The visualization of a graph semantics of imperative languages

This work aims to present the software support for teaching in the field of formal semantics of imperative programming languages. The main part focuses on a software tool that provides a visual representation of the individual steps of the calculation in categorical semantics, which can also be referred to as graph semantics. The use of software tools in teaching to visually represent computational steps considerably facilitates understanding by students and can also serve as a good basis for supporting distance learning. Our program works in the standard form: after reading the correct user input, a visual representation of the meaning of the program is generated in the form of a category of states, which is displayed as an oriented graph. For better extensibility, the program is implemented as a web application.

Some Possibilities of Modeling Colored Petri Nets

Petri nets are a mathematical apparatus for modelling dynamic discrete systems. Their feature is the ability to display parallelism, asynchrony and hierarchy. First was described by Karl Petri in 1962 [1,2,8]. The Petri net is a bipartite oriented graph consisting of two types of vertices - positions and transitions connected by arcs between each other; vertices of the same type cannot be directly connected. Positions can be placed by tags (markers) that can move around the network. [2] Petri Nets (PN) used for modelling real systems is sometimes referred to as Condition/Events nets. Places identify the conditions of the parts of the system (working, idling, queuing, and failing), and transitions describe the passage from one state to another (end of a task, failure, repair...). An event occurs (a transition fire) when all the conditions are satisfied (input places are marked) and give concession to the event. The occurrence of the event entirely or partially modifies the status of the conditions (marking). The number of tokens in a place can be used to identify the number of resources lying in the condition denoted by that place [1,2,8]. Coloured Petri nets (CPN) is a graphical oriented language for design, specification, simulation and verification of systems [3-6,9,15]. It is in particular well-suited for systems that consist of several processes which communicate and synchronize. Typical examples of application areas are communication protocols, distributed systems, automated production systems, workflow analysis and VLSI chips. In the Classical Petri Net, tokens do not differ; we can say that they are colourless. Unlike standard Petri nets in Colored Petri Net of a position can contain tokens of arbitrary complexity, such as lists, etc., that enables modelling to be more reliable. The article is devoted to the study of the possibilities of modelling Colored Petri nets. The article discusses the interrelation of languages of the Colored Petri nets and traditional formal languages. The Venn diagram, which the author has modified, shows the relationship between the languages of the Colored Petri nets and some traditional languages. The language class of the Colored Petri nets includes a whole class of Context-free languages and some other classes. The paper shows modelling the task synchronization Patil using Colored Petri net, which can't be modeled using well- known operations P and V or by classical Petri network, since the operations P and V and classical Petri networks have limited mathematical properties which do not allow to model the mechanisms in which the process should be synchronized with the optimal allocation of resources.

The Representation Theory of Neural Networks

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.

List-antimagic labeling of vertex-weighted graphs

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.

Ford-Fulkerson algorithm refinement for the cooperation effectiveness increase of intensive and low-density lines

The article describes tools of the railway control on intensive and low-density lines which is directed on the effectiveness increase of low-density line functioning, for the solution of perspective tasks of the railway network functioning and development. For the too realization the oriented graph with the Ford-Fulkerson algorithm which allows determining the maximum flow and the minimum cut for non-oriented graphs. Firstly as values of graph tops inverse values of the station rating and as graph edges inverse values of the railway line class are accepted. The use of this approach allows determining the maximum flow in the system and provides the clear view of relations of transportation capacities of railway lines and stations.

Vertex Turán problems for the oriented hypercube

In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph F → \vec F , determine the maximum cardinality e x v ( F → , Q → n ) e{x_v}\left( {\vec F,{{\vec Q}_n}} \right) of a subset U of the vertices of the oriented hypercube Q → n {\vec Q_n} such that the induced subgraph Q → n [ U ] {\vec Q_n}\left[ U \right] does not contain any copy of F → \vec F . We obtain the exact value of e x v ( P k , → Q n → ) e{x_v}\left( {\overrightarrow {{P_k},} \,\overrightarrow {{Q_n}} } \right) for the directed path P k → \overrightarrow {{P_k}} , the exact value of e x v ( V 2 → , Q n → ) e{x_v}\left( {\overrightarrow {{V_2}} ,\,\overrightarrow {{Q_n}} } \right) for the directed cherry V 2 → \overrightarrow {{V_2}} and the asymptotic value of e x v ( T → , Q n → ) e{x_v}\left( {\overrightarrow T ,\overrightarrow {{Q_n}} } \right) for any directed tree T → \vec T .

The diachromatic number of double star graph

We are interested in the extension for the concept of complete colouring for oriented graph G → that has been proposed in many different notions by several authors (Edwards, Sopena, and Araujo-Pardo in 2013, 2014, and 2018, respectively). An oriented colouring is complete if for every ordered pair of colours, at least one arc in G → whose endpoints are coloured with these colours. The diachromatic number, dac ( G → ) , is the greatest number of colours in a complete oriented colouring. In this paper, we establish the formula of diachromatic numbers for double star graph, k 1 , n , n → , over all possible orientations on the graph. In particular, if din (u) = 0 (resp. dout(u) = 0)and din (wi ) = 1 (resp. dout (w 1) = 1) for all i, then dac ( k 1 , n , n → ) = ⌊ n ⌋ + 1 , where u is the internal vertex and w i , i ∈ {1,…, n}, is the pendant vertices of the digraph.

Document-level medical relation extraction via edge-oriented graph neural network based on document structure and external knowledge

Objective Relation extraction (RE) is a fundamental task of natural language processing, which always draws plenty of attention from researchers, especially RE at the document-level. We aim to explore an effective novel method for document-level medical relation extraction. Methods We propose a novel edge-oriented graph neural network based on document structure and external knowledge for document-level medical RE, called SKEoG. This network has the ability to take full advantage of document structure and external knowledge. Results We evaluate SKEoG on two public datasets, that is, Chemical-Disease Relation (CDR) dataset and Chemical Reactions dataset (CHR) dataset, by comparing it with other state-of-the-art methods. SKEoG achieves the highest F1-score of 70.7 on the CDR dataset and F1-score of 91.4 on the CHR dataset. Conclusion The proposed SKEoG method achieves new state-of-the-art performance. Both document structure and external knowledge can bring performance improvement in the EoG framework. Selecting proper methods for knowledge node representation is also very important.

Lexical attention and aspect-oriented graph convolutional networks for aspect-based sentiment analysis

The purpose of aspect-based sentiment analysis is to predict the sentiment polarity of different aspects in a text. In previous work, while attention has been paid to the use of Graph Convolutional Networks (GCN) to encode syntactic dependencies in order to exploit syntactic information, previous models have tended to confuse opinion words from different aspects due to the complexity of language and the diversity of aspects. On the other hand, the effect of word lexicality on aspects’ sentiment polarity judgments has not been considered in previous studies. In this paper, we propose lexical attention and aspect-oriented GCN to solve the above problems. First, we construct an aspect-oriented dependency-parsed tree by analyzing and pruning the dependency-parsed tree of the sentence, then use the lexical attention mechanism to focus on the features of the lexical properties that play a key role in determining the sentiment polarity, and finally extract the aspect-oriented lexical weighted features by a GCN.Extensive experimental results on three benchmark datasets demonstrate the effectiveness of our approach.