A New Study of Parikh Matrices Restricted to Terms

2020 ◽  
Vol 31 (05) ◽  
pp. 621-638
Author(s):  
Zi Jing Chern ◽  
K. G. Subramanian ◽  
Azhana Ahmad ◽  
Wen Chean Teh
Keyword(s):  
Equivalence Classes ◽  
Graph Distance ◽  
Rewriting Rules

Parikh matrices as an extension of Parikh vectors are useful tools in arithmetizing words by numbers. This paper presents a further study of Parikh matrices by restricting the corresponding words to terms formed over a signature. Some [Formula: see text]-equivalence preserving rewriting rules for such terms are introduced. A characterization of terms that are only [Formula: see text]-equivalent to themselves is studied for binary signatures. Graphs associated to the equivalence classes of [Formula: see text]-equivalent terms are studied with respect to graph distance. Finally, the preservation of [Formula: see text]-equivalence under the term self-shuffle operator is studied.

scholarly journals Free mu-lattices

2000 ◽  
Vol 7 (28) ◽  
Author(s):  
Luigi Santocanale
Keyword(s):  
Ordered Set ◽  
Order Relation ◽  
Equivalence Classes ◽  
Complete Lattices ◽  
Models Of Computation ◽  
Fix Point

A mu-lattice is a lattice with the property that every unary <br />polynomial has both a least and a greatest fix-point. In this paper<br />we define the quasivariety of mu-lattices and, for a given partially<br />ordered set P, we construct a mu-lattice JP whose elements are<br />equivalence classes of games in a preordered class J (P). We prove<br />that the mu-lattice JP is free over the ordered set P and that the<br />order relation of JP is decidable if the order relation of P is <br />decidable. By means of this characterization of free mu-lattices we<br />infer that the class of complete lattices generates the quasivariety<br />of mu-lattices.<br />Keywords: mu-lattices, free mu-lattices, free lattices, bicompletion<br />of categories, models of computation, least and greatest fix-points,<br />mu-calculus, Rabin chain games.

scholarly journals On Hypergroups with a β-Class of Finite Height

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 168 ◽  
Author(s):  
Mario De Salvo ◽  
Dario Fasino ◽  
Domenico Freni ◽  
Giovanni Lo Faro
Keyword(s):  
Equivalence Classes ◽  
Fundamental Relation ◽  
Finite Height

In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms.

Solvable groups derived from hypergroups

2016 ◽  
Vol 15 (04) ◽  
pp. 1650067 ◽  
Author(s):  
M. Jafarpour ◽  
H. Aghabozorgi ◽  
B. Davvaz
Keyword(s):  
Solvable Group ◽  
Equivalence Relation ◽  
Solvable Groups ◽  
Equivalence Classes ◽  
Strongly Regular

In this paper, we introduce the smallest equivalence relation [Formula: see text] on a hypergroup [Formula: see text] such that the quotient [Formula: see text], the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relations is investigated and several results on the topic are presented.

scholarly journals On the Relation Between Schmidt Coefficients and Entanglement

2009 ◽  
Vol 16 (02n03) ◽  
pp. 127-143 ◽  
Author(s):  
Paolo Aniello ◽  
Cosmo Lupo
Keyword(s):  
Density Operator ◽  
Equivalence Classes ◽  
Symmetric Polynomials ◽  
Schmidt Decomposition ◽  
Open Problems ◽  
Density Operators ◽  
New Family ◽  
Separability Criteria ◽  
Bipartite States

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the 'realignment' or 'computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the 'symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.

scholarly journals MEASUREMENT OF DISTANCE BETWEEN REGULAR EVENTS FOR MULTITAPE AUTOMATA BASED ON A NEW CHARACTERIZATION OF EQUIVALENCE CLASSES

2021 ◽  
Vol 55 (1 (254)) ◽  
pp. 72-80
Author(s):  
Tigran A. Grigoryan ◽  
Murad S. Hayrapetyan
Keyword(s):  
Approximate Calculation ◽  
Finite Automata ◽  
Equivalence Classes ◽  
Regular Expressions

In this paper several problems related to the implementation of the method for the approximate calculation of distance between regular events for multitape finite automata are considered and resolved. An algorithm of matching for the considered regular expressions is suggested and results of the algorithm application to some specific regular expressions are adduced. The proposed method can be used not only for the mentioned implementation, but also separately.

