AUNVERITSISPREESMIATSN AN ALGEBRAIC APPROACH TO KAUPPI 'S CONCEPT THEORY Marko Junkkari and Marko Niinim?aki DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF TAMPERE REPORT A-1998-1 UNIVERSITY OF TAMPERE DEPARTMENT OF COMPUTER SCIENCE SERIES OF PUBLICATIONS A A-1998-1 , JANUARY 1998 AN ALGEBRAIC APPROACH TO KAUPPI 'S CONCEPT THEORY Marko Junkkari and Marko Niinim?aki University of Tampere Department of Computer Science P.O.Box 607 FIN-33101 Tampere , Finland ISBN 951-44-4288-1 ISSN 0783-6910 An Algebraic Approach to Kauppi 's Concept Theory Marko Junkkari , Marko Niinim?aki January 1 , 1998 Abstract The aim of this paper is to represent concept systems algebraically . We discuss concepts and their intensions , a formal concept system and its operations , and a description of how the operations and some basic notions can be represented in algebra . This approach enables us to handle the formal conceptual level of a representation system in an established and well-known formalism . 1 Introduction In this paper , some relationships between concept systems and algebraic operations of concepts are discussed . The paper brings together two research traditions ; concept analysis and conceptual modelling -LRB- Kauppi -LSB- 1967 -RSB- , Kangassalo -LSB- 1983 -RSB- , Kangassalo -LSB- 1993 -RSB- -RRB- and algebraic database de?nition languages -LRB- Niemi and J?arvelin -LSB- 1992 -RSB- -RRB- . In this paper , we understand concepts to be discrete knowledge units , that subsist relatively independently . In general , concepts can be studied by their extensions and intensions -LRB- see e.g. Palom?aki -LSB- 1994 -RSB- -RRB- . The extension of a concepts is the set of individuals to which the concept applies . The intension of a concept determines its extension -LRB- Kauppi -LSB- 1967 -RSB- , Bunge -LSB- 1967 -RSB- , Palom?aki -LSB- 1994 -RSB- -RRB- . When concepts are studied by their extensions , the study is limited , for example , to individuals , their properties , and relationships between individuals . Concepts are understood to be the results of generalizing and abstracting these individuals , properties and relations . However , we are more interested in the intensional approach , where concepts can be constructed without refering to states of individuals . In the intensional approach , the contents of concepts are considered . Thus , the representation -LRB- conceptual model -RRB- is not dependent on the momentary state or existence of speci?c objects . A concept always has an intension , but its extension can be -LRB- momentarily or permanently -RRB- empty . In the intensional approach , the we are interested in the intension of a concept . According to Bunge -LSB- 1967 -RSB- , the intension of a concept can be de?ned as a -LRB- possibly in?nite -RRB- set of properties and relationships contained in the concept . In Bunge 's -LRB- intensional -RRB- view , the problem is to ? nd the relevant properties and relationships for objects . The modeller discovers these by abstracting his or her observations in the Universe of Discourse . Another problem is that the structure of the intension is not explicit . The relationships among concepts can be studied according to Bunge 's idea , i.e. the generality of concept is de?ned on the basis of its intension . For instance the concept of centaur should be more general than the concept of greek centaur , provided that all the properties of centaur are properties of greek centaur , too . This is the case , of course , even though the extensions of both centaur and greek centaur are empty . The in?nity of intensions forms a practical problem . In this paper , we do not adopt the idea of in?nite intension . Bunge , too , presents a ? nite core intension , that consists of the so-called characteristics -LRB- earmarks -RRB- of the concept . We assume the characteristics of a concept C to be concepts that are contained in C. For instance , the characteristics for the concept of dog could be mammal , quadruped and carnivore . But in addition to those , it would include the characteristics of mammal , quadruped and carnivore , like living being , thing etc. . Bunge discusses the containment between intensions , but Kauppi -LSB- 1967 -RSB- bases the intensions on the relation of intensional contaiment . Kauppi developed her notion of the intensional containment independently of Bunge . Therefore , we only observe that Kauppi 's idea about the intensional containment resembles Bunge 's idea of comparions between generality of concepts . Moreover , the generality - relationship between concepts is very close to