Printed EAST-WEST International Conferense "Information technology in design", Moscow, Russia, 5-9 September, 1994.
Defining Information as an Object of Informational Technology. By N.A.Lumpov.
ABSTRACT: This paper is writen to discuss the problems of conseptual stage of designing applied systems functioning in the class of means of informational technologies. The notion "informational technologies" has a wide meaning and that why it is nessary to specify the features characterizing information assigment of these technologies. The following definition of information is given in this paper: the information is the result of reflection of objects belonging to one system (source) by the objects of another system (receptor).
The world seems to be nothing but nail to the man who has noting but hammer.
Abraham Maslow
The solution of problems of general theory and methodology is supposed to give answers to the most general questions which are connected in one or another way with all the others, more specific ones [1]. For the specialists in theory and practice of the development and use of informational technologies (IT), one of such problems consists in the definition of notions of the information.
The intuitive approach to the definition of the sense and content of information is in the majority of cases sufficient to be able to design and create fuctioning software packages that use one or another IT for collecting, updating, storing and processing the data, as well as for its output. By that this data, depending on the context and the domain of application may be interpreted as knowledge. A slightly different situation arises in such cases when one tries to estimate capacity and ability of an IT to provide for evolutionary development of data (knowledge) systems and software packages managing this systems.
The questions to be answered in this case are as follows:
- how to ensure the growth of data and knowledge about specific matter, about data systems?,
- is the interface linking the man and the system in user-friendly and well though out?
Аnd so on.
The functionality of any software built on of the base of an IT, depends in some extent upon the way the problems of program systems are solved. Moreover, one can say that the competitiveness and "vitality" of software directly depend of the way the problem of evolutianary development of the data and knowledge systems solved. Attempts that were aimed at some formalization of the problems of creation of developing and self-organizing systems of data has lead to the necessity to analyze the essence and notion of IT.
The essense and notion of IT are important for systems of automated design of software complexes and they are in any case related to the solutions of problems of typifying architectural ans systemotechnie concepts being used as the base of software packages. The formalization of designing such systems, unification of methods of data processing is mainly determined by the results of analysis of creative power of IT because of the fact that such property of IT as "informativity", has rather high semantic vagueness and therefore, deep content. In great extent it is explained by the absence of strict definition of information which would meet the needs of the IT analysis. That is why for defining the content of IT, it nessary first of all to consider the notion of information.
Let us define the technology as an aggegate of method and means enabling to collect, treat and process the information, and then we shall pass to exposure of the notion of information.
We will consider reflection as the capacity of any structure to reproduce in its changes the objects and structures interacting with it. Object will be understood as any essence of either ideal or material character, and speaking more exactly, "something" according to Hegel. System will be understood as an entity, composed of its parts. Another definition of system was given by Gaines [2], namely, then system is what is distinguished as system. In other words the possibility to interpret any object, phenomena or process as a system constitutes indispensable and sufficient condition for them to be considered as systems (evendtly these objects, phenomena, process should exist). Since the demonstration of existence of objects and systems, as well as the reality of the reflection process, belong to another domain of examination, their existence is assumed axiomatically indisputable.
Definition. The information is the result (!) of reflection of objects belonging to a system by the objects of another system.
Essential point of this definition consist in the presence of three sides, namely, the system interpeted as source, the system interpreted as receptor and the result which is alienated in favor of some third party. This situation may be represented by the chart given in figure 1a. At the same time, from the point of view of analysis of cause-and-effect links, the definition of information may be represented by the chart shown in figure 1b.
Characteristics, features, sighs inherent in the information which update and distinguish an information process, consist in such combination of forces, energy and masses that the expected fact, exactly correlated with some result, is alienated in one or another way, in favor of the third party. The nature of result is constitued by the fact that the result is the effect of possible to talk about a system of objects, the existence of each of them not being identical to that chaotic structures, but being identical conditioned by some principles of energetic, organizational and system character. On the other hand, the expectation of some fact is the property inherent in the structure of system-source, and although the result is sended by means of reflection, it depends on properties of system-receptor. The latter is one of the manifectations of relativist principle in the interpretation of the information properties, namely, the "observer" determines in some extent the result of reflection.
Fig. 1. Charts illustrating the definition of informarion.