THE CHINESE MONOID

2001 ◽  
Vol 11 (03) ◽  
pp. 301-334 ◽  
Author(s):  
JULIEN CASSAIGNE ◽  
MARC ESPIE ◽  
DANIEL KROB ◽  
JEAN-CHRISTOPHE NOVELLI ◽  
FLORENT HIVERT
Keyword(s):  
Nous Avons ◽  
Equivalence Class ◽  
Cross Section ◽  
Equivalence Classes ◽  
Nous Permet ◽  
Plactic Monoid ◽  
Relation Scheme ◽  
Section Theorem ◽  
Chinese Monoid

Résumé: Cet article présente une étude combinatoire du monoïde Chinois, un monoïde ternaire proche du monoïde plaxique, fondé sur le schéma cba≡bca≡cab. Un algorithme proche de l'algorithme de Schensted nous permet de caractériser les classes d'équivalence et d'exhiber une section du monoïde. Nous énonçons également une correspondance de Robinson–Schensted pour le monoïde Chinois avant de nous intéresser au calcul du cardinal de certaines classes. Ce travail a permis de développer de nouveaux outils combinatoires. Entre autres, nous avons trouvé un plongement de chacune des classes d'équivalence dans la plus grande classe. Quant à la dernière partie de cet article, elle présente l'étude des relations de conjugaison. This paper presents a combinatorial study of the Chinese monoid, a ternary monoid related to the plactic monoid and based on the relation scheme cba≡bca≡cab. An algorithm similar to Schensted's algorithm yields a characterization of the equivalence classes and a cross-section theorem. We also establish a Robinson–Schensted correspondence for the Chinese monoid before computing the order of specific Chinese classes. For this work, we had to develop some new combinatorial tools. Among other things we discovered an embedding of every equivalence class in the largest one. Finally, the end of this paper is devoted to the study of conjugacy classes.

Decomposable theories

2007 ◽  
Vol 7 (5) ◽  
pp. 583-632 ◽  
Author(s):  
KHALIL DJELLOUL
Keyword(s):  
General Algorithm ◽  
Boolean Combination ◽  
First Order ◽  
Free Variables ◽  
Infinite Trees ◽  
Decomposable Theory ◽  
Rewriting Rules

AbstractWe present in this paper a general algorithm for solving first-order formulas in particular theories called decomposable theories. First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory T. The algorithm is given in the form of five rewriting rules. It transforms a first-order formula ϕ, which can possibly contain free variables, into a conjunction φ of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if ϕ has no free variables then φ is either the formula true or ¬true. The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory ${\cal T}$ of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in ${\cal T}$ with more than 160 nested alternated quantifiers.

CONCURRENT AUTOMATA AND DOMAINS

1992 ◽  
Vol 03 (04) ◽  
pp. 389-418 ◽  
Author(s):  
MANFRED DROSTE
Keyword(s):  
Concurrent Systems ◽  
Equivalence Classes ◽  
Binary Relations ◽  
Transition Systems ◽  
Operational Model ◽  
Locally Finite ◽  
Natural Domain ◽  
Theoretic Characterization ◽  
Natural Way

We introduce an operational model of concurrent systems, called automata with concurrency relations. These are labeled transition systems [Formula: see text] in which the event set is endowed with a collection of symmetric binary relations which describe when two events at a particular state of [Formula: see text] commute. This model generalizes the recent concept of Stark’s trace automata. A permutation equivalence for computation sequences of [Formula: see text] arises canonically, and we obtain a natural domain [Formula: see text] comprising the induced equivalence classes. We give a complete order-theoretic characterization of all such partial orders [Formula: see text] which turn out to be particular finitary domains. The arising domains [Formula: see text] are particularly pleasant Scott-domains, if [Formula: see text] is assumed to be concurrent, i.e. if the concurrency relations of [Formula: see text] depend (in a natural way) locally on each other, but not necessarily globally. We show that both event domains and dI-domains arise, up to isomorphism, as domains [Formula: see text] with well-behaved such concurrent automata [Formula: see text]. We introduce a subautomaton relationship for concurrent automata and show that, given two concurrency domains (D, ≤), (D′, ≤), there exists a nice stable embedding-projection pair from D to D′ iff D, D′ can be generated by concurrent automata [Formula: see text] such that [Formula: see text] is a subautomaton of [Formula: see text]. Finally, we introduce the concept of locally finite concurrent automata as a limit of finite concurrent automata and show that there exists a universal homogeneous locally finite concurrent automaton, which is unique up to isomorphism.

scholarly journals A characterization of Markov equivalence classes for acyclic digraphs

1997 ◽  
Vol 25 (2) ◽  
pp. 505-541 ◽  
Author(s):  
Steen A. Andersson ◽  
David Madigan ◽  
Michael D. Perlman
Keyword(s):  
Equivalence Classes ◽  
Markov Equivalence

scholarly journals Counting $k$-Marked Durfee Symbols

10.37236/528 ◽  
2010 ◽  
Vol 18 (1) ◽  
Author(s):  
Kağan Kurşungöz
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Equivalence Classes ◽  
Alternative Characterization ◽  
Binomial Coefficient Identity

An alternative characterization of $k$-marked Durfee symbols defined by Andrews is given. Some identities involving generating functions of $k$-marked Durfee symbols are proven combinatorially by considering the symbols not individually, but in equivalence classes. Also, a related binomial coefficient identity is obtained in the course.

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