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1

Learning software for handling the mathematical expressions

100%

Steingartner W. , Yar-Muhamedov I.

Journal of Applied Mathematics and Computational Mechanics

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2018

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tom Vol. 17, nr 2

77--91

EN

Educating young software engineers and IT experts is a great challenge nowadays. Still new technologies are used in a practical approach and many of them come from formal methods. To help future software experts in the understanding of formal methods grounded in semantics, learning software that illustrates and visualizes important techniques seems to be very fruitful. In this paper, we present software, which handles the arithmetic and Boolean expressions, their analysis, evaluation, drawing the syntax tree and the other techniques with the expressions. This software is devoted as a teaching tool for teachers when explaining appropriate theory and for students for self-studying and making their own experiments. Furthermore, this software is an integral part of our software package for several semantic methods.

2

Coalgebras for modelling observable behaviour of programs

100%

Steingartner W. , Novitzka V.

Journal of Applied Mathematics and Computational Mechanics

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2017

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tom Vol. 16, nr 2

145--157

EN

A useful tool for modelling behaviour in theoretical computer science is the concept of coalgebras. Coalgebras enable one to model execution of programs step by step using categorical structures and polynomial endofunctors. In our paper, we start with a short introduction of basic notions and we use this method for modelling structural operational semantics of a simple imperative language.

3

The rôle of categorical structures in infinitesimal calculus

100%

Steingartner W. , Galinec D.

Journal of Applied Mathematics and Computational Mechanics

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2013

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tom Vol. 12, nr 1

107--119

EN

The development of mathematics stands as one of the most important achievements of humanity, and the development of the calculus, differential calculus and integral calculus is one of the most important achievements in mathematics. Differential calculus is about finding the slope of a tangent to the graph of a function, or equivalently, differential calculus is about finding the rate of change of one quantity with respect to another quantity. On the other hand, integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The category theory is a mathematical approach to the study of algebraic structure that has become an important tool in theoretical computing science, particularly for semantics-based research. The notion of a limit in category theory generalizes various types of universal constructions that occur in diverse areas of mathematics. In our paper we illustrate how to represent some parts of infinitesimal calculus in categorical structures.

DE

Die Theorie von Kategorien ist der Bereich von Mathematik und sie dient vor allem fur das Studium der algebraischen Strukturen. Sie wird aber sehr oft auch in Informatik geltend gemacht. Manche bedeutende mathematische Bereiche kann man mithilfe der Kategorien darstellen und das ermöglicht mit den mathematischen Strukturen viel einfacher zu arbeiten als ohne Anwendung der Kategorien. Der Grund der Infinitesimalrechnung bilden zwei duale Bereiche - Differential- und Integralrechnung. In unserem Beitrag orientieren wir uns auf die Konstruktion des Diagramms von Stammfunktionen zur Funktion Kosinus. Von diesen Funktionen konstruieren wir den kommutativen Kegel und wir finden seinen Grenzwert. In dem zweiten Teil des Artikels zeigen wir den Ausdruck der Derivationen von Funktionen in der Kommakategorie und wir konstruieren den kodomänen Funktor zwischen der Kategorie der Derivationen und der Kategorie der Mengen für differenzierbare Funktionen.

4

Some properties of coalgebras and their role in computer science

80%

Steingartner W. , Radakovic D. , Valkosak F. , Macko P.

Journal of Applied Mathematics and Computational Mechanics

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2016

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tom Vol. 15, nr 4

145-156

EN

This paper introduces basic theoretical knowledge of coalgebras in computer science. Coalgebras are, specifically in category theory, structures defined according to an endofunctor. For both algebra and coalgebra, a functor is a convenient and general way of defining a signature. We present practical usage of the coalgebras in an example. We observe a behavior of a simple Sequencer developed in SLGeometry framework. We model its behavior with the simple program written in Python, and we describe its behavior within coalgebra of endofunctor. The computation of the values stored in internal states is performed coinductively. Our approach can be used in the teaching process of formal methods for young software engineers.

5

Linear logic in computer science

80%

Steingartner W. , Poláková A. , Prazňák P. , Novitzká V.

Journal of Applied Mathematics and Computational Mechanics

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2015

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tom Vol. 14, nr 1

91--100

EN

Linear logic has many properties that make it suitable for application in various areas of computer science. It is able to describe dynamic processes, non-determinism, parallelism on syntactic level. In our paper we try to discuss resource oriented character of linear logic, its possibility to deal with such important resources for computer science as space (memory) and time. Handling with resources takes place in deduction system of linear logic. We show how special form of proofs, called designs, is constructed and we show the relationship between space and time in designs.