Introduction

Modern programming includes many different paradigms, approaches, techniques and programming languages. Object-Oriented Programming (OOP) is one of the famous and useful programming paradigm nowadays. Indeed, according to [1-3] the most popular programming languages are such languages, which support OOP.

As we know, object and class are the main concepts of OOP. According to [4], objects are the building blocks of an object-oriented program. We associate these blocks with the objects of real world, during developing programs. Concerning classes, they are blueprints, which we use as the basis for objects building. Every object is defined by two terms: attributes and behaviors. Attributes are properties of object, which describe it, and behaviors are procedures, functions (methods) which we can apply to this object and change its state, form and so on. According to [5], real world is created by objects, and OOP is the approach for description and simulation of this world or some his particular parts.

It is obvious, if person can develop the programs, which operate with models of objects of real world, their classes and so on, then this person precisely knows how to operate with them using his mind (intellect). Of course, sometimes people do this consciously, sometimes not, but it is important that they do this, and simple proof of this fact is that people have invented OOP. However, real world is very complicated, and consists of huge number of objects of different classes, that is why formalization of objects of real world into OOP objects and classes sometimes is nontrivial. Despite this, programming, in particular OOP, is developing day by day. People are finding more and more new practical applications for programming, and such area as Artificial Intelligence (AI) is not exception.

Modern AI includes many directions as knowledge representation, automated planning and scheduling, machine learning, robotics, computer vision, artificial creativity, natural language processing, and so on. Nevertheless, all these directions have something common among themselves. One of the main targets for all these directions is development of intelligent information systems (IIS) or artificial intelligent systems (AIS) for solving particular practical problems in these areas. Also very important aspect of AIS is intellectual analysis of information, in particular in real world or in some its parts. Ideologically this is very close to process of development of object-oriented programs. However, software development which is just tool for solving particular tasks, management by some process, etc. is easier then development of software which has some level of individuality and intellectuality and can do something bigger than just some computations. That is why questions about usability of development AIS using OOP approach appear.

Nowadays very often development of AIS is reduced to heuristic programming [6]. The advantage is that such approach gives an opportunity to solve corresponding practical problems, but the disadvantage is that partial solution does not guarantees general solution of corresponding problems. However, every AIS is based on some model of knowledge representation (KR). Nowadays, there is a variety of them. The most famous and common are Semantic Nets, Frames [7][8][9][10][11][12][13] and Logical Models [9][10][11][12][13][14] also Scripts [8][9][10], Conceptual Dependency [8,10], Search Spaces, Search Trees [7], Petri Nets [8], K-Lines, Memory Organization Packets [9], Semantic WEB, Computational Knowl-edge Discovery [11] are known. Furthermore, there are a few hybrid models of KR, in particular Brooks' Subsumption Architecture and Copycat [10].

Each of these models has own specifics and is useful in the particular domain. However, during developing certain AIS we need to implement a model of KR. Therefore, in this case, we have at least two levels of our AIS, first of them is a level of KR model and second one is a level of its practical implementation. In some cases, implementation of particular KR model can cause additional problems and difficulties. That is why, sometimes it leads to development of new programming paradigms and languages, which are invented for effective implementations of particular KR model. One of such examples is programming language PROLOG, which is logical programming language and it gives us opportunities for implementations of different Logical models of KR.

One of the main postulates of OOP is that objects create world. Development of OO-programs is formal description of objects from some world and relations between them, etc. That is why we can conclude that OOP can be one of the KR models, which represents (describes) knowledge about objects. Such model is very close to human perception of real world, because a person perceives real world by objects of this world. This fact gives us an opportunity to combine level of KR model and level of its practical implementation, during AIS development. In addition, a person also uses concept of class and set of objects in the process of thinking. Really, we use classes and sets in our mental activity during perception, analysis, comparison, retrieval, classification every day. We create classes and sets consciously or subconsciously, operate with them and apply a variety of operations to them. Moreover, set is the central concept in set theory and one of the most important for mathematics in general. However, questions about the origin of specific sets are emerging while analyzing the definition of this term, which is given in [15]. We can conclude that the "new" set can be obtained by set-theoretic operations over "existing" sets, analyzing the different systems of set theory [16,17], and it is really so. However, the questions about origin of these so-called "existing" sets, their number, their types and so on do not disappear. Our target is development of AIS for intellectual analysis of information, based on operating with sets of objects and their classes, using OOP. However, primarily we need to understand specifics of classes' and sets creation from different sides, i.e. sets in the set theory, sets and classes in OOP and in the process of human thinking.

