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as a ballpoint pen, a fountain pen, and a quill pen. A concept in linguistics is a noun or noun-phrase that serves as the subject of a to-be statement [8, 15, 16].
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International Journal of Semantic Computing Vol. 4, No. 3 (2010) 331–356 c World Scientific Publishing Company  DOI: 10.1142/S1793351X10001061

ON CONCEPT ALGEBRA FOR COMPUTING WITH WORDS (CWW)

YINGXU WANG Department of Electrical and Computer Engineering University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4 [email protected]

Computing with words (CWW) is an intelligent computing methodology for processing words, linguistic variables, and their semantics, which mimics the natural-language-based reasoning mechanisms of human beings in soft computing, semantic computing, and cognitive computing. The central objects in CWW techniques are words and linguistic variables, which may be formally modeled by abstract concepts that are a basic cognitive unit to identify and model a concrete entity in the real world and an abstract object in the perceived world. Therefore, concepts are the most fundamental linguistic entities that carries certain meanings in expression, thinking, reasoning, and system modeling, which may be formally modeled as an abstract and dynamic mathematical structure in denotational mathematics. This paper presents a formal theory for concept and knowledge manipulations in CWW known as concept algebra. The mathematical models of abstract and concrete concepts are developed based on the object-attributerelation (OAR) theory. The formal methodology for manipulating knowledge as a concept network is described. Case studies demonstrate that concept algebra provides a generic and formal knowledge manipulation means, which is capable of dealing with complex knowledge and their algebraic operations in CWW. Keywords: Computing with word; concept algebra; semantic computing; denotational mathematics; cognitive informatics; concept network; knowledge representation; OAR model; soft computing; cognitive computing; computational intelligence; case studies.

1. Introduction Computing with words (CWW) proposed by Lotfi A. Zadeh in 1999 is intended to provide a foundation for a computational theory of human and machine thought, reasoning, and causality analyses [36]. In his seminal paper, Zadeh described CWW as an emerging computing methodology for word and linguistic variable processing in soft computing, which mimics human natural-language-based reasoning [35, 36]. The central objects in CWW are words and linguistic variables that are used to be modeled by dictionary entries or WordNet in linguistics [26]. A denotational mathematical model of words and linguistic variables is known as abstract concept 331

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[20], which forms a mathematical structure for CWW and a basic unit of both knowledge and reasoning [1, 4–7, 9, 14, 16, 18, 19, 28, 35, 36]. Definition 1. A concept is a basic cognitive unit to identify and/or model a concrete entity in the real world and an abstract subject in the perceived world. A concept can be identified by its intension and extension [12, 27, 20]. The intension of a concept is the attributes or properties that a concept connotes; the extension of a concept is the members or instances that the concept denotes. For example, the intension of the concept pen connotes the attributes of being a writing tool, with a nib, and with ink. The extension of the pen denotes all kinds of pens that share the common attributes as specified in the intension of the concept, such as a ballpoint pen, a fountain pen, and a quill pen. A concept in linguistics is a noun or noun-phrase that serves as the subject of a to-be statement [8, 15, 16]. Concepts in cognitive informatics [15, 17, 19, 21, 25] are an abstract structure that carries certain meaning in almost all cognitive processes such as thinking, learning, and reasoning. In computing, a concept is an identifier or a name of a class. The intension of the class is a set of operational attributes of the class. The extension of the class is all its instantiations or objects and derived classes. Concept algebra provides a rigorous mathematical model and a formal semantics for object-oriented class modeling and analyses. The formal modeling of computational classes as a dynamic concept with predesigned behaviors may be referred to as system algebra [16, 24]. This paper presents a formal treatment of abstract concepts and a comprehensive set of algebraic operations for CWW. The mathematical model of abstract concepts is established. A mathematical structure known as concept algebra is developed for knowledge representation and manipulations in CWW. Based on concept algebra, a knowledge system is formally modeled as a concept network, where the methodology for knowledge manipulating toward CWW is presented. Case studies demonstrate that concept algebra provides a powerful denotational mathematical means for manipulating complicated abstract and concrete knowledge structures as well as their algebraic operations for CWW. 2. Concepts: The Basic Unit of Semantic Entities for CWW Concepts can be classified into two categories known as the concrete and abstract concepts. The former are proper concepts that identify and model real-world entities such as the sun, a pen, and a computer. The latter are virtual concepts that identify and model abstract subjects, which cannot be directly mapped to a real-world entity, such as the mind, a set, and an idea. The abstract concepts may be further classified into collective concepts such as collective nouns and complex concepts, or attributive concepts such as qualitative and quantitative adjectives. The concrete concepts are used to embody meanings of subjects in reasoning; while the abstract concepts are used as intermediate representatives or modifiers in reasoning. Based on concepts

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and their relations, meanings of real-world concrete entities may be represented and semantics of abstract subjects may be embodied. This section describes a formal treatment of abstract concepts and the new mathematical structure, concept algebra, in CWW and semantic computing. Before an abstract concept is defined, the semantic environment or context [5, 6, 8] in a given language, is introduced. Definition 2. Let O denote a finite or infinite nonempty set of objects, and A be a finite or infinite nonempty set of attributes, then a semantic environment or the universal context Θ is denoted as a triple, i.e.: ∧

Θ = (O, A, R) = R : O → O|O → A|A → O|A → A,

(1)

where R is a set of relations between O and A, and | denotes alternative relations. On the basis of Θ, an abstract concept is a composition of the above three elements as given below. Definition 3. An abstract concept c on Θ is a 5-tuple, i.e.: ∧

c = (A, O, Rc , Ri , Ro ),

(2)

where • A is a finite nonempty set of attributes (meronyms), A = {a1 , a2 , . . . , an } ⊆ ÞA, where Þ denotes a power set. • O is a finite nonempty set of objects (holonym), O = {o1 , o2 , . . . , om } ⊆ ÞO. • Rc = O × A is a set of internal relations. • Ri ⊆ A × A, A  C  ∧ A  c, is a set of input relations, where C  is a set of external concepts, C  ⊆ ΘC . For convenience, Ri ⊆ A × A may be simply denoted as Ri ⊆ C  × c. • Ro ⊆ c × C  is a set of output relations. Theorem 1. The dynamic and adaptive property of concepts states that an abstract concept is a dynamic mathematical structure that possesses the adaptive capability to interrelate itself to other concepts via Ri and Ro . Based on Definition 3, an object derived from a concept and the intension/extension of a concept can be formally defined as follows. Definition 4. An object of a concept o is a derived instantiation of the concept that implements an end product of the concept, o ⊂ O, i.e.: ∀ c(A, O, Rc , Ri , Ro ),

o = c · oi ,

oi ⊂ O,

Roo ≡ Ø

⇒ o(Ao , Roc , Roi |Ao ⊇ A, Roc = o × Ao , Roi = {(c, o)}, Roo = {(o, c)}).