the so-called is-a-relationship -LRB- see Brachman -LSB- 1983 -RSB- , Nilsson and Palom?aki -LSB- 1997 -RSB- -RRB- . Nilsson and Palom?aki consider the is-a relationship equal to the intensional containment . This is , however , only one of the presentations of intensional containment -LRB- see Kangassalo -LSB- 1996 -RSB- -RRB- . In this paper we ? rst introduce some basic notions for concepts and intensions . In section 2 we present some mathematical conventions . In section 3 , a simple concept calculus -LRB- in the sense of Kauppi -LSB- 1967 -RSB- -RRB- will be outlined . In section 4 we introduce our de?nition for the intension of concept and discuss how the concept system can be assumed to imitate the UoD . Section 5 gives a more formal treatment for concepts and concept algebra . Finally , section 6 contains a short summary and items for further study . 2 Mathematical Notational Conventions Similar objects can be collected into a set on the basis of some relevant principle . These objects are called the elements of the set . Two sets , A and B are equal -LRB- denoted as A = B -RRB- if and only if they have the same elements . Set A is said to be the subset of B , denoted as A ? B , if every element of A is also an element of B . The union A -LSB- B is a set that contains all the elements in A and all the elements in B . The intersection of A and B , A \ B , contains elements that are contained both in A and B . The difference A?B is a set that contains all those elements of A that are not elements in B. Given a set A , the power set of A , denoted by P -LRB- A -RRB- is the set consisting of all subsets of A . If x and y are two objects , we connect them together explicitly by ordered pair < x ; y > . Now we can form the set of ordered pairs from two individual sets . The set of all possible ordered pairs in which the ? rst member of the ordered pair is an element of A and the second member is an element of B is called the Cartesian product over A and B , denoted by A?B . It means the following set : A?B = f < x ; y > jx 2 A ; y 2 Bg : Sets A and B in A?B are called operand sets . A relation R from the set A to the set B is subset of A?B . If the relation is a subset of A?A we say that the relation R is on the set A . If the ordered pair < x ; y > is an element of the relation R , we use notations < x ; y > 2 R or xRy , alternatively . A two-placed relation -LRB- as R above -RRB- is also called a binary relation . A binary relation R on A is re?exive if , for every x 2 A , the ordered pair < x ; x > 2 R . The relation R is irre?exive if for each x 2 A , < x ; x > 62 R . We now de?ne the identity relation IdentA on the set A follows : IdentA = f < x ; x > jx 2 Ag : If R is a relation on A then R ? IdentA does not contain those ordered pairs of R in which the ? rst and second elements are same -LRB- i.e. re?exive ordered pairs -RRB- . A relation R is symmetric , if for every < x ; y > 2 R , there is < y ; x > 2 R . A relation R on A is antisymmetric if , for every x 2 A and y 2 A holds < x ; y > 2 R and < y ; x > 2 R , only if x = y . If x ; y and z are arbitrary elements in A so that < x ; y > 2 R and < y ; z > 2 R and if < x ; z > 2 R , then R is called transitive . Relation R is called a weak partial order , if it is re?exive , antisymmetric and transitive . R is called a strict partial order , if it is irre?exive , antisymmetric and transitive . If R is a partial order on A , A is called a partial order set or a poset . The composition of two relations R1 and R2 , denoted by R1 ffi R2 , contains such pairs < x ; z > for which there is an intermediate element y such that xR1y and yR2z ; R1 ffi R2 = f < x ; z > j < x ; y > 2 R1 ^ < y ; z > 2 R2g : The composed relation R ffi R , is denoted by R2 , R ffi R ffi R , is denoted by R3 etc. . For simplicity , we de?ne R1 = R. Generally , we denote by -LSB- ni = 1Ri the union R1 -LSB- R2 -LSB- : : -LSB- Rn . -LSB- ? i = 1Ri is known as the transitive closure of R . This is often denoted as R + . Intuitively , if < a ; b > 2 R + then an immediate or indirect relationship exists among a and b in R . The re?exive transitive closure of the relation R on A is the union of the transitive closure of R and the identity relation of on the set A : R ? = R + -LSB- IdentA : 3 Intensional containment and concept systems In this section , Kauppi 's theory -LSB- 1967 -RSB- is outlined in a very limited form . Moreover , some of the de?nitions have been changed to serve the purposes of this paper . There are only two ontological primitives in the theory : concept and the relation of intensional containment . This