Reasoning with Contexts

Reasoning with Multilevel Contexts in Semantic Metanetwork

The goal of this topic is to study formal way to represent knowledge within several levels of contexts which can also been interpreted as levels of experts’ competence.

Reference: Terziyan V., Puuronen S., Multilevel Context Representation Using Semantic Metanetwork, In: Context-97 - International and Interdisciplinary Conference on Modeling and Using Context, Rio de Janeiro, Brazil, Febr. 4-6, 1997, pp. 21-32.

A multilevel semantic network is proposed to be used to represent knowledge within several levels of contexts. The zero level of representation is semantic network that includes knowledge about basic domain objects and their relations. The first level of presentation uses semantic network to represent contexts and their relationships. The second level presents relationships of metacontexts i.e. contexts of contexts, and so on at the higher levels. The topmost level includes knowledge which is considered to be “truth” in all the contexts. Such representation allows to reason with contexts towards solution of the following problems:  to derive knowledge interpreted using all known levels of its context;  to derive unknown knowledge when interpretation of it in some context and the context itself are known;  to derive unknown knowledge about a context when it is known how the knowledge is interpreted in this context;  to transform knowledge from one context to another.

A Semantic Metanetwork

Formally metanetwork is a quadruple , where:

A is a set of objects which consists of the subset of basic domain 0 objects Ai ,i  1,...,n1, and several subsets of contexts (d) , and identifies Ai ,i  1,...,nd d  1,...,klev the level of the metanetwork where the context appears; (d) L is a set of unique names of relations Lk ,k  1,...,md and d  0,...,klev identifies the level of the metanetwork where the relation appears; (d) (d) (d) (d) S is the set of relations Sr  P(Ai , Lk , Aj ),r  1,...,ld (d) (d) composed in each level so that: S   Sr , and r P(A(d), L(d), A(d)) (d) i k j is true when there is the relation Lk L (d) (d) (d) (d) between the objects Ai and Aj ,(Ai , Aj A) at the level d; (d) (d 1) (d) D is the set of context predicates Dr  ist( Ai , Sr ) connecting contexts of the level d+1 to the relations of the (d 1) (d) (d) level d and ist( Ai , Sr ) is true if the relation Sr (d 1) holds in the context Ai .

A Semantic Metanetwork: An Example

A ' ' 2 L'' L'' 1 2 Second l evel A ' ' A ' ' 3 1

A ' 1 A ' 3 L' 2 A ' L' 4 1 First l evel

L' A ' 3 2

A 2 L L L 2 4 1 Zero l evel A A 1 L 3 3 A Semantic Metanetwork: An Example

L' A ' 3 2

A 2 L L L 2 4 1 Zero l evel A A 1 L 3 3

0 ' '' 0 ' ' ' ' ' '' '' '' '' A  A , A , A , A   A1, A2 , A3, A  A1, A2 , A3, A4, A  A1 , A2 , A3 are the objects’ set and its three subsets accordingly to levels;

0 ' '' 0 ' ' ' ' '' '' '' L  L , L , L , L  L1, L2 , L3, L4, L  L1, L2 , L3, L  L1, L2  are the set of relations’ names and its three subsets accordingly to levels;

0 ' '' 0 ' ' ' ' '' '' '' S  S ,S ,S  , S  S1,S2 ,S3,S4,S  S1,S2 ,S3,S  S1 ,S2 are the set of relations and its three subsets accordingly to levels;

S1  P(A1, L1, A2 ),S2  P(A2, L2, A3),S3  P(A1, L3, A3),S4  P(A3, L4, A3); ' ' ' ' ' ' ' ' ' ' ' ' S1  P(A1, L1, A2),S2  P(A2, L2, A3),S3  P(A2, L3, A4); '' '' '' '' '' '' '' '' S1  P(A1 , L1, A2 ),S2  P(A2, L2, A3 ) are the relations of all the three levels in a predicate form; A Semantic Metanetwork: An Example