(3)

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Equation (3) indicates that an object is a tailored end-product of a concrete concept where there is not any output-oriented relation to other concepts except its parent. Definition 5. The intension of a concept c = (A, O, Rc , Ri , Ro ), c∗ , is represented by its set of attributes A, i.e.: ∧

c∗ (A, O, Rc , Ri , Ro ) = A =

#O 

Aoj ⊆ ÞA,

(4)

i=1

where ÞA denotes a power set of A, and # is the cardinal operator that counts the number of elements in a given set. Definition 5 indicates that the narrow sense or the exact semantics of a concept is determined by the set of common attributes shared by all of its objects. On the contrary, the broad sense or the fuzzy semantics of a concept is referred to as the set of all attributes identified by any of its objects as defined below. Definition 6. The complete set of attributes of a concept c = (A, O, Rc , Ri , Ro ), or the instant attributes denoted by all objects of c, is a closure of all objects’ intensions, A∗ , i.e.: ∧

A∗ =

#o 

Aoj .

(5)

j=1

Definitions 5 and 6 specify that: (a) The intension of a concept is a finite set of objectively identifiable attributes at a given level of abstraction; and (b) the intension of a concept is dynamic. When more objects are denoted for the same concept, the domain of the intension is usually shrinking in order to accommodate the new objects in the same structure of the concept. Conventionally, the domain of a concept’s intension is used to be perceived as highly subjective in the literature [7]. In this approach, it is deemed that a concept connotes the attributes, which occur in the minds of people who use that concept, or where something must have in order to be denoted by the concept. Both the above informal perceptions are not objectively operational in defining a least complete and unambiguity set of intensions. To solve this fundamental problem, Definition 5 provides a unique and objective determination of the intension of any given concept, which is stable, unique, and objective. Definition 7. The extension of a concept c = (A, O, Rc , Ri , Ro ), c+ , is represented by its set of objects O, i.e.: ∧

c+ (A, O, Rc , Ri , Ro ) = O = {o1 , o2 , . . . , om } ⊆ ÞO.

(6)

A formal and objective definition of the domain of concept is provided below. Definition 8. The domain of a concept c = (A, O, Rc , Ri , Ro ) is a set of attributes with a narrow sense Dmin referring to its intension and a broad sense Dmax referring

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to its closure, i.e.:

∧

D(c) =

 #O    Dmin (c) = A = Aoj   j=1 #O     ∗  Aoj Dmax (c) = A =

.

335

(7)

j=1

It is noteworthy that in conventional literature, it is only perceived that the intension of a concept determines its extension [7]. However, Definition 8 reveals that the extension of a concept, particularly the common attributes elicited from the extension, determines its intension as well. Relationships between concepts in a concept hierarchy can be illustrated in Fig. 1 at three levels known as the knowledge, object, and attribute levels. The internal relations of concepts, Rc = O × A can be formally represented by concept matrices. Theorem 2. The nature of concept hierarchy states that for a given concept c = (A, O, Rc , Ri , Ro ) in an abstraction hierarchy, the higher the level of a concept in abstraction (α(c)), the greater the intension of the concept (Ac ); and vice versa, i.e.: α(c) ∝ Ac .

(8)

Theorem 3 can be proved by Definition 5. Example 1. A concrete concept network with c1 (pen), c2 (printer ), and c3 (stationery) can be illustrated in Fig. 2. The concept network may be dynamically

c3 c1

Knowledge level (K)

c2

Ri, Ro

O1 o11

o12

O2

o13

o21

Rc1

A1

Object level (U)

o22

Rc2 A2

a1

a2

a3

a4

A5

A6

…

an

Attribute level (M)

Fig. 1. The hierarchical relations of concepts and their internal structures.

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c3

pen c1

O1

fountain

ballpoint o11

o12

Knowledge level (K)

printer c2

stationery

O2

o13

o21 brush

o22

laser

Object level (U)

Ink-jet

A2

A1 a1

a2

a3

a writing using having tool ink a nib

a4

A5

A6

with an ink a printing using container tool papers

A7

…

Attribute level (M)

with a toner cartridge

Fig. 2. A concrete concept network for CWW.

extended along with the development of related knowledge such as to extend it to a more abstract concept network of c3 (stationery). In Fig. 2, the three concepts can be formally described in concept algebra as given below: ∧

c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o )  A1 = {a1 , a2 , a3 } = {a writing tool, using ink, having a nib}       O1 = {o11 , o12 , o13 } = {ballpoinbt, fountain, brush}    R1c = O1 × A1 = {(o11 , a1 ), (o11 , a2 ), (o11 , a3 )}∪   =

{(o12 , a1 ), (o12 , a2 ), (o12 , a3 )}∪     {(o13 , a1 ), (o13 , a2 ), (o13 , a3 )}     i  R1 = {(c3 , c1 ), (c2 , c1 )}     o R1 = {(c1 , c3 ), (c1 , c2 )}

(9)

∧

c2 (printer) = c2 (A2 , O2 , R2c , R2i , R2o )  A2 = {a5 , a6 } = {a printing tool, using papers}       O2 = {o21 , o22 } = {ink jet printer, laser printer} =

Rc = O2 × A2 = {(o21 , a5 ), (o21 , a6 )} ∪ {(o22 , a5 ), (o22 , a6 )}  2   i  R2 = {(c3 , c2 ), (c1 , c2 )}   o R2 = {(c2 , c3 ), (c2 , c1 )} (10)

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∧

c3 (stationery) = c3 (A3 , O3 , R3c , R3i , R3o )  A3 = A1 ∪ A2 = {a1 , a2 , a2 , a5 , a6 }       O3 = O1 ∪ O2 = {o11 , o12 , o13 , o21 , o22 } = R3c = O3 × A3    R3i = {(c1 , c3 ), (c2 , c3 )}    o R3 = {(c3 , c1 ), (c3 , c2 )}.

