization of text, we, however, are interested in diacritizing the text using a fewer number of diacritics. Studies have shown that a fully diacritized text is visually displeasing and slows down the reading. This article proposes a system to diacritize homographs using the least number of diacritics, thus the name “light.” There is a large class of words that fall under the homograph category, and we will be dealing with the class of words that share the spelling but not the meaning. With fewer diacritics, we do not expect any effect on reading speed, while eye strain is reduced. The system contains morphological analyzer and context similarities. The morphological analyzer is used to generate all word candidates for diacritics. Then, through a statistical approach and context similarities, we resolve the homographs. Experimentally, the system shows very promising results, and our best accuracy is 85.6%.

ON PSEUDOVARIETIES OF MONOUNARY ALGEBRAS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250003 ◽  
Author(s):  
Danica Jakubíková-Studenovská
Keyword(s):  
Direct Products ◽  
Constructive Description ◽  
Finite Algebras

A class of finite algebras is called a pseudovariety if it is closed under homomorphisms, subalgebras and direct products of finitely many members. We give a constructive description of members of all pseudovarieties of monounary algebras. Further, we show that each equational pseudovariety of monounary algebras except the pseudovariety of all finite monounary algebras can be generated by a single algebra.

scholarly journals Transitivity in Fuzzy Hyperspaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1862
Author(s):  
Daniel Jardón ◽  
Iván Sánchez ◽  
Manuel Sanchis
Keyword(s):  
Fuzzy Sets ◽  
Metric Space ◽  
Complete Metric Space ◽  
Discrete Dynamical System ◽  
Natural Extension ◽  
Classical Problem ◽  
The Family ◽  
Weakly Mixing ◽  
System F ◽  
Sendograph Metric

Given a metric space (X,d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f:(X,d)→(X,d) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension f^ of f to F(X), the family of all normal fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F(X) with different metrics: the supremum metric d∞, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other things, the following results are presented: (1) If (X,d) is a metric space, then the following conditions are equivalent: (a) (X,f) is weakly mixing, (b) ((F(X),d∞),f^) is transitive, (c) ((F(X),d0),f^) is transitive and (d) ((F(X),dS)),f^) is transitive, (2) if f:(X,d)→(X,d) is a continuous function, then the following hold: (a) if ((F(X),dS),f^) is transitive, then ((F(X),dE),f^) is transitive, (b) if ((F(X),dS),f^) is transitive, then (X,f) is transitive; and (3) if (X,d) be a complete metric space, then the following conditions are equivalent: (a) (X×X,f×f) is point-transitive and (b) ((F(X),d0) is point-transitive.

THE DOT-DEPTH OF A GENERATING CLASS OF APERIODIC MONOIDS IS COMPUTABLE

1992 ◽  
Vol 03 (04) ◽  
pp. 419-442 ◽  
Author(s):  
F. BLANCHET-SADRI
Keyword(s):  
Natural Extension ◽  
Finite Alphabet ◽  
Aperiodic Monoids ◽  
Positive Integers

Given a finite alphabet A and a sequence of positive integers [Formula: see text] congruences on A*, denoted by [Formula: see text] and related to a version of the Ehrenfeucht-Fraïssé game, have been defined by Thomas in order to give a new proof that the Brzozowski’s dot-depth hierarchy of star-free languages is infinite. A natural extension of some of the results of Thomas states that the monoid variety corresponding to level k of the Straubing hierarchy (the Straubing hierarchy is closely related to the Brzozowski’s dot-depth hierarchy) can be characterized in terms of the monoids [Formula: see text]. In this paper, it is shown that the dot-depth of the [Formula: see text]’s is computable.

scholarly journals Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 558
Author(s):  
Thodsaporn Kumduang ◽  
Sorasak Leeratanavalee
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Embedding Theorems ◽  
Transformation Semigroups ◽  
Strong Connection ◽  
Theoretical Computer ◽  
Full Transformation ◽  
Menger Algebra ◽  
Tree Languages ◽  
Novel Concept

The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups.

scholarly journals Patterns in Inversion Sequences I

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Sylvie Corteel ◽  
Megan A. Martinez ◽  
Carla D. Savage ◽  
Michael Weselcouch
Keyword(s):  
Computer Science ◽  
Fibonacci Numbers ◽  
Natural Extension ◽  
Great Depth ◽  
Permutation Patterns ◽  
Bell Numbers ◽  
Combinatorial Structures ◽  
Vast Array

Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i<i$ for all $i \in [n]$. Inversion sequences of length $n$ are in bijection with permutations of length $n$; an inversion sequence can be obtained from any permutation $\pi=\pi_1\pi_2\ldots \pi_n$ by setting $e_i = |\{j \ | \ j < i \ {\rm and} \ \pi_j > \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.

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