L' A ' 3 2

0 ' 0 ' ' ' ' D  D , D  , D  D1, D2 , D3, D4 , D5, D  D1, D2 , D3 are the set of context predicates and its two subsets accordingly to levels; ' ' ' D1  ist( A1, S1), D2  ist( A2 , S2 ), D3  ist( A3 , S3 ), ' ' D4  ist( A3 , S4 ), D5  ist( A4 , S4 ) are the context predicates describing context relationships between zero and first levels;

' '' ' ' '' ' ' '' ' D1  ist( A1 ,S1), D2  ist( A2 ,S2 ), D3  ist( A3 ,S3) are the context predicates describing context relationships between first and second levels. Some Concepts

Further in formulas we will use:

“  ” to mark the logical equivalence between any two predicates;

“  ” to mark the logical consequence between two predicates;

“  ” between names of relations or objects to mark equality of semantic meanings of these two names;

“  ” between names of relations to mark inequality of semantic meanings of these two names.

The main equation used in proofs:

(P( Ai , Lk , A j )  P( Ai , Lm , A j ))  Lk  Lm Semantic Constants: The Always True Relations - Relation of “Difference” and Relation of “Same”

We consider knowledge about the difference as semantic constant DIF which denotes the relation between every pair of different objects:

Ai , A j( j i) (P( Ai , DIF, A j )  true)

Semantic constant of equivalence SAME:

Ai (P( Ai ,SAME, Ai )  true)

The last formula means that if there is not any knowledge about properties of some object, than at least one property always holds - to be same to itself. Semantic Constants: The Always False Relation and Relation of “Similarity”

We define NIL relation as relation which never holds between any two objects or among properties of any object by the following way:

Ai , A j( j i) (P( Ai , DIF, A j )  P( Ai , NIL, A j )  false)

Ai (P( Ai ,SAME, Ai )  P( Ai , NIL, Ai )  false)

The relation of “similarity”:

(P( As , Lk , Ai )  P( As , Lk , A j ),As  A,Lk  L)  b  P( Ai ,SAME , A j ) b where SAME means the binary equivalent of SAME relation. This relation does not mean the negation of DIF relation, because it means only that two different objects have the same semantics. Semantic operations: Semantic inversion ~ Lk Lk Ai Aj  Ai Aj ~ P( Ai , Lk , A j )  P( A j , Lk , Ai ) ~ where Lk is the new inverse relation. It is named with unique symbol Lm and added to the set L.

For example, if Lk = , then Lm = ~ and Lm  Lk .

Similarly, let Ln = , then ~ Lq = and Lq  Ln .

In the case when Lk is a property of an object: ~ ~ P( Ai , Lk , Ai )  P( Ai , Lk , Ai ) and Lk  Lk .

The obvious property for the semantic inversion ~ operation is double inversion: Lk  Lk . Semantic Operations: Semantic Negation

The semantic negation operation means changing the name of a relation if the value of appropriate predicate is false:

P( Ai , Lk , A j )  P( Ai , Lk , A j ) where Lk is the new relation. It can be named with unique symbol Lm and added to the set L. If, for example:

P(, , ) = false,

then it means the same as:

P(, , ) = true.

Thus, if Lk = and Lm = , then Lm  Lk . Properties of Semantic Negation

 double negation: Lk  Lk ;

~ ~  negation of inversion: Lk  Lk ;

Proof:

~ ~ (P( Ai , Lk , Aj )  P( Ai , Lk , Aj )  P( Aj , Lk , Ai )  ~ ~ ~  P( Aj , Lk , Ai )  P( Ai , Lk , Aj ))  Lk  Lk

Semantic Operations: Semantic Multiplication (Composition) L As L L As L k n  k n Ai Aj Ai Aj Lk * Ln