(11)

It is noteworthy that, according to Definition 5, the intension of c1 (pen) does not include the attribute a4 , because it is not commonly shared by all objects of the given concept. However, A∗1 = {a1 , a2 , a3 , a4 } do include a4 in the closure of attributes of the given concept pen. Definition 9. The identification of a new concept c(A, O, Rc , Ri , Ro ) is the elicitation of its objects O, attributes A, and internal relations Rc , from the semantic environment Θ = (O, A, R), i.e.: ∧

c = (A, O, Rc , Ri , Ro | O ⊂ O, A ⊂ A, Rc = O × A, Ri = Ø, Rc = Ø).

(12)

In Definition 9, Ri = Ro = Ø denotes that the identification operation is an initialization of a newly created concept where the input and output relations may be determined later when it is put into a certain context of knowledge. Definition 10. A qualification of a concept c(A, O, Rc , Ri , Ro ), denoted by *c, is the identification of its domain, i.e.:  #O     Dmin (c) = A = Aoj   j=1 ∗ ∧ c = D(c) = . (13) #O     ∗ Dmax (c) = A = Aoj  j=1

Definition 11. A quantification of a concept c(A, O, Rc , Ri , Ro ), denoted by #c, is the cardinal evaluation of its domain in terms of the number of attributes included in it, i.e.:

 #O      #Dmin (c) = #A = # j=1 Aoj ∧

. # c = # D(c) = (14)  #O    ∗  Aoj #Dmax (c) = #A = # j=1

Example 2. According to Definitions 10 and 11, the qualification and quantification of concept c1 as given in Fig. 2 are as follows, respectively: Dmin (c1 ) = Ac1 = {a1 , a2 , a2 } ∗ c1 = D(c1 ) = Dmax (c1 ) = A∗c1 = {a1 , a2 , a3 , a4 }

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# c1 = # D(c1 ) =

#Dmin (c1 ) = #Ac1 = 3 #Dmax (c1 ) = #A∗c1 = 4

.

The formal models of abstract concepts for CWW provides a rigorous denotation of the architecture, intension, and extension of concepts, and their applications in modeling and manipulating of words and linguistic variables in CWW, semantic computing, cognitive computing, and computational linguistics.

3. Operations on Concepts in CWW by Concept Algebra On the basis of abstract concept and its mathematical models, as developed in the preceding section, a set of generic algebraic operators on concepts can be defined, which forms a mathematical structure of concept algebra. Concept algebra provides a denotational mathematical means for algebraic manipulations of abstract concepts toward CWW. Concept algebra can be used to model, specify, and manipulate generic “to be” type problems in natural-language-based computing such as CWW, semantic computing, and cognitive computing. Concept algebra is an abstract mathematical structure for the formal treatment of concepts and their algebraic relations, operations, and associative rules for composing complex concepts [19]. Definition 12. Concept algebra (CA) on the given semantic environment Θ is a triple, i.e.: ∧

CA = (C, OP, Θ) = ({A, O, Rc , Ri , Ro }, {•r, •c }, Θ),

(15)

where OP = {•r , •c } are the sets of relational and compositional operators on abstract concepts. Details of the relational and compositional operators of concept algebra will be elaborated in the following subsections.

3.1. Relational operators of concept algebra for CWW The relational operators of abstract concepts are static and comparative operations that do not change the concepts involved. It is recognized that relationships between concepts are solely determined by the relations of both their intensions A and extensions O. The relational operators of concept algebra [19] are summarized in Table 1. (a) Related/Independent Concept The related concepts are a pair of concepts that share some common attributes in their intensions; while the independent concepts are two concepts that their

Related

Independent

Subconcept

Superconcept

Synonym

Antonym

Consistence

Comparison

Definition

2

3

4

5

6

7

8

9

Operation

1

No.

Rc1 = O1 × A1 , Ri1 = Ri2 , Ro1 = R02 )

i=1

= c1 , A1 , O1 Rc1 , Ri1 , Ro1 |A1 ⊇ A2 , O1 ⊆ O2 , Rc1 = O1 × A1 , Ri1 = Ri2 , Ro1 = Ro2 ), c2  c1 c1 (O1 , A1 , Rc1 , Ri1 , Ro1 )  c2 (O2 , A2 , Rc2 , Ri2 , Ro2 )

∧

c1 (A1 , O1 , Rc1 , Ri1 , Ro1 )  c2 (A2 , O2 , Rc2 , Ri2 , Ro2 )



j=1

c1 ∼ c2 =

∧ #(A1 ∩ A2 ) #(A1 ∪ A2 )

= (A1 ⊂ A2 ) ∨ (A1 ⊃ A2 )

∧ c1 ∼ = c2 = (c1  c2 ) ∨ (c1 ≺ c2 )

)

O1 =!O2 = {!o21 , !o22 , . . . , !o2n }, Rc1 = O1 × A1 ,  ff n ff ni j Ri1 = R (ci , c1 ) , Ro1 = R (cj , c1 ) )

= c1 (A1 , O1 , Rc1 , Ri1 , R01 |A1 =!A2 = {!a21 , !a22 , . . . , !a2n },

∧

c1 (A1 , O1 , Rc1 , Ri1 , R01 )! = c2 (A2 , O2 , Rc2 , Ri2 , Ro2 )

—

—

Equivalence, meronym/hyponym, or holonym/hypernym

Antonym (!)

Synonym (=)

= c1 (A1 , O1 , Rc1 , Ri1 , R01 |A1 = A2 , O1 = O2 ,

∧

Holonym/hypernym

c1 (A1 , O1 , Rc1 , Ri1 , R01 ) = c2 (A2 , O2 , Rc2 , Ri2 , Ro2 )

Meronym/hyponym

—

—

Linguistic term (WordNet)

c 2  c 1 = A2 ⊃ A1

∧

c 1 ≺ c 2 = A1 ⊂ A2

∧

c1  c2 = A1 ∩ A2 = Ø

∧

∧

c1 ↔ c2 = A1 ∩ A2 = Ø

Definition

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∼

∼ =

!=

=



≺



↔

Symbol

Table 1. Relational operators of concepts in concept algebra for CWW.