basic relation is denoted by a > b ; and reads ? concept a intensionally contains concept b ? or ? concept b is intensionally contained in concept a ? . Also we can say that ? b is a characteristic of a ? or ? a has a characteric b ? . The ? rst axiom of the theory is re?exivity , so every concept contain intensionally itself , Axre f 8x -LRB- x > x -RRB- : The second axiom is transitivity ; i.e. if x contains intensionally y and y contains intensionally z , then x contains intensionally z , Axtr 8x8y8z -LRB- x > y ^ y > z ! x > z -RRB- : The intensional equivalence means conceptual equivalence and it is de?ned D f ? a ? b =d f a > b ^ b > a : The de?nition D f ? means that two concepts a and b are equivalent , if and only if a contains intensionally b and b contains intensionally a . So we can consider the relation ? > ? as an antisymmetric relation . From the axioms Axre f ; Axtr and the de?nition D f ? follow that the relation of intensional containment forms a partial order . Actually the de?nition D f ? is not suf?cient clause for antisymmetric relation , because it only dominates equivalence classes . In spite of that we assume that the relation of intensional containment is antisymmetric . With intensional containment Kauppi de?nes relation of strict containment . The concept a contains intensionally strictly the concept b if and only if a contains intensionally the concept b and a and b are not equivalent , D f > a > b =d f a > b ^ : -LRB- b ? a -RRB- : Kauppi does not present any de?nition for immediate containment , since her study applies to in?nite concept systems . In our study , only ? nite concept systems are considered , and for them the following de?nition applies . The concept a is immediately contained in a concept b -LRB- a Oic b -RRB- , if and only if a contains intensionally b strictly and if there is no concept between a and b. D fOic a Oic b =d f a > b ^ :9 x -LRB- a > x ^ x > b -RRB- : In algebraic approach we will use just the relation of immediately containment , but now we go ahead with basic relation intensional containment . Based on the relation of intensional containment between concepts , some further association relations between concepts have been de?ned . Two concepts share some characteristics if there exists a concept that is characteristic to both the concepts . In that case the concepts are comparable . It is marked by symbol H , and the de?nition is D fH aHb =d f 9x -LRB- a > x ^ b > x -RRB- : Concepts a and b are incomparable , if they have no common characteristics . It is marked by symbol , and the de?nition is D f a b =d f :9 x -LRB- a > x ^ b > x -RRB- : Another way to observe indirect connection between concepts a and b is compatibility . If there exist a concept x whose characteristics include a and b , it is said that concepts are compatible . It is marked by symbol j ^ and its de?nition is D f j ^ a j ^ b =d f 9x -LRB- x > a ^ x > b -RRB- : The opposite of compatibility is incompatibility . This means that there is no concept that could make a and b compatible . It is marked by symbol _ j , and the de?nition is D f_j a _ j b =d f :9 x -LRB- x > a ^ x > b -RRB- : If a concept a is contained intensionally in every concept , then a is equivalent with the so called general concept G , D fG a ? G =d f 8x -LRB- x > a -RRB- : According to Kauppi 's theory there exist a concept that is included intensionally in every concept , AxG 9x8y -LRB- y > x -RRB- : From the axiom AxG follows that any two concept are comparable . A special concept is a concept that is not contained intensionally in any other concept than itself . There can be several special concepts in Kauppi 's system . The set of special concepts is denoted by S . If a concept a is an element in S , this is de?ned as follows : D fS S -LRB- a -RRB- =d f 8x -LRB- x > a ! a > x -RRB- : For every concept y there must be one or several special concepts whose characteristic the concept y is , AxS 8y9x -LRB- S -LRB- x -RRB- ^ x > y -RRB- : As indicated earlier , Kauppi 's system includes exactly one general concept and one or several special concepts . The concept operations in the theory are intensional sum , intensional product , intensional negation , intensional reciprocal , intensional difference and intensional quotient . In the scope of this paper we introduce only intensional sum and product . The concept c is equivalent with intensional product of the concept a and b , if and only if c shares every common characteristic of a and b , but not any other characteristic except itself , D f ? c ? a?b =d f 8x -LRB- x > c $ x > a ^ x > b -RRB- : The axiom of intensional product is : if concepts a and b are comparable , then their product exists , Ax ? a j ^ b ! 