P( Ai , Lk , As )  P( As , Ln , A j )  P( Ai , Lk * Ln , A j ) , where Lk * Ln is the new relation. It can be named with unique symbol Lm and added to the set L. If, for example, it is true that: P(, , ) and P(, , ), then it is also true that: P(, , ). Thus, if: Lk = , Ln = ,

and Lm = , then Lm  Lk * Ln . Properties of Semantic Multiplication:

 non-commutativity : (Lk * Ln  Ln * Lk ), b Lk , Ln (Lk  Ln  DIF  SAME )  transitivity:

Lk *(Ln * Lm )  (Lk * Ln ) * Lm ; Proof:

(P( Ai , Lk *(Ln * Lm ), A j ) 

 P( Ai , Lk , As )  P( As , Ln * Lm , A j ) 

 P( Ai , Lk , As )  P( As , Ln , At )  P( At , Lm , A j ) 

 P( Ai , Lk * Ln , At )  P( At , Lm , A j ) 

 P( Ai ,(Lk * Ln ) * Lm , A j )) 

 Lk *(Ln * Lm )  (Lk * Ln ) * Lm  inversion over multiplication: ~ ~ ~ (Lk * Ln )  Ln * Lk ; Proof:

(P( Aj , Lk * Ln, Ai )  P( Aj , Lk , As)  P( As, Ln, Ai )  ~ ~  P( As, Lk , Aj )  P( Ai , Ln, As)  ~ ~  P( Ai , Ln, As)  P( As, Lk , Aj )  ~ ~ ~ ~ ~  P( Ai , Ln * Lk , Aj ))  (Lk * Ln)  Ln * Lk .

Semantic Operations: Semantic Addition Lk Lk  Ln Ai Aj  Ai Aj Ln

P( Ai , Lk , A j )  P( Ai , Ln , A j )  P( Ai , Lk  Ln , A j ) , where Lk  Ln is the new relation. It can be named with unique symbol Lm and added to the set L. If, for example, it is true that: P(, , ) and P(, , ), then: P(, , ) is also true. Thus, if: Lk = , Ln = , and Lm  Lk  Ln = . It is also possible to sum properties. If, for example, it is true that P(, , ) and P(, ,), then it is also true that: P(, , ).

Properties of Semantic Addition  commutativity: Lk  Ln  Ln  Lk ;

 transitivity: Lk  (Ln  Lm )  (Lk  Ln )  Lm ;

 reflexivity: Lk  Lk  Lk ; ~ ~ ~  inversion over sum: (Lk  Ln )  Lk  Ln ;  distributivity (left):

Lk *(Lm  Ln )  Lk * Lm  Lk * Ln ; Proof:

(P( Ai , Lk *(Lm  Ln), Aj )  P( Ai , Lk , As)  P( As, Lm  Ln, Aj ) 

 P( Ai , Lk , As)  P( As, Lm, Aj )  P( As, Ln , Aj ) 

 P( Ai , Lk , As)  P( Ai , Lk , As)   P( As, Lm, Aj )  P( As, Ln, Aj )  (P( Ai , Lk , As)  P( As, Lm, Aj )) 

 (P( Ai , Lk , As)  P( As, Ln, Aj )) 

 P( Ai , Lk * Lm, Aj )  P( Ai , Lk * Ln, Aj ) 

 P( Ai , Lk * Lm  Lk * Ln, Aj )) 

 Lk *(Lm  Ln)  Lk * Lm  Lk * Ln;

 distributivity (right)

(Lk  Lm ) * Ln  Lk * Ln  Lm * Ln .

Semantic Closeness We define the L-set L:L1, L2 ,... Ln as semantically closed set of relations’ names if the result of any semantic operation with any operands from the L-set also belongs to the L -set: ~ Lk  L,Lk  L ; Lk  L,Lk  L ;

Lk * Lm  L,Lk , Lm  L ;

Lk  Lm  L,Lk , Lm  L ; ~ ~ ~ ~ L:L1, L2 ,..., Ln  L ; L:L1, L2 ,... Ln  L ;

(Li * L):Li * L1, Li * L2 ,..., Li * Ln  L,Li  L ;

(L * Li ):L1 * Li , L2 * Li ,..., Ln * Li   L,Li  L ;

(Li  L):Li  L1, Li  L2 ,..., Li  Ln  L,Li  L .