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intensions are disjoint. The relational operations of related and independent concepts can be illustrated in the following example. Example 3. Using the concepts c1 (pen) and c2 (printer ) as given in Eqs. (8) and (9), the related/independent operation on c1 and c2 can be expressed as follows: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

c2 (printer) = c2 (A2 , O2 , R2c , R2i , R2o ),

A1 ∩ A2 = {a1 , a2 , a3 } ∩ {a5 , a6 } = Ø

(16)

⇒ c1 (pen)  c2 (printer). (b) Subconcept/Superconcept The subconcept of a given concept is a concept that its intension is a subset of the superconcept; while a superconcept over a subconcept is a concept that its intension is a superset of the subconcept. Example 4. Using the concepts c1 (pen) and c3 (stationery) as given in Eqs. (9) and (11), the subconcept/superconcept operations on c1 and c3 can be expressed as follows: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

c3 (stationery) = c3 (A3 , O3 , R3c , R3i , R3o ),

A1 = {a1 , a2 , a3 } ⊂ A3 = {a1 , a2 , a3 , a5 , a6 }

(17)

⇒ c1 (pen) ≺ c3 (stationery). Equation (17) indicates that c1 (pen) is a subconcept of c3 (stationery); in other words, c3 (stationery) is a superconcept of c1 (pen). (c) Synonyms The synonym concepts are two concepts that both their intensions and extensions are identical or similar. Example 5. Using the concept c1 (pen) and object o11 (ballpoint pen) as given in Eq. (9) and Fig. 2, the synonym operation on c1 and o11 can be expressed as follows: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

cs (pen ) = cs (As , Os , Rsc , Rsi , Rso ),

cs (As , Os , Rsc , Rsi , Rso ) = c1 (A1 , O1 , R1c , R1i , R1o )  As = A1       Os = O1 ∧ = Rsc = Os × As    Rsi = R1i     o Rs = R1o .

(18)

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(d) Antonyms The antonym concepts are two concepts that have opposite meanings characterized by negative intensions (!A) and extensions (!O). Example 6. Using the concept c1 (pen) as given in Eq. (9), the antonym operation of c1 can be expressed as follows: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

ca (not a pen) = ca (Aa , Oa , Rac , Rai , Rao ),

ca (Aa , Oa , Rac , Rai , Rao )! = c1 (A1 , O1 , R1c , R1i , R1o )   Aa = !A1 = {!a1 , !a2 , !a3 }     = {non writing tool, not usining ink, having no nib}      Oa = !O 1 = {!o11 , !o12 , !o13 } = {not a ballpoint pen,       not a fountain pen, not a brush pen}  ∧ c = R1 = O1 × A1 

   i ni   R1 = R (ci , c1 )    i=1  

  nj   R1o = R (c1 , cj ) . 

(19)

j=1

(e) Consistent Concept The consistent concepts are two concepts with a mutual relation of being either a sub- or super-concept. Example 7. Using the concepts c1 (pen) and c3 (stationery) as given in Eqs. (9) and (11), the consistency operation of c1 and c3 can be expressed as follows: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

c3 (stationery) = c3 (A3 , O3 , R3c , R3i , R3o ), A1 ⊂ A3 ⇒ c1 ≺ c3 ⇒ c1 (pen) ∼ = c3 (stationery). (20)

(f) Concept Comparison A comparison between two concepts is an operation that determines the equivalency or similarity level of their intensions. Concept comparison is implemented according to the definition in Table 1, where the range of similarity between two concepts is between 0 and 1, in which weight 0 means no similarity and weight 1 means a full similarity between the two given concepts. Example 8. Using the concepts c1 (pen), c2 (printer ), and o11 (ballpoint pen) as given in Eqs. (9) and (10) as well as Fig. 2, the comparison operation of c1 ∼ c2

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and c1 ∼ o11 can be expressed as follows, respectively: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

c2 (printer) = c2 (A2 , O2 , R2c , R2i , R2o ),

c i o11 (ballpoint pen) = o11 (A11 , R11 , R11 ), ∧

c1 ∼ o11 = ∧

c1 ∼ c2 =

#{a1 , a2 , a3 } 3 #(A1 ∩ A11 ) = = = 0.75 #(A1 ∪ A11 ) #{a1 , a2 , a3 , a4 } 4

(21)

#{} 0 #(A1 ∩ A2 ) = = = 0. #(A1 ∪ A2 ) #{a1 , a2 , a3 , a5 , a6 } 5

The comparison function of concepts is a fuzzy membership function that quantitatively evaluates the extent of similarity of a pair of given concepts. Equation (21) indicates that although the similarity between c1 (pen) and c2 (printer ) is 0, the similarity between c1 (pen) and o11 (ballpoint pen) is 0.75. (g) Concept Definition Concept definition is an association between two concepts where they are equivalent. Example 9. Using the concepts c1 (pen) ando11 (ballpoint pen) as given in Eq. (9) and Fig. 2, the definition operation of c1 and o11 can be expressed as follows: ∃ c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ),

c i o11 (ballpoint pen) = o11 (A11 , R11 , R11 ),

c i , R11 )  c1 (A1 , O1 , R1c , R1i , R1o ) o11 (A11 , R11  A11 ⊇ A1 = {a writing tool, using ink, having a nib}     c  R11 = o × A11 ∧ = i i  R11 = R1    o R11 = {(o11 , c1 }.

(22)

3.2. Compositional operators of concept algebra for CWW The compositional operations of concept algebra are dynamic and integrative operations that change all involved concepts in parallel. Compositional operations on concepts provide a set of fundamental mathematical means to construct complex concepts on the basis of simple ones or to derive new concepts on the basis of existing ones. The compositional concept operators [19] are summarized in Table 2. (a) Concept Inheritance Concept inheritance as defined in Table 2 is a compositional operation that derives a new concept from the existing one. Multiple inheritances are an associative

Inheritance

Extension

Tailoring

Substitution

Composition

2

3

4

5

Operation

1

No.