Another important point in the definition of information is constituted by the conditional character of information (see fig. 1b), namely, the effect as one of the sides updating of the information phenomena, is derectly connected with condition expressed by the presense of receptor capable to represent some result of reflection in its structures. Conditionally of the system-receptor is obvious, as we could assume its presence only by the result displaying itself as the effect.
Before doing on to more concrete questions, let us note that from the positions of philosofic examination, the systematism of the source constitutes a sufficiently limiting condition for the information defifnition. Maybe more general from philosofic posistions is the following definition of information.
Definition. Information is the result of reflection of particular by the objects of the system.
Taking into consideration only three sides participating in the formation of the information phenomena, is certainly a theoretical idealization. The point is that the system which by the facts of its exestence and through its activity (e.g. movement) form a "sign", is the cause of appearence of this sign. But this cause, in its turn, could be the effect which results from formation of some condition not considered in the ternary presented in fig. 1b. It means that this source is not the primary reason and it displays itself in its turn due to the activity of the other systems. An attempt to take into account all the cause-and-effect chain leads to the necessity of examining the whole series of systems preceding the source in this chain, as well as the systems sensing the result.Composition of schemes of reflection could last without end. The degree of complexity of the model of information process presented as compositions of reflection, is determined by the formulation of the study objectives in the one hand and by expediency and necessary level of generality of information process details under consideration, on the other hand. The analisys gives an opportunity to infinitely divide phenomena and processes into their components, so that every seemingly indivisible process of reflection may be represented as a composition of some subprocesses. Moreover, in the definition of information there is another opportunuty of decomposition of the reflection process. The property of systematism, laid into the base of this defifnition, enables to examinate separate "channels" of reflection linking the parts of the source with those of the receptor within the limits of simple ternary.
To give more exact and formal interpretation of the definition of information and to study the properties of the information phenomena, there is proposed to use methodological tools of Category Theory [3, 4]. Let us give an interpretation of our definition of information from the positions of Category Theory. For that let us pick out and descride principle types of ternary (source-receptor-result) as different types of categories.
1. Category 2. Two-object category with three arrows. The category with three arrows. This category represents the category description (fig. 2) of one bit of information. Some
a defined by its arrow
1a is reflected in another
b which is, in its turn, defined by its unitary arrow
1b. The presence of arrow defines the state
1, while arrow less chart defines state
0.
Fig. 2. Diagram of category 2.
2. Category
N. One-object category with several arrows (fig. 3). This category describes the mapping of objects into itself and following interpretation of category
N may be given. By the use of arrows there is realized the objectification of multitude of values of one of the object's properties. Each arrow defines some values inherent in object
N only, and the set of all arrows
constitutes a set of values defined on object
N. The category
N has natural interpretation of monoid, the arrows being elements of monoid and their composition being an operation defined on it.
Fig. 3. Diagram of category N.
3. Category
AB. Two-object category having more than three arrows (fig. 4). This category is the development of the preceding one and it may be said that the category
AB enables to describe the situation where the properties of object
A are reflected by the properties of object
B.
Fig. 4. Diagram of category AB.
4. Category
A0. This category has one object which is the end of
n arrows except for identical one (fig. 5). The source is considered in this category as an entity. The specific of object
A is reflected by the objects of recepror interpreted as a class (set) of object which are the ends of arrows having their origin at the object
A.
Fig. 5. Diagram of category A0.
The spesific cases of
A0 for which the following conditions are met, also to be considered: the object A is an initial one; category containing A-cone; category containing extreme A-cone.
With the help of
A0 category following situation may be described: epistomology, when the structure of the source is unknown and that of receptor is hypothetical; specialization (inverce to the process of abstraction) when an abstract object is specified by given set of properties and relations defined of them.
5. Category
A1. It has an object which is the origin of
n arrow except for identical one (fig. 6).
Fig. 6. Diagram of category A1.
The receptor is represented by one object. the source is described by a class (set) of objects which are the origins of arrows having the objects
B as their end.
Similar to the category
A0, the interpretation of the category
A1 should include the following: epistomology, i.e. the source is not recognized, the object exists as one the forms of ignorance; abstracting. i.e. the concrete nature of an object, phenomena or process is ignored and only some general properties, signs are hold.
As a spesific case the category
A1 may fulfil the following conditions: the object
B is a finite one; the category contains B-cocone; the category contains co-extreme B-cocone.
6. Category
O, the category of "general" type (fig.7).
Fig. 7. Diagram of category O.