Properties of Semantic Constants The negation operation defines the main relationship between the two main constants as follows: DIF  SAME  NIL

Properties of DIF:

 neutrality to semantic sum: DIF  Lk  Lk . Proof: (P( A , DIF  L , A )  P( A , DIF, A )  P( A , L , A )  i k j   i   j i k j true ;  P( Ai , Lk , Aj ))  DIF  Lk  Lk  elimination in semantic multiplication:

DIF * Lk  Lk * DIF  DIF ; ~  inversion of DIF: (DIF)  DIF . Properties of Semantic Constants

Properties of SAME:

 inversion of SAME ~ (SAME)  SAME ;

 neutrality to semantic sum SAME  Lk  Lk ;

Properties of Semantic Constants Properties of SAME b : b b  inversion of SAME b : ~ (SAME )  SAME ;

b  neutrality to semantic sum: SAME  Lk  Lk ;

 neutrality to semantic multiplication:

b b Lk * SAME  SAME * Lk  Lk ;

b ~ SAME , for relations;  annihilation: a) Lk  Lk   Lk , for properties. ~ ~ b b) Lk * Lk  Lk * Lk  SAME .

 elimination (left): Lk  Lk * Lm  Lk * Lm . Proof: b Lk  Lk * Lm  Lk * SAME  Lk * Lm  ; b  Lk *(SAME  Lm)  Lk * Lm

 elimination (right): Lk  Lm * Lk  Lm * Lk .

Semantic Equations 1 Lx 2 Lx F1(L )  F2(L ) where F1 , F2 are functions, which use semantic operations and they have known operands respectively subsets L1, L2 of the L-set and one unknown operand Lx, which is defined on the whole L-set.

We use “=” in semantic equations to tell the difference between semantic equations and semantic identities where we use “  ”.

The solution of such equation is the relation Lw, which, being substituted in an equation instead of the operand Lx will transform it to an identity. We will use notation (equation) Lx  Lw as a statement that relation Lw is the solution of the equation.

Semantic Equations: Basic Properties 1 2 Lx  Lw (F1(L )  F2 (L ))  ~ 1 ~ 2 Lx  Lw  (F1(L )  F2 (L ))

1 2 Lx  Lw (F1(L )  F2 (L )) 

1 2 Lx  Lw  (F1(L )  F2 (L ))

1 2 Lx  Lw (F1(L )  F2(L ))  1 2 Lx  Lw  (F1(L )  Lk  F2 (L )  Lk ) ,Lk  L

1 2 Lx  Lw (F1(L )  F2 (L ))  1 2 Lx  Lw  (F1(L ) * Lk  F2 (L ) * Lk ) ,(Lk  DIF  L)

1 2 Lx  Lw (F1(L )  F2(L ))  1 2 Lx  Lw  (Lk * F1(L )  Lk * F2(L )) ,(Lk  DIF  L) Semantic Equations: Solution of Basic Equations

~ ~ ~ ~ Lx  Li  Lx  Li  Lx  Li ;

Lx  Li  Lx  Li  Lx  Li ;

~ ~ Lx  Li  Lj  Lx  Li  Li  Lj  Li  ; b ~ ~  Lx  SAME  Lj  Li  Lx  Lj  Li

~ ~ Lx * Li  Lj  Lx * Li * Li  Lj * Li  ; b ~ ~  Lx * SAME  Lj * Li  Lx  Lj * Li

~ ~ Li * Lx  Lj  Li * Li * Lx  Li * Lj  . b ~ ~  SAME * Lx  Li * Lj  Lx  Li * Lj Operation of Semantic Interpretation

a) ' ' Ln Ln Lk Lk Ai Aj  Ai Aj ' Al

b) ' Ln ' Ln ' L Al k Ai  Ai Lk

' ' ' ' P( Ai , Lk , Aj )  P( Al , Ln, Al )  ist( Al , P( Ai , Lk , Aj ))  ' Ln  P( Ai , Lk , Aj ) ' where Ln denotes the interpretation of the Lk knowledge Lk using knowledge L'n about the context ' Al . This can be named and included to the set L.