Ro2 = Ro1 ∪ {(c2 , c1 )})



Ri2 = Ri1 ∪ {(c1 , c2 )}, Ro2 = Ro1 ∪ {(c2 , c1 )})

Ri2 = Ri1 ∪ {(c2 , c1 )}, Ro2 = Ro1 ∪ {(c2 , c1 )})

Rc2 = O2 × A2 , Ri2 = Ri1 ∪ {(c1 , c2 )}, Ro2 = Ro1 ∪ {(c2 , c1 )})

i=1



j=1



i=1

n S

Aci , Rc =

j=1

n S

„

Rcci ∪ {(c, cj ), (ci , c)},

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j=1

 R cj (Oj , Aj , Rcj , Rij , Roj |Rjj = Rij ∪ {(c, cj )}, Roj = Roj ∪ {(cj , c)})

n

n S

O ci , A = i=1 « n n S i S o Ri = Rci , Ro = Rci

= c(O, A, Rc , Ri , Ro |O =

∧

i=1

c(O, A, Rc , Ri , Ro )  R ci

n

− c1 (O1 , A1 , Rc1 , Ri1 , R01 |Ri1 = Ri1 ∪ {(c2 , c1 )}, Ro1 = Ro1 ∪ {(c2 , c1 )})

 ) ∪ O  , A = (A \A ) ∪ A , = c2 (O2 , A2 , Rc2 , Ri2 , Ro2 |O2 = (O1 \Oc1 2 1 c2 c1 c2

∧

c1 (O1 , A1 , Rc1 , Ri1 , Ro1 ) ⇒ c2 (O2 , A2 , Rc2 , Ri2 , Ro2 )

∼

c1 (O1 , A1 , Rc1 , Ri1 , R01 |Ri1 = Ri1 ∪ {(c2 , c1 )}, Ro1 = Ro1 ∪ {(c1 , c2 )})

= c2 (O2 , A2 , Rc2 , Ri2 , Ro2 |O2 = O1 \O  , A2 = A1 \A , Rc2 = O2 × A2 ⊂ Rc1 ,

∧

c1 (O1 , A1 , Rc1 , Ri1 , Ro1 ) ⇒ c2 (O2 , A2 , Rc2 , Ri2 , Ro2 )

−

c1 (O1 , A1 , Rc1 , Ri1 , R01 |Ri1 = Ri1 ∪ {(c2 , c1 )}), Ro1 = Ro1 ∪ {(c1 , c2 )})

∧

= c2 (O2 , A2 , Rc2 , Ri2 , Ro2 |O2 = O1 ∪ O  , A2 = A1 ∪ A , Ro2 = O2 × A2 ⊃ Rc1 ,

+ c2 (O2 , A2 , Rc , Ri , Ro ) c1 (O1 , A1 , Rc1 , Ri1 , Ro1 )⇒ 2 2 2

c1 (O1 , A1 , Rc1 , Ri1 , Ro1 |Ri1 = Ri1 ∪ Ri1 {(c2 , c1 )}), Ro1 = Ro1 {(c2 , c1 )})



= c2 (O2 , A2 , Rc2 , Ri2 , Ro2 |O2 ⊆ O1 , A2 ⊆ A1 , Rc2 = O2 × A2 , Ri2 = Ri1 ∪ {(c1 , c2 )},

∧

c1 (O1 , A1 , Rc1 , Ri1 , Ro1 ) ⇒ c2 (O2 , A2 , Rc2 , Ri2 , Ro2 )

Definition

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⇒

∼

⇒

−

⇒

+

⇒

Symbol

Table 2. Compositional operators of concepts in concept algebra for CWW.

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Decomposition

Aggregation/ generalization

Specification

Instantiation

7

8

9

Operation

6

No.

→







Symbol

n

i=1

j=1







i=1

Definition



Ri1 = Ri2 ∪ {(c2 , c1 )}), Ro1 = Ro2 ∪ {(c1 , c2 )})

Ri2 = Ri1 ∪ {(c1 , c2 )}), Ro2 = Ro1 ∪ {(c2 , c1 )})













c(O, A, Rc , Ri , Ro |Ri = Ri ∪ {(o, c)}, Ro = Ro ∪ {(c, o)})



Rio = Rio ∪ {(c, o)}, Roo = {(o, c)})



= o(Ao , Rco , Rio |o ⊂ O, Ao = A, Rco = o × A,

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∧

c(O, A, Rc , Ri , Ro ) → o(Ao , Rco , Rio )

c1 (O1 , A1 , Rc1 , Ri1 , R01 |Ri1 = Ri1 ∪ {(c, c1 )}), Ro1 = Ro1 ∪ {(c1 , c)})



= c2 (O2 , A2 , Rc2 , Ri2 , Ro2 |O2 ⊂ O1 , A2 ⊂ A1 , Rc2 = (O2 × A2 ) ∪ {(c2 , c1 ), (c1 , c2 )}

∧

c1 (O1 , A1 , Rc1 , Ri1 , Ro1 )  c2 (O2 , A2 , Rc2 , Ri2 , Ro2 )

c2 (O2 , A2 , Rc2 , Ri2 , R02 |Ri2 = Ri2 ∪ {(c1 , c2 )}), Ro2 = Ro2 ∪ {(c2 , c1 )})



= c1 (O1 , A1 , Rc1 , Ri1 , Ro1 |O1 ⊃ O2 , A1 ⊃ A2 , Rc1 = (O1 × A1 ) ∪ {(c1 , c2 ), (c2 , c1 )},

∧

c1 (O1 , A1 , Rc1 , Ri1 , Ro1 )  c2 (O2 , A2 , Rc2 , Ri2 , Ro2 )

= R ci (Oj , Aj , Rcj , Rij , Roj |Rij = Rij ∪ {(c, cj )}, Roj = Roj ∪ {(c, cj )}) „ j=1 n n n S S S     c O, A, Rc , Rj , Ro |O = O ci , A = Aci , Rci = (Rcci \{(c, cj )(cj , c)}), j=1  ffj=1  j=1 ff« n n   Rj = Ri ∪ R (cj , c) , Ro = ∪ R (c, cj )