A category of this type may include as subcategories above mentioned categories. The main pithy component of category
O is constituted by its capacity to reflect the relations definable on the objects of the source. Such capacity may be, in particular, formalized with the help of category of sub-arrows, the objects of which being the arrows of original category. The objects and compositions of arrows in this category are described by the chart similar to one represented in fig. 7b, the object of arrows category (pair of arrows
<h,k>) and composition being determined by the rule
<j,l>*<h,k>=<j*h, l*k>.
With the help of category
O it becomes possible to represent the process which in their turn are described by arrows of
<h,k> type.
7. Functor reflection
Funct. Every system is represented by category, the reflection being described by functor. The functor description of reflection is the most natural from point of view of transmitting the systematism of the receptor by the objects of the source, although in number of cases the definition of functor is limiting from the positions of generally of description of information processes. Indeed, it requires that to every object of category-source corresponds an object of category-receptor, and that to each arrow of category-source corresponds an arrow of category-receptor.
Development of typology of representation of information processes should be continued by the study of reflection's compositions, the systems playing the role of objects in charts describing the compositions.
To describe the reflection's compositions, let us introduce a category of preorder, designating it by
PP. As the objects of this category, let us define the systems playing the role of source and their end was in the receptor. There will be examined only such compositions for which
ObPP would be a finite set. Such an approach is justified when considering computerized methods of data processing.
A relationship of preorder
R will be defined on object of the category
PP in following way:
- aRb if a is the source and b is the receptor;
- aRc if the system b exists, as well the reflections of the a to b and b to c (or aRb and bRc);
- aRa for any a ∈ ObPP.
Then let us draw a chart composed of all objects of the category PP and arrows joining them. Supposing that the category PP is composed of all objects and arrows which take part in maximum one-tie sub-graph of the graph represented by the chart of objects and arrows of the category PP.
This assumption is not limiting since of the graph is composed of several components, each of them is analyzed independently. Let us introduce a relation of equivalency Q on ObPP: if aRb and bRa then aQb. After examining the factor-set ObPP/Q and natural mapping natQ (ObPP→ObPP/Q), it becomes evident that the relation R on ObPP will be preordering and on ObPP/Q it will be ordering.
The analysis of compositions of reflections is performed in such a manner that while abstracting from content and internal scheme of reflection, there are picked out the systems-causes and the systems which are "pure" effects. Then, the link between these two classes of systems is to be drawn. From this point of view there five types of charts to be mentioned.
Fig.8. Charts type of reflections.
In fig. 8 these types are shown by corresponding charts and designated by P2, Pm, nP, nPm, EQL. By that categories Pm, nP, nPm:
- there could exist arrows between some objects ai (bi), and if for source ai, aj (bi, bj) an arrow f: ai →aj (u: bi →bj) exist, then the arrow g: aj → ai (v: bj → bi) exists as well,
- for any ai there is no arrow f: bj → ai.
Statement. There is functor which maps category PP into one of categories P2, Pm, nP, nPm, EQL.
To prove this statement let us construct a functor mapping category PP into one of above-mentioned categories, there will also be introdused the following sets:
S={b ∈ ObPP: xRb for any x ∈ ObPP and comparable with b},
I={a ∈ ObPP: aRx for any x ∈ ObPP and comparable with a},
SQ={b ∈ ObPP/Q:
xRb for any
x ∈ ObPP/Q and comparable with
b},
IQ={a
∈ ObPP/Q:
aRx for any
x ∈ ObPP/Q and comparable with
a}.
It should be noted that: the set ObPP/Q is not empty since the category PP is constructed; the sets S, I, SQ, IQ are also not empty.
It is evident that the objects belonging to the set
I are the "candidates" on the role of the first objects in the composition of reflections, the objects of the set
S being candidates on the role of the last objects in the composition. A correlation
r may be defined on the elements of the set
S∪ I in such a manner the
(x, y) ∈ r, if
x ∈ I, y ∈ S and
xRy. It could be shown that the graph of correlation
r is isomorphic to one of the charts of fig. 8.
Let us define a characterization of category
PP in accordance with the value of following signs:
П1={
natQ isomorphism},
П2={
PP/Q one-element},
П3={
I one-element},
П4={
S one-element}.
