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1

Optimized Method based on Lattice Sequences for Multidimensional Integrals in Neural Networks

Todorov Venelin, Dimov Ivan, Fidanova Stefka

Annals of Computer Science and Information Systems

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2021

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Vol. 25

243--246

EN

In this work we investigate advanced stochastic methods for solving a specific multidimensional problem related to neural networks. Monte Carlo and quasi-Monte Carlo techniques have been developed over many years in a range of different fields, but have only recently been applied to the problems in neural networks. As well as providing a consistent framework for statistical pattern recognition, the stochastic approach offers a number of practical advantages including a solution to the problem for higher dimensions. For the first time multidimensional integrals up to 100 dimensions related to this area will be discussed in our numerical study.

2

Approximation Algorithms for Three Dimensional Protein Folding

Shaw D. L., Karmaker S., Islam A. S. M. S, Rahman M. S.

Fundamenta Informaticae

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2018

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Vol. 163, nr 4

375--393

EN

Predicting the secondary structure of a protein using a lattice model is one of the most studied computational problems in bioinformatics. Here the secondary structure or three dimensional structure of a protein is predicted from its amino acid sequence. The secondary structure refers to the local sub-structures of a protein. Simplified energy models have been proposed in the literature on the basis of interaction of amino acid residues in proteins. We focus on a well researched model known as the Hydrophobic-Polar (HP) energy model. In this paper, we propose the hexagonal prism lattice with diagonals that can overcome the problems of other lattice structures, e.g., parity problem. We give two approximation algorithms for protein folding on this lattice using HP model. Our first algorithm leads us to a similar structure of helix structure that is commonly found in a protein structure. This motivates us to propose the next algorithm with a better approximation ratio. Finally, we analyze the algorithms on the basis of intensity of the chemical forces along the different types of edges of hexagonal prism lattice with diagonals.

3

Lattice Theory for Rough Sets : A Case Study with Mizar

Grabowski A.

Fundamenta Informaticae

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2016

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Vol. 147, nr 2/3

223--240

EN

Rough sets offer a well-known approach to incomplete or imprecise data. In the paper I briefly report how this framework was successfully encoded by means of one of the leading computer proof-assistants in the world. The general approach is essentially based on binary relations, and all natural properties of approximation operators can be obtained via adjectives added to underlying relations. I focus on lattice-theoretical aspects of rough sets to enable the application of external theorem provers like EQP or Prover9 as well as to translate them into TPTP format widely recognized in the world of automated proof search. I wanted to have a clearly written, possibly formal, although informal as a rule, paper authored by a specialist from the discipline another than lattice theory. It appeared that Lattice theory for rough sets by Jouni Järvinen (called LTRS for short) was quite a reasonable choice to be a testbed for the current formalisation both of lattices and of rough sets. A popular computerised proof-assistant Mizar was used as a tool, hence all the efforts are available in one of the largest repositories of computer-checked mathematical knowledge, called Mizar Mathematical Library.

4

Inverted Fuzzy Implications in Approximate Reasoning

Suraj Z., Lasek A., Lasek P.

Fundamenta Informaticae

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2016

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Vol. 143, nr 1/2

151--171

EN

In 1973 Lotfi Zadeh introduced the theory of fuzzy logic [17]. Fuzzy logic was an extension of Boolean logic so that it allowed using not only Boolean values to express reality. One kind of basic logical operations in fuzzy logic are so-called fuzzy implications. From over eight decades a number of different fuzzy implications have been described [3] - [16]. In the family of all fuzzy implications the partial order induced from [0,1] interval exists. Pairs of incomparable fuzzy implications can generate new fuzzy implications by usingmin(inf) andmax(sup) operations. As a result the structure of lattice is created ([1], page 186). This leads to the following question: how to choose the correct functions among basic fuzzy implications and other generated as described above. In our paper, we propose a new method for choosing implications. Our method allows to compare two fuzzy implications. If the truth value of the antecedent and the truth value of the implication are given, by means of inverse fuzzy implications we can easily optimize the truth value of the implication consequent. In other words, we can choose the fuzzy implication, which has the greatest or the smallest truth value of the implication consequent or which has greater or smaller truth value than another implication. Primary results regarding this problem are included in the paper [14].