∧

c(O, A, Rc , Ri , Ro )  R ci

n

344

Table 2. (Continued )

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operation where a concept is derived from multiple concepts. The inherence operation on concepts can be illustrated in the following example. Example 10. Using the concept c1 (pen) as given in Eq. (9), the inherence operation of a new concept c(pen  ) can be expressed as follows: c1 (A1 , O1 , R1c , R1i , R1o ) ⇒ c(A, O, Rc , Ri , Ro )   A = A1 = {a1 , a2 , a3 } = {a writing tool, using ink, having a nib} = O = O1 = {o11 , o12 , o13 } = {ballpoint, fountain, brush}   c (23) R = R1c = O × A. (b) Concept Tailoring, Extension, and Substitution Concept tailoring as defined in Table 2 is a special inheritance operation on concepts that reduces some inherited attributes or objects in the derived concept. Concept extension is associative operation that carries out a special concept inheritance with the introduction of additional attributes and/or objects in the derived concept. Concept substitution is an inheritance operation that results in the replacement or overload of attributes and/or objects by locally defined ones. Example 11. Using the concept c1 (pen) as given in Eq. (9), the tailoring, extension, and substitution operations of a new concept c(pen  ) can be expressed as follows, respectively: −

c1 (A1 , O1 , R1c , R1i , R1o ) ⇒ c(A, O, Rc , Ri , Ro )   A = A1 = {a1 , a2 , a3 } = {a writing tool, using ink, having a nib} = O = O1 = {o11 , o12 , o13 } = {ballpoint, fountain, brush}   c (24) R = R1c = O × A, +

c1 (A1 , O1 , R1c , R1i , R1o ) ⇒ c(A, O, Rc , Ri , Ro )  A = A1 = {a1 , a2 , a3 , a4 } = {a writing tool, using ink, having a nib,      with an ink container} = O = O1 = {o11 , o12 , o13 } = {ballpoint, fountain, brush}     c (25) R = R1c = O × A, ∼

c1 (A1 , O1 , R1c , R1i , R1o ) ⇒ c(A, O, Rc , Ri , Ro )  A = A1 = {a1 , a2 , a3 , a4 } = {a writing tool, using ink, having a nib,      with an ink container} =  O = O = {o , o , o } = {ballpoint, fountain, brush}  1 11 12 13    c (26) R = R1c = O × A.

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(c) Concept Composition Concept composition as defined in Table 2 is an associative operation that integrates multiple concepts in order to form a complex superconcept. Example 12. Using the concepts c1 (pen) and c2 (printer ) as given in Eqs. (9) and (10), the composition of c1 and c2 results in a superconcept c3 (stationery) as follows: ∧

c3 (stationery) = c3 (A, O, Rc , Ri , Ro ) = c1 (A1 , O1 , R1c , R1i , R1o )  A3              = O3              c R3

c2 (A2 , O2 , R2c , R2i , R2o ) = A1 ∪ A2 = {a1 , a2 , a3 } ∪ {a5 , a6 } = {a writing tool, using ink, having a nib, a printing tool, using papers}

(27)

= O1 ∪ O2 = {o11 , a12 , a13 } ∪ {o21 , o22 } = {ballpoint pen, fountain pen, brush pen, ink jet printer, laser printer} c = R1c ∪ R2c ∪ ∆R12 = R1c ∪ R2c ∪ {(c, c1 ), (c, c2 ), (c1 , c), (c2 , c)}.

c In the superconcept c3 (stationery), the newly created internal relations ∆R12 by the composition operation generate a set of new connections between the superconcept and the subconcepts.

(d) Concept Decomposition Concept decomposition as defined in Table 2 is an inverse operation of concept composition that results in two or multiple separated subconcepts from a superconcept. Example 13. Using the same layout of Example 12, a superconcept c3 (stationery) may be decomposed into two subconcepts c1 (pen) and c2 (printer ) as follows: 2

c3 (stationery) = c3 (A, O, Rc , Ri , Ro )  ci (Ai , Oi , Ric , Rii , Rio ) ∧

i=1

= c1 (A1 , O1 , R1c , R1i , R1o )  c2 (A2 , O2 , R2c , R2i , R2o )   A1 = {a1 , a2 , a3 } = {a writing tool, using ink, having a nib} = O1 = {o11 , a12 , a13 } = {ballpoint pen, fountain pen, brush pen}   c R1 = O1 × A1   A2 = {a5 , a6 } = {a printing tool, using papers}  O2 = {o21 , o22 } = {ink jet printer, laser printer}   c R2 = O2 × A2 .

(28)

During the decomposition operation on the superconcept c3 (stationery), the c = {(c, c1 ), (c, c2 ), (c1 , c), (c2 , c)} has been lost, incremental relations such as ∆R12 because they do not belong to any of these subconcepts.

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(e) Concept Aggregation Concept aggregation as defined in Table 2 is a compositional operation that assembles a complex concept by using components collected from multiple subconcepts. Example 14. Using the concepts c1 (pen) and c2 (printer ) as given in Eqs. (9) and (10), the aggregation operation assembles a superconcept c3 (stationery) from c1 and c2 as follows: 2

c3 (A3 , O3 , R3c , R3i , R3o )  R ci (Ai , Oi , Ric , Rii , Rio ) i=1  A3 ⊃ A1 ∪ A2 = {a writing tool, using ink, having a nib,       a printing tool, using papers}      O ⊃ O  1 ∪ O2 = {ballpoint pen, fountain pen, brush pen,  3 ∧ = ink jet printer, laser printer}   Rc = Rc ∪ Rc ∪ ∆Rc = Rc ∪ Rc ∪ {(c , c ), (c , c ), (c , c ), (c , c )}  3 1 1 3 3 2 2 3  3 1 2 12 1 2   i  i i  = {R ∪ (c , c )} ∪ {R ∪ (c , c )} R  1 3 1 3 3 1 2    o R3 = {R1o ∪ (c3 , c1 )} ∪ {R2o ∪ (c3 , c2 )}. (29) (f) Concept Specification Concept specification as defined in Table 2 is an inverse operation of concept aggregation that refines a concept by one or multiple subconcepts with more specific and precise attributes. Example 15. Using the concepts c1 (pen) and c3 (stationery) as given in Eqs. (9) and (11), the specification operation derives a concept from c3 to c1 as follows: c3 (O3 , A3 , R3c , R3i , R3o )  c1 (O1 , A1 , R1c , R1i , R1o )  A1 ⊂ A3 = {a writing tool, using ink, having a nib}       O1 ⊂ O3 = {ballpoint pen, fountain pen, brush pen} ∧

=

R1c = (O1 × A1 ) ∪ {(c3 , c1 ), (c1 , c3 )}     R1i = R3i ∪ {(c1 , c3 )}    o R1 = R3o ∪ {(c3 , c1 )}.