Objects and Classes

Objects. We know that each set consists of elements, which form it. Everything, phenomena of our imagination or of our world can be the elements of the set. It is convenient for us to call them objects. Let consider such object as "natural number". It is clear that every natural number must be integer and positive. These are characteristic properties of natural numbers. It is obvious, that 11 is really a natural number, but −16 and 9.52, for example, are not natural numbers. We can conclude that each object has certain properties, which define it as some essence while analyzing this fact. Usually in OOP [4,5], we can consider properties of objects separately from objects. Such form of consideration is very close to concept of object-oriented class (OO class). Actually, objects and their properties cannot exist separately, because if we assume the opposite, we will have contradiction. On the one hand, object cannot exists separately from its properties, because without properties we cannot imagine and cannot describe it. On the other hand, object's properties cannot exist separately from object, because without object we cannot see and cannot perceive them. That is why, we cannot consider them separately, and there are few variants of the definitions order. It means that we cannot introduce definition of object without definition of its properties and vice versa. Therefore, we decided to introduce concept of object's properties firstly.

Globally we can divide properties of objects into two types -quantitative and qualitative. We will define these two types of object's properties formally, but their semantics has intuitive nature.

Definition 1. Quantitative property of objectA is a tuple ))) ( ( )), ( ( ( ) ( A p u A p v A p i i i  , where n i , 1  , )) ( ( A p v i is an quantitative value of ) ( A p i ,and)) ( ( A p u i are units of measure of quantitative value of ) ( A p i .

Let us consider some examples of quantitative properties of objects. Suppose we have an apple, and one of its properties is weight. We can present this property as follows We can conclude that, such approach gives an opportunity to combine description of property and its verification in the one function, i.e. verification function is a verification function and a description of property at the same time. Therefore, different algorithms can be verifiers and descriptors of property simultaneously.  .

))) ( ( )), (( ( )

Classes of Objects.

In general, we can divide objects on concrete and abstract, and does not matter when or how someone created each particular object. It is material implementation of its abstract imagea prototype. This prototype is essentially an abstract specification for creation the future real objects. Besides properties of objects, we should allocate operations (methods) which we can apply to objects, considering the features of their specifications. Really, we can apply some operations (methods) to objects for their changing and for operating with them. That is why, it will be useful to define concept of object's operation (method).

Definition 9. Operation (method) of objectA is a function ) ( A f

, which we can apply to object A considering the features of its specification.

For example, for such objects as natural numbers n , m we can define operations "+" and "*". In OOP [4,5,19,20], programmers operate with specifications and methods of objects without objects, and they call it a type or a class of object. It consists of fields and methods. Fields of class, essentially, are specification of class. Methods are functions, which we can apply to objects of this class for their changing and for operating with them. Concept of OO class is similar to universal algebra [21,22], where carrier of algebra consists of objects and signature consists of methods of objects. That is why, henceforth sometimes we will use title signature of class for methods of class. Let us define concept of object's signature.

Definition 10. Signature of objectA is a vector )) ( ),..., ( ( ) ( 1 A f A f A F m 

, where

) (A f i , m i , 1 

is an operation (method) of object A .

Generally, signature of particular object can consist of different quantity of operations, but in practice, especially in programming, usually we are considering finite signatures of objects.

According to definition of object, every object has some specification, which defines it as some essence. There are some objects, which have similar specifications. It means that we can apply the same methods to them. Let us define similar objects. Definition 11. Objects A and B are similar objects, if and only if, when they have the same dimension and equivalent specifications.

