The current formal as well as not formal definitions of the concept "Information” are presented in 
the paper

Markov, Krassimir

Mitov, Ilia

Velikova-Bandova, Evgeniya

Bulgarian Digital Mathematics Library at IMI-BAS

International Journal "Information Theories & Applications" Vol.10  5THE INFORMATION K. Markov, K. Ivanova, I. Mitov, E. Velikova-Bandova  Abstract: The current formal as well as not formal definitions of the concept "Information” are presented in the paper.  Keywords: Information, General Information Theory, and Philosophy of Informatics   1. Introduction The fundamental notion of the General Information Theory is the concept "Information". All other concepts are defined based on this definition. The first not formal definition of the concept of Information was published in [Markov, 1988]. The main philosophical explanations were published in [Markov et al, 1993]. The first variant of the formal definitions were introduced in [Markov et al, 2003]. This work refines the philosophical basis and represents the more precise variant of the formal definition of the concept “information”.  1.1. Entity In our examination, we consider the real world as a space of entities. The entities are built by other entities, connected with relationships. The entities and relationships between them form the internal structure of the entity they build. To create the entity of a certain structural level of the world, it is necessary to have: − the entities of the lower structural level; − establishing of the forming relationship. The entity can dialectically be considered as a relationship between its entities of all internal structural levels. The forming relationship has a representative significance for the entity. The destruction of this essential relationship causes its disintegration. The establishment of forming relationship between already existing entities has a determine significance for the emerging of the new entity. The forming relationship is the reason for the emergence of individual properties, which distinguish the new entity from the forming ones.  The relationships form and present the entity.   1.2. Impact, Interaction, Reflection Building the relationship between the entities is a result of the contact among them. During the contact, one entity impacts on the other entity and vice versa. In some cases the opposite impact may not exist, but, in general, the contact may be considered as two mutually opposite impacts which occur in the same time. The set of contacts between entities forms their interaction. The interaction is a specific interactive relationship between entities which take part in it. The contacts of the given structural level are processes of interaction of the entities on the lower levels.  During the establishing of the contact, the impact of an entity changes temporally or permanently the internal structure of the impacted entity. In other words, the realisation of the relationships between entities changes, temporary or permanently, their internal structure at one or at few levels.  The internal change in the entity, which is due to impact of the other entity we denote with the notion "direct reflection". Every entity has its own level of sensibility. This means that the internal changes occur when the external influence is over the boundary of the sensibility of the entity. The "reflection impulse" for given entity is the amount of the external influence needed for transition from one state to the reflection one. International Journal "Information Theories & Applications" Vol.10  6 The entities of the world interact continuously. It is possible, after one interaction may be realised another. In this case, the changes received by any entity, during the first interaction, may be reflected by the new entity.  This means the secondary (transitive external) reflection exists.  The chain of the transitive reflections is not limited. In general, the concept “transitive impact” (respectively “transitive reflection”) of the first entity on the third entity through the second one will denote every chain of impacts (reflections) which start from first entity and ends in the third entity, and include the second entity in any internal place of the chain. One special case is the external transitive self-reflection where the entity reflects its own relationships as a secondary reflection during any external interaction.  Some entities have an opportunity of internal self-reflection. The internal self-reflection is possible only for very high levels of organisation of the entities, i.e. for entities with very large and complicated structure. The self-reflection (self-change) of the entity leads to the creating of new relationships and corresponding entities in it. Of course, the internal self-reflection is a result of the interaction provided between entities in the low levels of the structure of the entity. Such kind of entities has relatively free sub-entities with own behaviour in the frame of self-preservation of the whole entity. As a result of the self-reflection some relationships and corresponding sub-entities are created or changed in the entity.  