(30)

Example 16. The relationships between a series of specifications and aggregations of related concepts at different levels of abstraction are a pair of inverse operations, i.e.: c  c1  c2  · · ·  cn ⇒ c  c1  c2  · · ·  cn

(31)

given a series of concepts, say, animals as follows: (animal  mammal  f eline  tiger) ⇒ (animal  mammal  f eline  tiger). (32)

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(g) Concept Instantiation Concept instantiation as defined in Table 2 is a special compositional operation that derives an object or instance on the basis of the inherited concept. Example 17. Using the concept c1 (pen) as given in Eq. (9), the instantiation operation produces a derived object o11 (ballpoint pen) from the given concept c1 as follows: c1 (pen) = c1 (A1 , O1 , R1c , R1i , R1o ) → o11 (ballpoint pen) c i , R11 ) = o11 (A11 , R11   A11 = A1 = {a1 , a2 , a3 , a4 }       = {a writing tool, using ink, having a nib,        with an ink container} ∧ = c   = o × A11 = {(o, a1 ), (o, a2 ), (o, a3 ), (o, a4 )} R11       i  R11 = {(c3 , c1 ), (c2 , c1 )} ∪ {(c1 , o)}       Ro = {(o, c )}. 1 11

(33)

Contrasting the operators of concept instantiation and specification, it can be seen that concept instantiation is a special case of concept specification as given in Example 15, where the derived entity is a concrete object rather than an abstract or concrete concept.

4. Implementation of CWW by Concept Algebra It is found that although a word may often be ambiguous, the concepts elicited from a given word are always unique and rigorous. A wide range of applications of concept algebra has been found in CWW, semantic computing, and cognitive computing [22]. On the basis of a CWW support tool, this section describes the applications of concept algebra in the manipulations of abstract models of knowledge and the methodology for knowledge representation in CWW.

4.1. The formal model of knowledge for CWW In cognitive informatics [15, 17, 23, 25], particularly the OAR model [18] on internal knowledge representation in the brain, human knowledge is modeled as a concept network, where concept algebra is applied as a set of rules for knowledge composition in order to construct complex and dynamic concept networks. Definition 13. A generic knowledge K is an n-nary relation  among a set of n concepts, Ci , 1 ≤ i ≤ n, and the entire set of existing concepts C acquired in the

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brain, i.e.:  K =:

n

X Ci

i=1

 →C

n

=  Ci , i=1

−

+

(34)

∼

where  = •c = {⇒, ⇒, ⇒, ⇒, , , , , →}. According to Definition 13, the simplest knowledge k is a binary relation  between two concepts in C, i.e.: k =  : C × C → C.

(35)

Definition 13 indicates that the compositional operations of concept algebra, •c , provide a set of coherent mathematical means and rules for knowledge manipulations for CWW. Because the relations between concepts are transitive, the generic topology of knowledge is a hierarchical network as shown in Figs. 2 and 3. Theorem 3. The generic topology of the abstract knowledge system, K, is a hierarchical concept network. Theorem 3 can be proved by the nine compositional rules in concept algebra, particularly the composition/decomposition and aggregation/specification operations as described in Sec. 3. Corollary 1. The property of the hierarchical knowledge architecture K in the form of concept networks are as follows: (a) Dynamic: The knowledge network may be updated dynamically along with information acquisition and learning without destroying the existing concept nodes and relational links. (b) Evolvable: The knowledge network may grow adaptively without changing the overall and existing structure of the hierarchical network. 4.2. The hierarchical model of concept networks for CWW A concept network in CWW as a generic knowledge model has been widely studied in linguistics, computing, and cognitive informatics. The notion of the semantic network for knowledge representation is first proposed by Quillian in 1968 [7, 10, 11], where the semantic memory is perceived as information represented in network structures with conceptual nodes and interrelations. The meaning of a given concept depends on other concepts to which it is connected in the network. The semantic network has been extended by a number of models such as the hypothetical network [4], the adaptive control of thought-star (ADT*) model [1, 2]. The latter proposes that all cognition processes in thought are controlled by unitary network models.

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These empirical semantic networks can be formally modeled by the concept network based on concept algebra and the OAR model [18] for knowledge representation in CWW, which treat a concept as a basic and adaptive unit for knowledge representation and thinking. Definition 14. A concept network CN is a hierarchical network of concepts interlinked by the set of nine composing rules  in concept algebra, i.e.: n

n

i=1

j=1

CN =  : X Ci → X Cj .

(36)

Theorem 4. In a concept network, CN, the abstract levels of concepts lc form a partial order of a series of superconcepts, i.e.: c = (Ø  c1  c2  · · ·  cn  · · ·  Ω),

(37)

where Ω is the universal concept, Ω = (O, A), and Ø the empty concept, Ø = (⊥, ⊥). According to Theorem 4 and Definition 14, a hierarchical structure of concepts in a given semantic environment Θ can be formally described by concept algebra. The algebraic relations and compositional operators of concept algebra enable the construction of hierarchical concept networks in a dynamic process. Example 18. An abstract concept network for CWW that is formed by the composition and aggregation of a set of related concepts c0 through c9 , as well as objects o1 through o3 , can be expressed in Fig. 3. In CWW, a formal description corresponding to the above concept network can be carried out using a set of equivalent concept expressions in algebra as given below.

c9

c1

c3

-

c6

o1

+

c4

c0

C2

c5

o2

c7

c8

o3

Fig. 3. An abstract concept network for modeling CWW.