Expansion of Definitions for Semantic Constants:

SAME 1. Lk  Lk . This means an abstract case when the context of a relation contains only the universal property SAME. In such case the context cannot change an interpretation of this relation;

Lk 2. SAME  Lk . This means also an abstract case when the knowledge being interpreted contains only the universal property SAME. In such case the only knowledge obtained as a result of interpretation is the knowledge about context;

b Lk b 3. (SAME )  Lk . This means that interpretation of the equivalence relation between two objects inherits the properties of context Lk producing its binary b equivalent Lk ; Expansion of Definitions for Semantic Constants: L m Lm 4. (Lk  Ln )  (Ln  Lk ) - axiom of extracting knowledge.

This means that if some knowledge Ln has been obtained as interaction of knowledge Lk and property Lm of some context, then removal of that property from the context of the interpretation result leads to the restoration of initial knowledge; Consequence: Lk b ; Lk  SAME

Lm Ln 5. (Lk  Ln )  (Lk  Lm ) - axiom of extracting context.

This means that if some knowledge Ln has been obtained as interaction of knowledge Lk and property Lm of some context, then removal of the initial knowledge Lk in the context of the interpretation result leads to the restoration of initial context. L Consequence: k b . Lk  SAME Transformations with contexts in the Algebra

Inversion in the context. The inverse result of interpretation a relation in a context is equal to the result of interpretation the inverse relation in the same context. Formally:

~ Lm ~Lm (Lk )  Lk .

Negation in the context. The negative result of interpretation a relation in a context is equal to the result of interpretation the negative relation in the same context, and also it equals to the result of interpretation of this relation in a negative context. Formally:

Lm Lm Lm . Lk  Lk  Lk Transformations with contexts in the Algebra

Addition in the context. The result of interpretation of the sum of two not- conflicting relations in a context is equal to the semantic sum of these two relations interpreted separately in the same context. Formally:

Ln Ln Ln (Lk  Lm )  Lk  Lm , when Lk  Lm. Multiplication in the context. The result of interpretation of the semantic multiplication of two not-conflicting relations in a context is equal to the semantic multiplication of these two relations interpreted separately in the same context. Formally:

Ln Ln Ln (Lk * Lm)  Lk * Lm , when Lk  Lm . Transformations with contexts in the Algebra

Interpretation in the sum of contexts. The result of interpretation of a relation in a semantic sum of two not-conflicting contexts is equal to the semantic sum of this relation interpreted separately in these two contexts. Formally:

Lm Ln Lm Ln Lk  Lk  Lk , when Lm  Ln . Addition of interpreted relations. The result of a semantic sum of two not- conflicting relations interpreted separately in two not-conflicting contexts is equal to the semantic sum of these relations interpreted in the semantic sum of these contexts:

Lm Ln Lm Ln Lk  Lr  (Lk  Lr ) , when Lk  Lr, Lm  Ln . Transformations with contexts in the Algebra

Multiplication of interpreted relations. The result of a semantic multiplication of two not-conflicting relations interpreted separately in two not-conflicting contexts is equal to the semantic multiplication of these relations interpreted in the semantic sum of these contexts:

Lm Ln Lm Ln Lk * Lr  (Lk * Lr ) , when Lk  Lr, Lm  Ln.

