# Logical Operations in Algebraic System of Aggregates for Multimodal Data Representation and Processing

## Authors

• Ivan A. Dychka Igor Sikorsky Kyiv Polytechnic Institute, Ukraine
• Yevgeniya S. Sulema Igor Sikorsky Kyiv Polytechnic Institute, Ukraine

## Keywords:

Algebraic system of aggregates, Multi-image, Multimodal data, Logical operations

## Abstract

Background. The range of computer software applications which operate with multimodal data becomes wider and wider. To develop efficient algorithms for data processing, the representation of multimodal data as complex structures is required. Existing approaches are mostly oriented to independent data sets representation and they are not efficient for complex representation of multimodal data.

Objective. The development of a mathematical approach which can be used for complex multimodal data representation and processing in computer systems.

Methods. The Algebraic System of Aggregates is developed for enabling complex representation of multimodal data. There are logical, ordering and arithmetical operations in the Algebraic System of Aggregates. Logical operations can be used for preparing complex data representations in a form of aggregates which are supposed to be a subject of further data analysis. The precondition for such representation is that data sequences are of different modalities and they are recorded with respect to time.

Results. The logical operations on aggregates are proposed and described. They allow to construct different compositions of multimodal data what in its turn enables complex data representation for compound description of objects and processes in different areas including healthcare. The multi-image concept which enables overall description of an object, a subject, or a process of observation carried out in the course of time is also proposed and discussed.

Conclusions. The Algebraic System of Aggregates is a tool for complex data representation. It enables multimodal data processing by using a range of operations and relations. The key difference in the fulfilment of logical operations in the Algebraic System of Aggregates and logical operations on sets is that an aggregate is a form of complex representation of data sequences where the order of data in each sequence of each type is important and influences on the order of data in the result of a certain logical operation.

## Author Biographies

### Ivan A. Dychka, Igor Sikorsky Kyiv Polytechnic Institute

Іван Андрійович Дичка

### Yevgeniya S. Sulema, Igor Sikorsky Kyiv Polytechnic Institute

Євгенія Станіславівна Сулема

## References

A.A. Fraenkel et al., Foundations of Set Theory. Elsevier, 1973, 415 p.

B.A. Kulik et al., Algebraic Approach to Intellectual Processing of Data and Knowledge. Saint Petersburg, Russia: SPbPU Publ., 2010, 235 p.

V.G. Kanovey and V.A. Liubetskii, Modern Theory of Sets: Borel and Projective Sets. Moscow, Russia: MTsNMO, 320 p., 2010 (in Russian).

B.A. Kulik, “Generalized approach to modelling and analysis of intellectual systems based on tuples algebra”, in Proc. VI Int. Conf. “Identification of systems and management tasks” SICPRO’07, IPU RAN, 2007, pp. 679–715.

A.B. Petrovsky, “An axiomatic approach to metrization of multiset space”, in Proc. X Int. Conf. Multiple Criteria Decision Making, Taipei, Taiwan, 1992, vol. 1, pp. 381–390.

A.B. Petrovsky, et al., “An axiomatic approach to metrization of multiset space”, in Multiple Criteria Decision Making. G.H. Tzeng et al., eds. New York: Springer-Verlag, 1994, pp. 129–140. doi: 10.1007/978-1-4612-2666-6_14

A.B. Petrovsky et al., “Structuring techniques in multiset spaces”, in Multiple Criteria Decision Making. Lecture Notes in Economics and Mathematical Systems, vol. 448, G. Fandel and T. Gal, eds. Berlin, Heidelberg, Germany: Springer-Verlag, 1997, pp. 174–184. doi: 10.1007/978-3-642-59132-7_20

A.B. Petrovsky et al., “Multi-attribute sorting of qualitative objects in multiset spaces”, in Multiple Criteria Decision Making in New Millennium. Lecture Notes in Economics and Mathematical Systems, vol. 507, M. Köksalan and S. Zionts, eds. Berlin, Heidelberg, Germany: Springer-Verlag, 2001, pp. 124–131. doi: 10.1007/978-3-642-56680-6_11

A.B. Petrovsky, “Constructing a general decision rule for contradictory expert classification of multi-attribute objects”, Pattern Recognition and Image Analysis, vol. 11, no. 1, pp. 73–76, 2001.

A.B. Petrovsky, Space of Sets and Multi-Sets. Moscow, SU: Editorial URSS, 2003, 248 p.

A.I. Maltsev, Algebraic Systems. Moscow, SU: Nauka, 1970, 392 p.

Ye. Sulema, “ASAMPL: Programming language for mulsemedia data processing based on algebraic system of aggregates”, Interactive Mobile Communication Technologies and Learning. IMCL 2017. Advances in Intelligent Systems and Computing, vol. 725, M. Auer and T. Tsiatsos, eds. Cham, Switzerland: Springer, 2018, pp. 431–442. doi: 10.1007/978-3-319-75175-7_43