Let us designate by
P2, Pm, nP, nPm, EQL the statements becoming equal to
1 if the graph of correlation
r is isomorphic to the charts
P2, Pm, nP, nPm, EQL correspondingly. Therefore, the characterization presented in fig. 8 and the table of truth (tab. 1) are correct for the category
PP (1 corresponds to true, 0 - to false).
According to the table 1 it may be written
EQL = ¬ П1 ∧ П2,
P2 = (П1 ∨ ¬П2) ∧ П3 ∧ П4,
Pm = (П1 ∨ ¬П2) ∧ П3 ∧ (¬П4),
nP = (П1 ∨ ¬П2) ∧ (¬П3) ∧ П4,
nPm = (П1 ∨ ¬П2) ∧ (¬П3) ∧ (¬П4).
It is evident that
EQL∨ P2 ∨ Pm ∨ nP ∨ nPm = 1 for any
П1-П4 what proves the fact that the characterization of the category
PP with the use of signs
П1-П4 is complete.
Table 1.
П1
|
П2
|
П3
|
П4
|
Name
|
П1
|
П2
|
П3
|
П4
|
Name
|
1
|
1
|
1
|
1
|
P2
|
0
|
1
|
1
|
1
|
EQL
|
1
|
1
|
1
|
0
|
Pm
|
0
|
1
|
1
|
0
|
EQL
|
1
|
1
|
0
|
1
|
nP
|
0
|
1
|
0
|
1
|
EQL
|
1
|
1
|
0
|
0
|
nPm
|
0
|
1
|
0
|
0
|
EQL
|
1
|
0
|
1
|
1
|
P2
|
0
|
0
|
1
|
1
|
P2
|
1
|
0
|
1
|
0
|
Pm
|
0
|
0
|
1
|
0
|
Pm
|
1
|
0
|
0
|
1
|
nP
|
0
|
0
|
0
|
1
|
nP
|
1
|
0
|
0
|
0
|
nPm
|
0
|
0
|
0
|
0
|
nPm
|
Let us that the charts
P2, Pm, nP, nPm, EQL correspond to the expressions
P2, Pm, nP, nPm, EQL. If
П=1, then the preordering
R introduced on the category
PP becomes the ordering because
aQb is correct then and only then if
a=b, then set
S(I) being upper (lower) cone for the set
ObPP. In vertue of assumption about "monocomponent" structure of the graph of the set
ObPP, the set
S(I) is confinal (coinitial). Thus, for any
x∈I is there in
y∈S and the arrow
f: x→y, besides,
S∩I=∅. Then, depending of the number of elements in the set
S and
I the following relationships take place:
- P2 if I and S have only one element each;
- nP if I is composed of n elements and S has only one;
- Pm is one-element set and S is composed of m elements;
- nPm if I is composed of n elements and S has m elements.
Let us examine the case of П1=0 and take the factot-set ObPP/Q. If the latter is composed of the element (П2=1), then for any x, y there exist arrows f: x → y, g: y → x in the category PP. Choosing any two representers of the set ObPP, it is easy to obtain the relationship EQL. If ObPP/Q is composed of more than one element (П2=0), then it becomes possible to apply to it reasoning of previous paragraph and to construct correspondences P2, Pm, nP, nPm on the set ObPP/Q. For these correspondences let us replace the factor-set objects by the objects of the category PP which are the prototypes for mapping natQ, Since S∩ I=0, such substitution is correct and there exist correspondences of P2, Pm, nP, nPm type on the set ObPP.
Taking account of constructions elaborated above, it becomes easy to define a functor mapping the category PP into one of categories P2, Pm, nP, nPm. The concrete type of category which is the image of functor, is determined by the structure of the category PP. In particular, the functor F may be determined by following relationships:
It should be noted that there is an ambiguity while choosing z, but that has no influence upon the demonstration of our statement. The demonstration is terminated.
The interpretation of information process from positions of theory of categories, enables to represent the result of reflection by means of arrows defined on the objects of the set S or, more precisely it may be said that any result of the reflections composition is described by the arrow f: I →S where f∈Hom(I,S).
The definition of information introduced in this paper, is in considerable extent based upon the tools of Category Theory. Moreover, within the limits of interpretation based on this theory, the form of information is represented by arrow and that is, possibly, an obstacle for an extrapolation of our definition of information onto another domains of its manifestation. When a category describing an information process, is being constructed, it is assumed that the whole aggregate of essences under consideration is divided into two aggregates according to following signs. Some objects (essences) play the role of the origin of arrow describing the result, others serve as arrow's end.