9x -LRB- x ? a?b -RRB- : Because we have assumed existence of the general concept , the intensional product between any two concept naturally exists . Another intensional operation considered here is intensional sum . Concept c is equal to the intensional sum of concepts a and b , if and only if c has both characteristics of a and charasteristics of b , but no other characteristics except itself , D f ? c ? a?b =d f 8x -LRB- x > c $ x > a ^ x > b -RRB- : The axiom of intensional sum is that if concepts a and b are compatible , then their intensional sum exists , Ax ? a _ j b ! 9x -LRB- x ? a?b -RRB- : The axioms Ax ? and Ax ? bring some demands to form of concept system . For example the form of picture 1 is forbidden , because there is no intensional product for a and b , and no intensional sum for c and d . In the pictures 1 and 2 any upper level concept contains intensionally lower level concepts . c d a b e G Picture 1 . An illegal structure . There is also an axiom related with distributivity in Kauppi 's theory . We do not assume that , because we want to accept the structure like in picture 2 . G b c a d Picture 2 . A non-distributive structure . With some additions , it can be shown that Kauppi 's original theory forms a distributive structure , a boolean semilattice -LRB- Palom?aki -LSB- 1994 -RSB- -RRB- . The system we consider is not as strong as a pure Kauppi 's system . 4 The intension of a concept In this section we de?ne the intension of a concept as a structure based on the intensional containment relation . This resembles Kangassalo 's de?nition in -LSB- Kangassalo 1982 -RSB- . The starting point is to bring together Kauppi 's -LSB- 1967 -RSB- and Bunge 's -LSB- 1967 -RSB- views of characteristics for concepts . Bunge -LSB- 1967 -RSB- de?nes the intension of a concept as ? the set of properties and relations Pi subsumed under the concept , or which the concept , so to speak , synthesizes . ? This is possibly an in?nite set . Bunge proposes , however , that we can study a subset of this set . The ? nite core intension , Icore -LRB- C -RRB- , consists of the characteristics contained in the given concept C : Icore -LRB- C -RRB- = fP1 ; P2 ; : : ; Png : We assume in this paper that these characteristics are concepts . We also assume , that the concept C belongs to the set of characteristics . The set of characteristics is ? nite , but the problem for conceptual modelling is that it is unsure if all the characteristics can be found . A characteristics for a concept C is , for example , a property or a concept that is more general than C. Thus , being a mammal is a necessary attribute for the concept of dog . In this way , there should be a structure in the set of characteristics , but by using a simply set of characteristics we can not present this . Next we describe the structure for the characteristics of a concept . A concept C intensionally contains a set of concepts and we assume that these contained concepts correspond to Bunge 's characteristics . Here the characteristics , in this paper , are the concept intensionally contained in C. Contrary to Bunge , our de?nition of intension is based on the relation of intensional containment among concepts . Moreover , the intension is not a set of ? characteristics ? but a set of ordered pairs of concepts . In this respect the de?nition is more accurate than Bunge 's . It also incorporates the fact that the intension of concept C contains C itself -LRB- according to Kauppi 's axiom Axre f -RRB- . The intension of concept C , I -LRB- C -RRB- denotes set of ordered pairs < a ; b > such that the intensional containment relation holds between a and b . The intension of C is thus the relation > on the set of characteristics , Icore -LRB- C -RRB- for C : I -LRB- C -RRB- = f < x ; y > jx ; y 2 Icore -LRB- C -RRB- ^ x > yg : The intension of the concept C is ? nite and it is a subset of a Cartesian product I -LRB- C -RRB- ? Icore -LRB- C -RRB- ? Icore -LRB- C -RRB- : Kangassalo -LSB- 1982 -RSB- suggests a quite similar kind of de?nition for the intension of concept . His de?nition includes other relations in addition to intensional containment . In conceptual modelling , a limited part of the real world is relevant from the view point at hand . This relevant part is called the Universe of Discourse -LRB- UoD -RRB- . Let us assume that the set C-setUoD is the set of all concepts that subsist in the UoD . The intension of a concept C in the UoD , denoted by IUoD -LRB- C -RRB- , is