The combination of the internal and external self-reflection is possible.  1.3. Information The reflection could not be detected by the entity that contains it. This is dialectical behaviour of the reflection - it is only an internal change caused by the interaction.  During as well as after the interaction between two entities, they may interact with other entities from the environment. If any third entity contains reflections of given entity received by two different ways: 1. by transitive impact of the first entity on the third one through the second entity, 2. by impact of the first entity on the third one which is different from the transitive one, i.e. it can be direct impact or transitive impact through another entity then the third entity became as an external relationship between entities and their reflections - it became as “reflection evidence”.  We may say that the reflection of the first entity in the second one is “information” for the first entity if there is corresponded reflection evidence. The generalisation of this idea leads to assertion that every reflection can be considered as information, if there is corresponding reflection evidence.  2. Formal Definitions 2.1. Entity Definition 1. The entity A  is the couple A=(EA,RA)  where: EA  is a collection of sub-sets of a set MA ;  RA={ri | i∈I,  I  is a set}  is a nonempty set of relations in EA , i.e. r i⊂EA×EA={(X,Y)|X,Y∈EA}  is a relation and  ř i=r i∪{(X,Y)|(Y,X)∈r i)}, ∀i∈I ;  and: 1. ∅∈EA ; 2. MA =∪X, X∈EA ; 3. ∀r∈RA  and ∀X,Y∈EA ⇒ ((∃(X,Y)∈ř ) or  (∃Z1,. . ,Zp∈EA, Zk≠∅,k=1,.. ,p : (X,Z1)∈ř ,(Z1,Z2)∈ř ,…,(Zp,Y)∈ř) ■ The condition 3  means that EA  is internally connected in respect to every relation r∈RA . MA  is called forming set of A . The intersection ρA  of all relations in RA  is called forming relation of A , i.e. ρA=ρ(A)=∩r i ,  i∈I,  r i∈RA .  International Journal "Information Theories & Applications" Vol.10  7When relationship (X,Y) belongs to the relation r  we will denote by (X→Y)r . Every element X  can be considered as an entity (EX,RX) , for which MX={X}, EX={∅,MX}  and RX={ρ} , where ρ={(∅→MX)} . Because of this, the elements of every forming set M  can be assumed as entities. Definition 2. Let A=(EA,RA)  is an entity. B  is a sub-entity of A  if B=(EB,RB) , where MB⊂MA , ∀X∈EB  ∃Y∈EA: X⊂Y  and ∀rB∈RB  ∃rA∈RA:rB⊂rA  ■  2.2. Impact and Reflection Definition 3.  Let A = (EA,RA)  and B = (EB,RB) , A≠B  and f∈RA . A direct impact  ψ of A  on B  concerning the relation f  is a nonempty subset of EA×EB  such that ψ=ψf=(A→B)ψ⊂EA×EB={(X,Y)|X∈EA,Y∈EB} , ψf≠∅ for which: if Xi , Xj∈EA  and Yk,Yl∈EB , such that  (Xi→Xj)f , (Xi,Yk)∈ψf ,  (Xj,Yl)∈ψf   then exists g∈RB:(Yk→Yl)g  ■  If (X,Y)∈ψf  we will denote (X→Y)ψ. Let remember that ∅∈EA  as well as ∅∈EB . This means that for every f∈RA  and g∈RB  there exists zero direct impact  οf=(A→B)ο={(∅,∅)}.    Definition 4. Let A=(EA,RA)  and B=(EB,RB)  and ψf=(A→B)ψ is an direct impact of A  on B . A reflection of A  into B  realised by the direct impact ψ is the couple  Fψ=F=(EF,RF)  for which EF={Y∈EB|∃X∈EA:(X,Y)∈(A→B)ψ}  and RF={rψ},  rψ=rψ, f={(Y1,Y2)|∃X1,X2∈EA:(X1→X2)f;  (X1→Y1)ψ;  (X2→Y2)ψ} ■  rψ,f is a reflection of the relationship f  by direct impact ψ.  The reflection Fψ will be assumed as a sub-entity of B .  2.3. Transitive Impact and Transitive Reflection Definition 5.  Let: A=(EA,RA), B=(EB,RB), C=(EC,RC) ; f∈RA, g∈RB;  ϕf=(A→B)ϕ is a direct impact of A  on B  concerning the relation f ; ψg=(B→C)ψ is a direct impact of B  on C  concerning the relation g . We will say that the direct impacts ϕf  and ψg  can be composed if the reflection of the relationship f  by direct impact ϕ is a subset of g , i.e. rϕ,f⊂g .   Definition 6. If ϕf  and ψg  can be composed than the couple {ϕf ,ψg}  is a transitive impact  ξ=ξf= ψg○ϕf  = (A→B→C)ξ  of A  on C  through B ■  It is clear, that every transitive impact is a chain of at least two composed direct impacts. In general, this chain can contain more than two composed direct and/or transitive impacts. Such chain is transitive impact, too.  Definition 7. Let A=(EA,RA), B=(EB,RB), C=(EC,RC).  Let ξ={ϕf ,ψg}=ξf ,g=(A→B→C)ξ  is transitive impact of A  on C  through B.  International Journal "Information Theories & Applications" Vol.10  8 The transitive reflection of A  into C  through B  realised by the impact ξ is the couple Gξ=G=(EG,RG)  for which  EG={Z∈EC|∃X∈EA  and  ∃Y∈EB:(X,Y)∈(A→B)ϕ and  (Y,Z)∈(B→C)ψ} and RG={rξ} ,  rξ=rξ, f={(Z1,Z2)|∃X1,X2∈EA: (X1→X2)f;  (X1→Y1)ϕ;  (X2→Y2)ϕ;  (Y1→Z1)ψ;  (Y2→Z2)ψ} ■  If A and C  are equal than: - the transitive impact ξ will be called self-impact; - the transitive reflection Gξ=G=(EG,RG)  for transitive self-impact ξ(A→B→A)  will be called self-reflection.   