By that,
- it is not indicated what plays the role of object - unitary, element, set, class or system;
- it is not carried out discrimination of objects neither from the point of view of their sameness, not from the positions of generality or relationship between the whole and its parts;
- it is not absolutized the truthfulness of the description of objects of the category-sources, instead of that the adequacy of forms of representation is studied.
The information is an entity of three essences;
- reflected;
- "copy" of reflected
- result-arrow.
Among them only the arrow is a "self-dependence" essence, the origin and the end of it being conditioned by the existence of arrow. The interpretation of the notion of information from the positions of Category Theory, narrows the generality of this notion, but this interpretation is not rigid because there is a sufficient number of representations which are far from being trivial. In particular, even at which level of abstraction as an arrow, there is a possibility to define in different ways its properties and only limited number of special types of categories allow the single definition of arrow linking any two objects. Another ambiguity in the interpretation of arrow's properties manifests itself in the compositional schemes of reflection. The point is that within the limits of the category PP the objects of sets S and I are connected by the composition of arrows. In general case for objects a∈I and b∈S there is such an arrow f: a → b and such a composition of arrows wv..u: a → b, that f = wv..u, i.e. a chart constructed on these objects is not commutative.
In practical applications there is sufficient freedom in interpreting compositional scheme of information process and in determining the degree of detailing (one of the forms of composition) of the reflection process. That leads to the situation where a series of properties of information are determined by the manner in which the compositional scheme was constructed and by the choice of systems to be used as source and recepror. Formally the series of the information properties are determined by the objects belonging to the set
I and
S, and by the set of morphisms
Hom(I, S) which are defined on the objects of these sets. Along with that, the freedom of choosing compositional schemes, is not absolute, being limited by objective factors which do not depend on subject. The relationship beetween the general and particular may be defined as follows.
Let us information process having a subset
S0 ⊂ S be described by the thernary
<I, S, Hom(I,S)>. After constructing a subset
I0 ⊂ I composed of objects being the origins of arrows which have their ends in
S0, and the set of arrows
Hom(I0,S0) ⊂ Hom(I0,S0), the ternary
<I0, S0, Hom(I0, S0)> will describe the information process inserted into another process
<I, S, Hom(I, S)>. With an assumption allowing the ternary
<I, S, Hom(I, S)> to describe some information system representing maximum (within the limits of our study) set of information processes. This ternary can not be observed in absolute sense because of the diversity of properties of real phenomena, processes and things. The set
I, S may be infinite themselves and
Hom(I, S) can be the proper class. In this case the information process described by the ternary
<I0, S0, Hom(I0, S0)> will be represent "observable" aggregate of information relations. A limitation of the set of observable objects may be caused by several reasons, among then there could be indicated pragmatic considerations, incomplete, limited knowledge, finite and restricted character of cognitive tools, finite time of observation. The ternary
<I0, S0, Hom(I0, S0)> could describe different cognitive situations, developing information systems, system's degradation in information sense. Let us call observer the pair
<S0, Hom(I0, S0)> since it describes the set of reflection results connected with a subset of objects interpreted as receptor of information.
An analysis of character of information processes enables to affirm the relativity of representation of the reflection result which is formally determined by different ways of choosing observer described by the pair
<S0, Hom(I0, S0)>. Analysis of pithy side of reasons for relativity of the reflection result enables to determine more factors influencing upon the relativity. Among the latter, the following ones have the influence of the interpretation of properties of information.
- the structure, properties and characteristics of receptor;
- the "depth" of perception of the reflection results by different systems which is characterized by total ordered set of the set ObPP;
- the "breadth" of the semantics representation of the reflection result characterized by the power of S0 set.
Thus, the feature of relativity is determined by the observer described by the pair
<S0, Hom(I0, S0)>, set
S0 and properties of its elements. Generalizing, we would like to note that the reasons of such relativity have several sides. Methodological side is defined by the interpretation of information from the positions of Category Theory. Logico-gnosiological side is determined by the relativity of our knowledge. Ontological side is determined by the form of the receptor existence and its internal properties. Finally, methodical side is determined by the limitations imposed by the scales of representation.