If certain two objects are similar, we can conclude that these objects have the same type or class. Now we can introduce concept of object's class.

Definition 12. Object's class T is a tuple )) ( ), ( ( T F T P T 

, where ) (T P is abstract specification of some quantity of objects, and ) (T F is their signature.

When we talk about class of objects, we mean properties of these objects and methods, which we can apply to them. Class of objects is a generalized form of consideration of objects and operations on them, without these objects.

As an example, let us describe type Int in programming language C++, using concept of similar objects and object's class. Let Considering concept of OO class, we can conclude that class is a prototype for particular objects, and all objects of class are described by class description. It means that every particular object of the class has the same specification and signature. That in turn, imposes some constraints on description of objects from real world. There are many different objects of real world, which belong to different classes, and if we need to work with them, we can describe them, using new class for each new type of objects. Especially, if we work with not very big quantity of different types of objects, we can do it without any fears. However, if we need to work with huge quantity of different types, for example, with a few thousands of different types, just a process of description of such types is very complex and time-consuming not to mention size of code and performance of such programs.

According to definitions of object and object's class, we can conclude two points. Firstly, every object is a member of at least one class of objects, and secondly, objects and their classes cannot exist separately. Furthermore, some objects are members of few classes simultaneously. For example, such objects as natural numbers

Universal Operation on Objects and Runtime Class Generation

One of the most important aspects of OOP is an opportunity to work only with classes, which are described before program execution. Of course, some programs provide us with a possibility to work with new classes, which we can obtained using basic classes. However, we cannot obtain new classes, which are not inheritors of basic classes, during program execution. In modern programming, this process is known as runtime class generation (RCG) or runtime class creation (RCC). Nowadays, there are some approaches for implementation of this task for some OOP-languages, in particular for Java [24] and C# [25]. These tools based on manipulating with bytecode and implemented for such platforms of programming as Java and .NET. However, now we will not discuss practical implementations of RCG for concrete platforms of programming, but will focus on specifics of this process.

It is known, that logic of program provides access and work with some classes of objects, during program execution, depending on particular program scenario. Moreover, it can provide RCG, based on some constructors of classes. According to this, we will propose some constructors of classes, defining some universal operations on objects.

As it was mentioned before, in OOP, objects have methods. Usually they are functions, which we can execute for these objects. We can divide methods of objects on two types, depending on character of their action. They are modifiers and exploiters. Modifiers are functions, which can change objects, in particular some fields of objects. Exploiters are functions, which use objects as arguments and cannot change them. However, majority of methods of objects are local with respect to objects, and cannot be applied to objects of different types. Of course, there are some methods, which we can apply to objects of different types, but usually we need to use overloading operator for this. Nevertheless, we will define some universal operations for objects, which can be applied to any objects. Let us do it using the concept of object and object's class.

Definition 15. Union  of 2  n

arbitrary objects is a new set of objects S , which obtain in the following way

) ( / } ,..., { ) ( / ... ) ( / 1 1 1 S T A A A T A A T A S n n n    

, where n A A ,..., 1 are objects, such that

j i A A , 

, where

n j i , 1 , andj i  , 0 ) , (  j i A A Eq , ) ( i A T , n i , 1  is a class of object i A and ) (S T

is a class of new set of objects S and n is its cardinality.

Let us consider such geometrical objects as triangle, square and trapeze. It is obvious that these objects belong to different classes of geometrical figures. Let us denote triangle as A , square as B , trapeze as C , and describe their classes as follows )) ( ),..., ( ( ) (

4 1 A p A p A P  , )) ( ),..., ( ( ) ( 3 1 B p B p B P  , )) ( ),..., ( ( ) ( 4 1 C p C p C P  , )) ( ), ( ( ) ( 2 1 A f A f A F  , )) ( ), ( ( ) ( 2 1 B f B f B F  and )) ( ), ( ( ) ( 2 1 C f C f C F  . Properties ) ( 1 A p , ) ( 1 B p , ) ( 1 CA f A p A pr  , )) ( ( ) ( 2 2 B f B pr  , )) ( ), ( ( ) ( 2 4 3 C f C p C pr 

. Essentially, set of objects S is the set of trian-

  A f A f Eq A f A F i i i  . DifferenceA f A p C A pr  .