5

Monteiro Spaces and Rough Sets Determined by Quasiorder Relations : Models for Nelson algebras

Järvinen J., Radeleczki S.

Fundamenta Informaticae

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2014

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Vol. 131, nr 2

205--215

EN

The theory of rough sets provides a widely used modern tool, and in particular, rough sets induced by quasiorders are in the focus of the current interest, because they are strongly interrelated with the applications of preference relations and intuitionistic logic. In this paper, a structural characterisation of rough sets induced by quasiorders is given. These rough sets form Nelson algebras defined on algebraic lattices. We prove that any Nelson algebra can be represented as a subalgebra of an algebra defined on rough sets induced by a suitable quasiorder. We also show that Monteiro spaces, rough sets induced by quasiorders and Nelson algebras defined on T0-spaces that are Alexandrov topologies can be considered as equivalent structures, because they determine each other up to isomorphism.

6

Rough Net Structures : Example of Information System

Czaja L.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

81--97

EN

Information system of net structures based on their calculus (a distributive lattice) is introduced and, in this context, basic notions of rough set theory are re-formulated and exemplified.

7

Generalized Hybrid Encoding of Polyhierarchical Structures

Lewandowski J., Rybiński H.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

461--477

EN

Polyhierarchical structures play an important role in artificial intelligence, especially in knowledge representation. The main problem with using them efficiently is lack of efficient methods of accessing related nodes, which limits the practical applications. The proposed hybrid indexing approach generalizes various methods and makes possible combining them in a uniform manner within one index, which adapts to a particular topology of the data structure. This gives rise to a balance between compactness of the index and fast responses to the search requests. The correctness of the proposed method is formally shown, and its performance is evaluated. The results prove its high efficiency.

8

Convex sublattice based reliability theory

Pang Y., Huang H. Z., He L., Wang Z., Xiao N. C.

Eksploatacja i Niezawodność

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2011

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nr 3

56-61

EN

Classical probability theory has been widely used in reliability analysis; however, it is hard to handle when the system is lack of` adequate and sufficient data. Nowadays, alternative approaches such as possibility theory and fuzzy set theory have also been proposed to analyze vagueness and epistemic uncertainty regarding reliability aspects of complex and large systems. The model presented in this paper is based upon possibility theory and multistate assumption. Convex sublattice is addressed on congruence relation regarding the complete lattice of structure functions. The relations between the equivalence classes on the congruence relation and the set of all structure functions are established. Furthermore, important reliability bounds can be derived under the notion of convex sublattice. Finally, a numerical example is given to illustrate the results.

PL

Klasyczna teoria prawdopodobieństwa ma szerokie zastosowanie w analizie niezawodności, jednak trudno jest się nią posługiwać, kiedy brak jest wystarczających i odpowiednich danych na temat systemu. Obecnie, proponuje się alternatywne podejścia, takie jak teoria możliwości czy teoria zbiorów rozmytych, za pomocą których można analizować niepewność epistemiczną oraz nieostrość w odniesieniu do aspektów niezawodności złożonych i dużych systemów. Model przedstawiony w niniejszym artykule oparto na teorii możliwości oraz na założeniu wielostanowości. Podkratę wklęsłą opisano na relacji kongruencji, odnoszącej się do całej kraty funkcji struktury. Ustalono relacje pomiędzy klasami równoważności na relacji kongruencji a zbiorem wszystkich funkcji struktury. Ponadto posługując się pojęciem podkraty wypukłej można wyprowadzać istotne kresy niezawodności. Wyniki zilustrowano przykładem numerycznym.