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+

c0  ((c1 ⇒ c3 ⇒ ¯ c6 → o1 )(c1 ⇒ c4 → o2 )) (c2  c5  (c7 (c8 → o3 ))

(38)

c9  (c6 c7 ). The case studies in Examples 1 through 18 presented in this work demonstrate that concept algebra and concept network are a generic and formal knowledge manipulation means for CWW, which are capable of dealing with complex abstract or concrete knowledge structures and their algebraic operations. Further detailed concept operations of concept algebra may be extended into a set of inference processes, which can be formally described by real-time process algebra (RTPA) [16, 20] as a set of behavioral processes. 4.3. Implementation of a CWW tool based on concept algebra In order to support a working CWW system, a concept-algebra-based CWW tool [13] has been developed in Java on Windows XP platform as shown in Fig. 4. The tool implements all concepts operations as defined in concept algebra by a set of Java classes.

Fig. 4. Design of the CWW support tool.

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Knowledge Representation Tool

Concept Parser

Knowledge Base

Knowledge Complier

Knowledge Representor

Fig. 5. The architecture of the CWW support tool.

The CWW support tool is composed by the concept parser, knowledge compiler, knowledge base, and knowledge representor as illustrated in Fig. 5. In the CWW support tool, the concept parser identifies new concepts from the input information, which is an automatic concept gatherer. Based on given rules, it recognizes new concepts from the input information stream. When structured data streams are received, tokens such as objects, attributes, internal relations, and external relations of the concept are distilled. Identified concepts will be compared with existing concepts in the knowledge base in order to check whether it is a new concept. If it is a new concept, attributes/objects and relations will be used to decide which layer the concept belongs to. Finally, the new concept and its attribute are stored into the concept database. The knowledge complier recognizes new knowledge from output data streams of the concept parser. It also collects relations among new and existing concepts in order to form knowledge. Objects, attributes and layer information of a given concept will be used by the knowledge complier to analyze implied relations. The knowledge base consists of a set of relation tables based on the OAR model [18], which incorporates the entire knowledge of acquired concepts and knowledge. The knowledge base component functions as a support center for the concept parser, the knowledge complier, and their intermediate work products. The knowledge representor of the tool represents and displays knowledge in different ways such as diagrams and tables. The knowledge representor visualizes the consumed knowledge acquired by machines stored in the knowledge base. The CWW tool for knowledge representation and concept-based reasoning not only implements a set of functional algorithms, but also considers its run-time performance, memory efficiency, and system portability. A linked digraph model is adopted to support concept operations in CWW. The internal knowledge structure

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Fig. 6. A case study on concept manipulation using the CWW tool.

is also flexible to guarantee query response time. Coding theories, as well as layer and inheritance information, are introduced in system design in order to effectively improve processing speed. For example, a set of concepts contains information pertaining to vehicles and modes of transportation. The dataset as shown in Fig. 6 encompasses seven different concepts, where the concepts “a car” and “a truck” share the same superconcepts of “a vehicle” and “transportation” from the bottom up. The concept “a car” has a certain number of four attributes such as ride, color, doors, and tires; so do the “a truck”. The CWW system can be sued to transfer the concept algebra theory in denotational mathematics into a rigorous knowledge and concept-based reasoning methodology. It is also adopted as a powerful tool for implementing knowledge engineering and autonomous machine learning systems in CWW. 5. Conclusions CWW is an emerging computational technology for words, linguistic variables, concept, and knowledge processing for supporting human and machine reasoning and causality analyses. Abstract concepts play an important role in CWW semantic computing, and cognitive computing. This paper has presented a formal treatment of abstract concepts and a comprehensive set of algebraic operations for CWW.

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The mathematical model of abstract concepts and an abstract mathematical structure known as concept algebra have been developed for knowledge representation and manipulations in CWW. Based on concept algebra, a knowledge system has been formally modeled as a concept network, where the methodology for knowledge manipulating towards CWW has been presented. Concept algebra have provided a powerful denotational mathematical means for manipulating complicated abstract and concrete knowledge structures as well as their algebraic operations in CWW. A CWW support tool has been developed to mimic and simulate human language processing and concept-centered reasoning. The case studies provided in this paper have demonstrated that concept algebra is not only a powerful conceptual modeling methodology for CWW, but also a functional specification methodology for knowledge manipulations in semantic computing and cognitive computing. The case studies have also illustrated that any concrete concept structures can be rigorously designed, modeled, and manipulated in CWW using concept algebra. Acknowledgments A number of notions in this work have been inspired by Prof. Lotfi A. Zadeh during my sabbatical leave at BISC, UC Berkeley as a visiting professor. I am most grateful to Prof. Zadeh for his vision, insight, and kind support. The author acknowledges the Natural Science and Engineering Council of Canada (NSERC) for its partial support of this work. The author would like to thank the anonymous reviewers for their valuable comments and suggestions. References [1] J. R. Anderson, The Architecture of Cognition (Harvard Univ. Press, Cambridge, MA, 1983). [2] J. R. Anderson, Is human cognition adaptive? Behavioral and Brain Science 14 (1991) 71–517. [3] R. Codin, R. Missaoui and H. Alaoui, Incremental concept formation algorithms based on galois (concept) lattices, Computational Intelligence 11(2) (1995) 246–267. [4] A. M. Colins and E. F. Loftus, A spreading-activation theory of semantic memory, Psychological Review 82 (1975) 407–428. [5] B. Ganter and R. Wille, Formal Concept Analysis (Springer, Berlin, 1999). [6] J. A. Hampton, Psychological Representation of Concepts of Memory (Psychology Press, Hove, 1997), pp. 81–110. [7] M. W. Matlin, Cognition, 4th edn. (Harcourt Brace College Pub., New York, 1998). [8] D. L. Medin and E. J. Shoben, Context and structure in conceptual combination, Cognitive Psychology 20 (1988) 158–190. [9] G. L. Murphy, Theories and concept formation, in I.V. Mechelen et al. (eds.), Categories and Concepts, Theoretical Views and Inductive Data Analysis (Academic Press, New York, 1993), pp. 173–200. [10] M. R. Quillian, Semantic memory, in M. Minsky (ed.), Semantic Information Processing (MIT Press, Cambridge, MA, 1968).

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