On Improving Model Checking of Time Petri Nets and Its Application to the Formal Verification

Time Petri nets (TPN) are successfully used in the specification and analysis of distributed systems that involve explicit timing constraints. Especially, model checking TPN is a hopeful method for the formal verification of such complex systems. For this, it is promising to lean to the construction of an optimized version of the state space. The well-known methods of state space abstraction are SCG (state class graph) and ZBG (graph based on zones). For ZBG, a symbolic state represents the real evaluations of the clocks of the TPN; it is thus possible to directly check quantitative time properties. However, this method suffers from the state space explosion. To attenuate this problem, the authors propose in this paper to combine the ZBG approach with the partial order reduction technique based on stubborn set, leading thus to the proposal of a new state space abstraction called reduced zone-based graph (RZBG). The authors show via case studies the efficiency of the RZBG which is implemented and integrated within the 〖TPN-TCTL〗_h^∆ model checking in the model checker Romeo.

The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group of some order for and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

The conjugacy class graphs of non-abelian 3-groups

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to if it satisfies for some elements in with its inverse . A conjugacy class graph of a group is defined when its vertex set is the set of non-central conjugacy classes of . Two distinct vertices and are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of and have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

Constructive and object-oriented modeling text for detection of text borrowings

The scientific community is encouraged to use such models and data structures as arrays of LERP-RSA (the longest expected duplicate array of reduced suffix templates), tag classifier-a model based on Stanford NER's three-class, structures based on DN-sequences, graph representations, etc. The following algorithms are used: GreedyString-Tiling, ARPAD, shingle, statistical methods, genetic algorithms, and others. It should also be noted that much attention is paid to morphological analysis and lemmatization, pre-processing of texts. Models and algorithms only partly have program realization.The purpose of this work is to develop a text model to identify borrowings and bring it to program implementation. The task is to develop the object-oriented model and program implementation of a graph text model, with the application of the problem of detection of borrowing. As well as obtaining timeframes for program implementation work for further evaluation of the possibility of its use in the academic environment.The main idea of the graph model is to present the text as a weighted oriented graph. The vertex weight is a character or sequence of characters. Edge weight is the set of numbers of paths into which the edge enters. To formalize the model will use the apparatus of constructive-synthesizing modeling. To create graphs, a constructor and its components are defined: carrier, signature, multiple statements of information support for design. Transformations are made for the constructor: specialization, interpretation and concretization.On the basis of this model, the object-oriented model is constructed. it includes three classes: vertex, graph and work .The object of class Work presents the text as a set of objects of class Graph. The correspondences between the components of the presented models are established.The object-oriented model is implemented by software. Data are given about the execution time of graph construction and texts comparison.At this stage, software implementation of the model has shown acceptable time performance. Further research in this direction is promising. Directions for improving the model and program are proposed.

Associate class graph of zero-divisors of a commutative ring

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring [Formula: see text], denoted by [Formula: see text]. Some properties of [Formula: see text], including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

On Some Graphs of Finite Metabelian Groups of Order Less Than 24

In this work, a non-abelian metabelian group is represented by G while represents conjugacy class graph. Conjugacy class graph of a group is that graph associated with the conjugacy classes of the group. Its vertices are the non-central conjugacy classes of the group, and two distinct vertices are joined by an edge if their cardinalities are not coprime. A group is referred to as metabelian if there exits an abelian normal subgroup in which the factor group is also abelian. It has been proven earlier that 25 non-abelian metabelian groups which have order less than 24, which are considered in this work, exist. In this article, the conjugacy class graphs of non-abelian metabelian groups of order less than 24 are determined as well as examples of some finite groups associated to other graphs are given.

The Laplacian Energy of Conjugacy Class Graph of Some Finite Groups

Let G be a dihedral group and its conjugacy class graph. The Laplacian energy of the graph, is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.