The relativity of properties, characteristics and sighs of information follow from the relativity of reflection result, since this latter constitutes the essence of information within the limits of the definition introduced. The relativity of properties and characteristics of information is examined in relation to the observer and the depth and breadth of its methodological paradigm, as well as in relation to internal properties inherent in the systems of observer. On the one hand the relativity is of "subjective" character, while on the other hand such subjectivity can not, in certain sence, be wider than objectivity determined by the objects of the set
I. To estimate the limitation imposed by this set on the degree of relativity of information, one should evaluate the limitation imposed by this set on the degree of relativity of information, one should evaluate the power of the set of objects belonging to
I, as well as to the elements of this set, and the power of the set composed of all the morphisms defined on these objects. We think that in the majority of cases the power of the set is such that the limitation imposed by the deversity represented by the objects of the source, does not narrow the feature of relativity in the reflection result. It should be also pointed out that our views on relativity of information presented in this paper differ from those exposed in [7] concerning the statement that the objective character of information can manifest itself only through the result of reflection, and that the properties of the result determine corresponding degree of relativity of information.
The relativity of the information properties reveals itself boldly when quantitative and qualitative aspects of information are examined.
The relativity of determination of qualitative aspect of information is in a great extent connected with the fact that this aspect is determined by the properties of the system-receptor. Every shade of semantic, pragmatic, truthfulness character requires the system of objects which would reflect this shade by their existence. By that, the objective nature of information is not refuted, but it is manifested in diffent ways depending on interpreting "subject".
The relativity of determination of quantitative aspect of information is first of all conditioned by relativity in interpretation of qualitative properties of information, and then, by different approaches to estimation of its quantity. The most "simple" way to measure this quantity consists in constructing an arrow joining each result with a category of estimating objects which describe the quantity of information inherent in each result. Such arrow could represent the quantity of information either by means of its "name", or through the object to be estimated which is indicated by this arrow. On the category of "estimation" it is easy to define analogs of statistical formulae used for evaluating the quantity of information. Another approach to be determitation of quantity of information relates to the use of notions and constructions of Category Theory. There is also possible to note several "categories" approaches:
- based upon the analysis of power of the set of morphisms linking the objects of source and receptor;
- based upon the analysis of the structure of sub-objects of the source, which are defined with the respect to the arrows joining the source and receptor;
- based upon the analysis of the structure of factor-objects defined with the respect to the arrows joining the source;
- based upon the analysis of the structure of charts composed by the arrow of result and by monomorphic and (or) epimorphic arrows which are joined to the arrows of result (this approach is in some extent a geometric interpretation of approaches b) and c)).
It may be noted that enumerated approaches are not the only possible ones. For instance, the analysis of retractions and co-retractions enables to develop more subtle methods for evaluating the quantity of information. The plurality of approaches to the solution of this problem has its explanation. Every estimate represents in some extent the degree of variety and differences and not of absolute diversity, but of the one interpreted through one or another form of adequacy. These forms in their turn could be interpereted by means of Category Theory. They are defined by the categorial properties of objects and arrows on then, and could be represented by different constructions of Category Theory. For example, such forms of adequacy as monomorphity and epimorphity enable to carry out an analysis of relationship of preordering defined on sub-objects (factor-objects) belonging to sub-category which describes the objects of the source (receptor), and to estimate the quantity of reflected diversity by means of concrete arrows of results. By that the sub-objects characterize the diversity of the source and factor-objects do so with the diversity that can be reflected by the receptor. It can be noted that within the limits of categorial definition of the quantity of information there is a place for topological, combinatory and algorithmic approaches to the solution of this problem.
Thus, the relativity of properties and that of the information characteristics is determined by the posibility of different interpretations of the result of reflection. The reasons for that are determined by:
- the form of the interpretation of information based upon Category Theory, and within the limits the model of this theory - by greater freedom of choosing properties of arrows and their compositions (methodological reasons);
- the relativity of knowledge expressed in the fact that the idea about the information process is formed by the observer <S0, Hom(I0, S0)> (logico-gnosiological reasons);
- the structure, internal properties and characteristics of receptor and by the way of its existence (ontological reasons);
- the way of choosing compositional medium, by the breadth of representation and the depth of perception of the reflection result (methodical reasons).
All this enables to advance the following principle.
Principle of relativism. Within the limits of the model of information process based upon Category Theory, the properties, signs and characteristics of this process are determined in respect of the observer which is described by the pair <S0, Hom(I0, S0)>, where I0 is the set of source (systems) of reflections, S0 is the set of receptors (systems) of information, Hom(I0, S0) is the set of process (results) of reflection from I0 to S0.