de?ned as follows : IUoD -LRB- C -RRB- = f < x ; y > jx ; y 2C-setUoD ^ < x ; y > 2 I -LRB- C -RRB- g : Axiomatized concept systems are abstract structures that force some constraints in the relationships among concepts . But there is a connection between concept systems and concepts that are abstracted from the UoD . The concept system can be a conceptual model of the UoD . Suppose we have a UoD called UoD1 and a concept system CS1 that contains concepts abstracted from UoD1 . We take a strong metaphysical approach that there really are concepts in UoD 's and that they subsist relatively independently of us and can be modelled -LRB- see Kauppi -LSB- 1967 -RSB- -RRB- . Having this approach , it is natural to think that the concepts of UoD1 form a -LRB- possibly in?nite or very large -RRB- concept system . For practical reasons we can not handle that large concept systems . But let us assume that the modeller has discovered a set of concepts , C-setCS1 from UoD1 . It is reasonable to assume that the intensional containment relation on C-setCS1 is a subset of the intensional containment relation on the concepts of UoD1 . 5 Algebraic representation of concept systems In this section we represent an exact formalism that expresses the same axiomatization that was given in section 3 . The presentation language of this section is algebraic . The signature of a function f is denoted by f : A ! B , where A de?nes a set of values to which the function can be applied and B de?nes the set of values , which the result of the function inheres in . C-set is the set of concepts related to a speci?c concept system . As an example we consider the set C-set1 = fG ; gadget ; radio ; clock ; clock radiog : In each concept system there is an intensional relation I-rel on the set C-set . This relation is a binary relation that corresponds to immediate containing of concepts presented in section 3 . Formally , I-rel is a subset of the Cartesian product of C-set : I-rel ? C-set ? C - set . The interpretation of I-rel is : if < a ; b > 2 I-rel , then concept a immediately contains concept b in the intensional sense .1 In our example , the relation I-rel1 on the set C-set1 is the following binary relation : I-rel1 = fhgadget ; Gi ; hradio ; gadgeti ; hclock ; gadgeti ; hclock radio ; radioi ; hclock radio ; clockig . Given an I-rel , we can compute the corresponding I-rel ? , which is the re?exive transitive closure of I-rel . The relation I-rel ? corresponds to the intensional containment relation considered in section 4 . In this approach we de?ne a CONCEPT SYSTEM to be an I-rel on a C-set . The re?exive transitive closure of I-rel1 is I-rel ? 1 = fhgadget ; Gi , hradio ; Gi , hclock ; Gi , hclock radio ; Gi , hradio ; gadgeti , hclock ; gadgeti , hclock radio ; gadgeti , hclock radio ; radioi , hclock radio ; clocki , hG ; Gi , hgadget ; gadgeti , hclock ; clocki , hradio ; radioi , hclock radio ; clock radioig . In the following functions , I-rel and C-set are considered global , when they are not given explicitly . The function genuses computes the set of all such concepts that no other concept in the concept system is contained in then immediately or indirectly . Genuses is a function from the power set of the Cartesian product C-set ? C-set to the power set of C-set . genuses : P -LRB- C-set ? C-set -RRB- ! P -LRB- C-set -RRB- genuses -LRB- I-rel -RRB- = fxj :9 y 2C-set : < x ; y > 2 I-relg : In our example , genuses -LRB- I-rel1 -RRB- produces the set fGg : Respectively , the function specie computes the set of all those concepts that are contained in no other concepts than themselves in the concept system . 1In the following examples , the names of concepts will be written in italics , not in boldface . This is due to the convention that names are -LRB- terminal -RRB- symbols of a given language . specie : P -LRB- C-set ? C-set -RRB- ! P -LRB- C-set -RRB- specie -LRB- I-rel -RRB- = fxj :9 y 2C-set : < y ; x > 2 I-relg : Related to our example , specie -LRB- I-rel1 -RRB- = fclock radiog : Boolean functions compatible and comparable check whether two concepts are compatible or comparable in the sense of Kauppi -LRB- see section 3 -RRB- . We de?ne functions incompatible and incomparable on their basis . compatible : C-set ? C-set ! Boolean compatible -LRB- a ; b -RRB- = ae true , if jfxj < x ; a > 2 I-rel ? ^ < x ; b > 2 I-rel ? gj > ; false , otherwise . In our example , compatible -LRB- clock , radio -RRB- is true , since clock radio intensionally contains both clock and radio . incompatible : C-set ? C-set ! Boolean incompatible -LRB- a ; b -RRB- = ae true , if not compatible -LRB- a ; b -RRB- ; false , otherwise . comparable : C-set ? C-set ! Boolean comparable -LRB- a ; b -RRB- = ae true , if jfxj < a ; x > 2 I-rel ? ^ < b ; x > 2 I-rel ? gj > ; false , otherwise . In our example , comparable -LRB- clock , radio -RRB- is true , since G and gadget are intensionally contained in both clock and radio . incomparable : C-set ? C-set ! Boolean incomparable -LRB- a ; b -RRB- = ae true , if not comparable -LRB- a ; b -RRB- ; false , otherwise . Intensional product and intensional sum for any two concepts in a I-rel can be expressed in following two functions : int sum : C-set ? C-set ! P -LRB- C-set -RRB- int sum -LRB- a ; b -RRB- = fcj8x 2C-set : -LRB- < x ; a > 2 I-rel ? ^ < x ; a > 2 I-rel ? $ < x ; c > 2 I-rel ? -RRB- g : int prod : C-set ? C-set ! P -LRB- C-set -RRB- int prod -LRB- a ; b -RRB- = fcj8x 2C-set : -LRB- < a ; x > 2 I-rel ? ^ < b ; x > 2 I-rel ? $ < c ; x > 2 I-rel ? -RRB- g : The concept system is said to be operation-legal if and only if for any two concepts a and b in C-set : ffl if the concepts are comparable , there is exactly one intensional sum of a and b and ffl if the concepts are compatible , there is exactly one intensional product of a and b . The function legal check operations checks if the concept system is operationlegal for given two concepts : legal check operations : C-set ? C-set ! Boolean legal check operations -LRB- a ; b -RRB- = 8 < : true , if compatible -LRB- a ; b -RRB- ! jint prod -LRB- a ; b -RRB- j = 1 ^ comparable -LRB- a ; b -RRB- ! jint sum -LRB- a ; b -RRB- j = 1 ; false , otherwise . Now we can construct all of the legal checks for a concepts system . Having the axioms we presented in section 3 , there can be only one member in the set generated by the function genuses in the concept system . For that follows also , that if the concept system is seen as a graph , it must be connected . Moreover there must be checking for concept operations in the legal check function . Function legal check checks if the concept system given by its I-rel is legal . legal check : C-set ? C-set ! Boolean legal check -LRB- I-rel -RRB- = 8 < : true , if 8a 2C-set ; 8b 2C-set : legal check operations -LRB- a ; b -RRB- ^ jgenuses -LRB- I-rel -RRB- j = 1 ; false , otherwise . We can easily see that I-rel1 is legal . In this section we have presented an abstract implementation for a part of Kauppi 's theory . We have done that because we have also wanted to give the basic implementations for different structures for concept systems . 6 Discussion In this paper , we have discussed connections between conceptual modelling , Kauppi 's concept theory and its abstract implementation . We have presented only a part -LRB- core -RRB- of Kauppi 's theory . C-set and I-rel are our contributions to the theory and together they can be interpreted to represents a known part -LRB- closed world -RRB- of concepts and their intensional containment . In this paper we have not discussed obvious connections with set theorical interpretations of concept structures . From the structure of a legal I-rel follows that a?b = fxj < a ; x > g \ fyj < b ; y > g . This makes computation of intensional product obvious for legal I-rels . For further study , we plan to examine different presentations of intensional containment . J?arvelin and Niemi -LSB- 1993 -RSB- have presented that algebraic approach is suitable to management relationships , which are very close to is-a-relation , which we assume to be one presentation of intensional containment . We also plan to study what kind of set of axioms would be fruitful in practical conceptual modelling . 7 References -LSB- Bunge , 1967 -RSB- Mario Bunge , Scienti?c Research I , The Search for Systems . Vol . 3/I , Springer-Verlag , 1967 . -LSB- Brachman , 1983 -RSB- Ronald J. Brachman , What IS-A is and is n't : An analysis of taxonomic links in semantic networks , IEEE Computer . Vol . 16 , No. 10 , 1983 . pp. 30-36 . -LSB- J?arvelin and Niemi , 1993 -RSB- Kalervo J?arvelin and Timo Niemi , Deductive information retrival based on classi?cations , Journal of the American Society for Information Science . 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O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O B O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O B O O O B O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O B O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O B O O B O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O B I O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

algebra
concept
intension
intensional containment