As in the case of intersection, we have obtained new class of objects, which describes some type of geometric figures, however, unlike the previous case we can say that this new type of geometric figures is a triangle, although it is described using smaller specification.

Definition 18. The symmetrical difference  of two arbitrary objects 1

A and 2 A is a class of objects )) ( ), ( (

) ( A F A P A T  , where 0 )) ( ), ( ( | )) ( ), ( ( ) ( 2 1 2 1 2 1 2 1   A p A p Eq A p A p A P i i i i  , 0 )) ( ), ( ( | )) ( ), ( ( ) ( 2 1 2 1 2 1 2 1   A f A f Eq A f A f A F i i i i  . Symmetrical differenceC A pr C A pr C A T     , where )) ( ), ( ( ) ( 2 4 1 A f A p C A pr   and )) ( ), ( ( ) ( 2 4 2 C f C p C A pr  

. As in the case of difference, we have obtained new class of objects, which describes a type of geometric figures, however, as in the case of intersection, we do not know exactly which type of geometrical figures this class describes. Nevertheless, we can say that this class describes two types of geometrical figures, and one of them is triangle.

1 1 1 S T C B A A S T B A C T C A T A S T S C T C A T A S        .

As we can see, creation of sets and multisets of objects entails creation of new classes of objects, in particular inhomogeneous, that causes some problems related to description of such classes in OOP. At first glance it may seems, that there is no necessity to work with such classes, and we can use just concept of homogeneous class, that implemented within OOP. However, if we analyze some aspects of human mind, especially mechanisms of its thinking and analysis, we can conclude that concept of set of objects is one of the basic and very important concepts for them. Really, let us consider situation, when you need to find some book among big amount of different books, which lying on the shelves of bookcase. If you know how exactly this book looks, you can imagine this book and you can distinguish it from other books in this bookcase. Finally, during searching for this book, you perform certain exhaustive search, and at the same time, you create set of books, which you have checked. Let us imagine another situation, when you need to count money, which you have in your wallet. During counting, you create at least two sets, set of banknotes and set of coins. In addition, we can consider situation when you want to play chess, and after opening the box with chess, you need to make initial arrangement of figures on the chessboard. During figures placement, you create set of white and set of black figures from set of all figures. These are just a few simple examples from our daily activity. Very often in such situations and similar to them we operate with sets of objects subconsciously, and as the result, we pay little attention to this, but it is happening permanently. That is why we can conclude that creation of sets and classes of objects is normal activity of our mind, and if we want to simulate and reproduce this process, using computer system, we need to have certain opportunities to work with sets of objects and inhomogeneous classes of objects. It means we need appropriate tools within certain OOP-language for working with such abstractions.

Nowadays, there are a few different implementations of tools for working with sets within some OOP-languages, in particular set in STL for C++ [26], HashSet in Java [20], HashSet, SortedSet and ISet in C# [27], set and frozenset in Python [28]. These tools allow sets creation, executing set-theoretic operations, membership checking, adding and removing of elements and checking of equivalence between sets, etc. However, we have not opportunities for working with classes of sets and multisets, and this is one of disadvantages of modern OOP.

Conclusions

In this paper we analyzed such basic concepts of OOP as object and class and their relation with set theory and AI. According to this, definitions of object and its properties, homogeneous and inhomogeneous classes of objects were proposed. We defined such operations on objects as union, intersection, difference, symmetric difference and cloning. Besides, we considered process of runtime class generation and showed that intersection, difference, symmetric difference of objects are examples of classes' generators.

Process of sets and multisets creation was considered in details from different sides, in particular mathematical set theory, OOP and AI. After that, constructive definition of set of objects, multiset of objects and methods of their creation were proposed. Relation between sets of objects and classes of objects was also shown. The proposed approach allows not only to create (generate) sets and multisets of objects, but also to classify them. It also allows considering the problem of object classification and identification in another way.