The substantiation of this statement depends in great extent upon the result of logico-gnosiological analysis and it consists in the necessity to show the posibility of the description of information process with the use of Category Theory. Secondly, the posibility of different interpretation of the reflection result is also to be shown. The later was already considered above, the first is easier to refute by a contr-example that to prove by some formal method. Some confidence is inspired by the fact that Category Theory enables to compose a great number of representations, each of them being able to describe some type of information process. Such situation is explained by the fact that the axioms of categories set very meagre abstraction [5]. That gives the ground for the formulation of a hypothesis about the representation which allows, besides all, to define the generally of our definition of information.
Hypothesis about representation. An information process meeting the condition of associativity of composition of reflections, may be represented by a category which is isomorphic to one of the categories O, Funct, nPm, EQL.
The maximalism of this hypothesis could prove to ruinous for itself, and this circumstance is in some extent connected with the generality of Category Theory as the base of the nathematics.
The relativity of the information properties examined in application to IT manifests itself in two ways. On the one hand it becomes difficult to estimate the degree of the reflection adequacy, on the other hand, it becomes possible to organize "economical" scheme of reflection. That find its expression in following problem of IT: to search and organize such forms of reflection in computer systems which would ensure an adequate representation of objects of receptor an acceptable way which is determined by the level of expenses and by the complexity of this representation. The acceptability of the method of representation is taken into account when choosing one or another form of adequacy which, in its turn is determined by means of universal construction of Category Theory. In this connection it is to be noted that Category Theory enables to build universal constructions which can be defined using only some of the properties connected with the arrows between the objects belonging to such construction. There are many constructions like that, for instance, the arrow itself is the most simple one. The product and co-product, the limit and co-limit, equalizer and co-equalizer, universe mapping and amalgam, the cone and co-cone, topos, etc. are more complex constructions. Such structures being universal among all objects similar to them, enable to talk about formalization of architecture of software packages on their base.
In the light of above-said the following important problems is the formalized description of the adequacy realtion, of properties and characteristics of adequate reflection. In this paper there will be examined a relation of properties of adequacy of reflection for the simplest construction, an arrow. Comformably to this construction there may be given following hierarchy of the adequacy forms:
- identity (sameness);
- isomorphism;
- bimorphism;
- monomorphism;
- epimorphism.
Such form of adequacy as sameness is a trivial one, and the isomorphism is already sufficiently studied.
In application to IT, a study of such forms of adequacy as monomorphism and epimorphism, is more constructive. In the constructions of Category Theory monomorphism and epimorphism correspond to the property of arrow cancelling. If the arrow is monomorphic, then in composition it can be canceled from the left, if it is epimorphic, then it can be canceled from the right. On the other hand, for a series of constructions monomorphism means that the different entries of the receptor will correspond to different exits of the source, and epimorphism means that for any object in the receptor there is an object which is prototype of this object. If an information model is considered as a developing structure, then the absence of heredity of monomorphism leads to the situation where different objects could become identical and there is no mechanism enabling to edentify the deversity. At the same time the requirement of epimorphism of arrows means that there is no object in the receptor which have no denotates (designates) in the source. The property of monomorphism is quite important when passing from one model of data to another one, more "developed" from the point of view of initial model. Monomorphism of the transition arrow enables to reduce the number of links defined on data, picking out only the most essential ones. Epimorphism of arrow ensures that the receptor represents (describe) the source by all its objects using the power of the set of objects in "optimal" way.
In application to above-examined types of categories, let us note that the categories 2, N, AB, A0, the arrow are monomorphic. For the categories A1, O, Funct the monomorphism of arrows requires that the same objects and the same relations on objects should be reflected in the receptor is identical arrows. In the categories 2, N, AB, A1 the arrows are epimorphic. In the categories A0, O, Funct the epimorphism of arrows determined that the equal arrows linking the source and the receptor always correspond to equal objects of source.