2.4. Interaction and Interactive Reflections Let denote by ΩA B the set of all direct or transitive impacts of A  on B. Definition 8. Let A=(EA,RA)  and B=(EB,RB) . An interaction between A  and B  is a set  ΔA B={Δi  |  Δi∈ΩA B∪ΩB A, i=1,. . ,h } ■  Definition 9. Let: A=(EA,RA)  and B=(EB,RB) ; Δ=ΔA B={Δi  |  i=1,. . ,h}  is an interaction between A  and B; FΔi=Fi=(EFi ,RF i)  is reflection realised by the impact Δi ;  An interactive reflection VAB between A  and B realised by the interaction ΔA B  is the set of all reflections FΔi ; i.e.  VAB={FΔi | i=1,. . ,h} ■  The self-interaction ΔA A  is the interaction from A  to A  where all impacts are self-impacts. In such case the corresponding interactive reflection is called interactive self-reflection.  2.5. Information Definition 10. Let: A=(EA,RA)  and B=(EB,RB) ; τ is an impact of A  on B , i.e. τ=(A→B) τ.  ,  τ ∈ ΩAB ;  ∃ entity C=(EC,RC):C≠A, C≠B;  ∃ ψ=(B→C)ψ which can be composed with τ=(A→B) τ ; ∃  transitive impact ξ={τ,ψ}=(A→B→C)ξ  ;  ∃ impact ϕ=(A→C)ϕ; , ϕ ∈ ΩAC  and ϕ≠ξ; Fϕ is a reflection of the impact ϕ and Fξ is a reflection of the impact ξ. F τ is information for A  in B  if ∃r∈RC:(Fϕ→Fξ) r  ■  The entity A  is called source, the entity B  is called recipient. The relation r∈RC  for which (Fϕ→Fξ) r  is called reflection evidence and the entity C  is called information evidence.  If VAB is an interactive reflection of between entities A  and B, and entity C contains reflection evidences for all reflections of VAB than C  is called information witness.  Every reflection may be considered as information iff there is corresponded information evidence or information witness.   International Journal "Information Theories & Applications" Vol.10  9For practical needs, it is more convenient to follow the next consideration. The reflection in the recipient represents both the relationships and the sub-entities of the source. From other point of view, the relationships build up and present the entities. Because of this, the reflected relationships are the essence of the reflection. In other words, if a reflection evidence exists then the reflection of the forming relationship may be considered as "information" for reflected entity.  Therefore, in the sense that the evidence exists to point what relationship (between what entities) is reflected and where it is done, we may say that "the information is reflected relationship".  Conclusion The translation of the philosophical theory into the formal one is a good approach for verification of the scientific ideas. The concept “Information” of the GIT was presented formally in this paper. The definition given above is a first step for building the formal part of the GIT. Together with the philosophical explanations, it gives us a useful tool for investigation of the information phenomena in the real world.  Acknowledgements This paper is based on the ideas considered during very creative discussions at the Int. Conf. "KDS 1997", September 1997, Yalta, Ukraine, and at the Int. Conf. "ITA 2000", September 2000, Varna, Bulgaria, as well as at the others scientific meetings organised by the International Workgroup on Data Base Intellectualisation (IWGDBI). Special discussion was provided at the Int. Conf. “Knowledge-Dialog-Solution”, Varna, Bulgaria, 2003.   Authors are very grateful to all participants in these fruitful discussions.  This work is partially financed by project ITHEA-XXI of the Institute of Information Theories and Applications FOI ITHEA.  Bibliography [Markov, 1988] Kr.Markov. From the past to the future of the definition of the concept of Information. Proceedings "PROGRAMMING '88", BAS, Varna 1988, p.150. (In Bulgarian). [Markov et al, 1993] Kr.Markov, Kr.Ivanova, I.Mitov. Basic Concepts of a General Information Theory. IJ "Information Theories and Applications". FOI ITHEA, Sofia, 1993, Vol.1,No.10, pp.3-10 [Markov et al, 2003] Kr.Markov, Kr.Ivanova, I.Mitov. General Information Theory. Basic Formulations. FOI ITHEA, Sofia, 2003.   Author information Krassimir Markov  - ITHEA - FOI Institute of Information Theories and Applications; Institute of Mathematics and Informatics, BAS;  P.O.Box: 775, Sofia-1090, Bulgaria; e-mail: foi@nlcv.net Krassimira Ivanova - Institute of Mathematics and Informatics, BAS, Acad.G.Bonthev St., bl.8, Sofia-1113, Bulgaria; e-mail: foi@nlcv.net Ilia Mitov - ITHEA - FOI Institute of Information Theories and Applications, P.O.Box: 775, Sofia-1090, Bulgaria;  e-mail: foi@nlcv.net  Evgeniya Velikova-Bandova – Sofia University, Faculty of Mathematics and Informatics.  e-mail: velikova@fmi.uni-sofia.bg     

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