More general view on such problems of adequacy as monomorphism and epimorphism consists in the following. In overpowering majority of cognitive situations we can not ensure isomorphic reflection of structure and properties of receptor, limiting our examination by the reflection of only some objects of the source by the objects of the receptor. Monomorphism of reflection ensures by that a univalent transmission of a part of objects to be reflected and the set of their morphisms, considered as essential for one another reason. The requirement imposed on the receptor to ensure the existence of minimum set of objects representing the ends of arrows while reflecting the objects of the source (under construction that there is given a set of morphism linking the source and receptor), leads to the requirement of epimorphism of arrows linking the source and receptor. If now both requirements are formulated simultaneously (univalency and minimality), then we obtain the requirement of epimorphism of arrows linking the source and receptor. Here there is possible following situation: bimorphic arrrows may be iso-arrow or not. The latter occur, for example, while reflecting sll the parts of a system, but not the property of emergencity (entity) inherent in the system being reflected.
Conditionally speaking, monomorphism of arrows describing the reflection, enables to express the internel content of the source represented by its objects and by relations defined on them. Epimorphism ensures the transmission of the "volume" of source, of its external form through setting a correspondence between the receptor objects and relations defined on them, and the source objects. With some kind of conditionality it may be stated that monomorphism expresses intensive properties of reflected and epimorphism expresses the extensive ones.
If we examine a cognitive situation, then monomorphism and epimorphism find their place in inductive and deductive forms of cognition. With the help of monomorphic arrows the receptor is recreated as a "reflection" of the source (the whole or its parts). Using epimorphic arrows we obtain a deductive system wich expresses the structure, properties and relations, and it is substantiated as a "reflection" of the source. Besides, epimorphism realizes some form of the "Okkam blade" principle, i.e. the number of objects of the source can not be greater than needed.
The properties of epimorphism and monomorphism is very important for the analysis of diversity reflected. With the help of monomorphic arrows the structure of the source could be devided into sub-objects, by means of epimorphic arrows the structure of the receptor could be devided into factor-objects. All that enables to estimate the diversity inherent in the source and that inherent in the receptor. Their relationship, in its turn make possible the determination of reduncy, degree of adequacy, noise level and other characteristics the concrete content of wich is determined by corresponding methodological directions.
Without doubt, monomorphism and epimorphism have much greater number of pithy interpretations form the positions of adequate reflection of information. Moreover, epimorphism and monomorphism participate in the formation of complex constructions of Categoty Theory (e.g. equalizer, inverse image, etc.). In particular, in special types of categories monomorphism and epimorphism correspond to injective (categories of set, topological spaces, the same types of universal algebra, in particular, algebra of groups, rings) and to surjective (category of set, topological spaces, groups) functions of Category Theory. This enables to make a deduction about fundamental character of properties of monomorphism and epimorphism ans allows to cosider them as essential principles (forms) of adequate reflection of information.
In conclusion we would like to determine the role and place of above-given definition of information in application to IT. As the subject of examination of information technology, the information determines the methods and means inherent in this technology. In light of our definition IT can and must have its interpretation from the position of Category Theory. One of such interpretations consists in the presentation of IT as and receptor interacting by its entry will the source. Such interaction in general way is described by means of the category of AB type (with two objects I, S), or with the help of the category EQL. IT plays the system forming role consisting in the fact that by means of IT the result of reflection is formed not only because there is reflection between the system, but by the possibility to process them quickly in different aspects (this is determined by means of the depth of perception), to present them in different aggregates and to use them immediately. It is to be noted that such problems could be solved without using Category Theory. However, this theory enables to represent in clear manner such requirements to IT as ensuring the possibility for designing different schemes of data transformation with minimum expenses on the part of the human being, ensuring representation of any object and relations defined on it. Successful solution of information problems connected with formalization of creative aspects, depends in a great extent on the diversity of constructions provided by IT with limited number of their types. Different degree of generality and different degree of abstraction are inherent in the constructions of Category Theory with comparatively small number of constructions. This circumstance pre-determines in a great extent their sufficiency for the creation of different representations within the limits of IT with the use of methods of Category Theory. With a fairly great optimism there could be affirmed that the pithy problems can be represented with the help of formal constructions of Category Theory. Along with that it would better not to make illusions about the easiness of designing in IT constructed wit the use of the Category Theory. Category Theory is a special tool requiring refined thinking and its abstractions are so meagre that in general case an information technology designed on the base of Category Theory represents rather the search for commensurability and proportionality of objects and morphisms that a formal realization of a series of procedures. Such a technology, however, should deliver the man, the user from all that routine of data processing and leave to him only the estimation of proportionality, harmonic combination of things and ideal essences, expediency, consonance of patterns.
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