Multilevel interpretation. The result of a relation interpretation in several levels of contexts does not depend on the order of interpretation:

Ln Lm Ln (Lm ) . (Lk )  Lk , when Lm  Lk , Lm  Ln Equations with Contexts

Basic rules:

1 2 Lx  Lw (F1(L )  F2 (L ))  1 Lk 2 Lk Lx  Lw  (F1(L )  F2(L ) ) ,Lk  L

1 2 Lx  Lw (F1(L )  F2 (L )) 

1 2 F1( L ) F2 ( L ) Lx  Lw  (Lk  Lk ) ,Lk  L

Solution of basic equations:

Li Li Li Li Lx  Lj  (Lx )  Lj 

Li ( Li ) Li SAME Li Li  Lx  Lj  Lx  Lj  Lx  Lj ;

Lx Lx ( Li ) Lj Li  Lj  Li  Li  L L L Li Lx j Lx j j  (Li )  Li  SAME  Li  Lx  Li . Semantic Equations: An Example

~ ~ ~ ~ ( L *L *L  L ) L6 L1 * L2 3 x 4 5 * L7  L8  L9 

~ ~ ~ L6 ~ ( L3*Lx *L4  L5) ~  L1 * L2 * L7  L9  L8 

~ ~ ~ ( L *L *L  L ) L6 ~ ~  L2 3 x 4 5 * L7  L1 *(L9  L8 ) 

~ ~ ~ L6 ( L3*Lx *L4  L5) ~ ~  L2  L1 *(L9  L8) * L7 

~ ~ ~ ( L *L *L  L ) ~ ~ L6  L2 3 x 4 5  (L1 *(L9  L8 ) * L7 ) 

~ ~ ~ L6 ~ ~ ( L1*( L9  L8)*L7 )  L3 * Lx * L4  L5  L2 

~ ~ ~ ~ ~ ( L *( L  L )*L ) L6 ~  L3 * Lx * L4  L2 1 9 8 7  L5 

~ ~ ~ ~ L6 ~ ~ ( L1*( L9  L8 )*L7 ) ~  Lx * L4  L3 *(L2  L5) 

~ ~ ~ ~ ( L *( L  L )*L ) L6 ~ ~  Lx  L3 *(L2 1 9 8 7  L5) * L4 

~ ~ ~ ~ ( L *( L  L )*L ) L6 ~ ~  Lx  (L3 * (L2 1 9 8 7  L5) * L4 ). Reasoning with Contexts Deriving an interpreted knowledge (decontextualization). Deriving of interpreted knowledge means here deriving the formula for knowledge interpreted using all known levels of its context. L We define the knowledge of a relation Ai Aj between any pairs of objects ( Aj , Ak ) from the same level of a semantic metanetwork as a semantic sum over all possible paths between these objects ( Aj , Ak ) that exist at this level of the metanetwork. L We define the knowledge Ai of an object Ai of a semantic metanetwork in one level as a semantic sum over all knowledge of the relations that connect this object with all objects of the same level, including the object itself: L  L Ai  Ai Aj . j The interpreted knowledge of any relation, considering all contexts and metacontexts, is derived by the following schema:  interpreted knowledge  

...knowl. about metacont. of nth level   knowledge knowl. about context  .

Example of Decontextualization (1)

L' A ' 3 2

Let us derive the interpreted knowledge of the relation between A1 and A3 . We start from the top level of the metanetwork and define '' '' '' knowledge about metacontexts A1 , A2, A3 :

L ''  L '' ''  L '' ''  L '' ''  A1 A1  A2 A1  A3 A1  A1 '' '' '' '' '' '' ''  L1  L1 * L2  SAME  L1 *(SAME  L2 )  L1 * L2 ; ~'' '' ~'' ~'' L ''  L1  L2 ; L ''  L2 * L1 . A2 A3 Example of Decontextualization (2)

L' A ' 3 2

A 2 L L L 2 4 1 Zero l evel A A 1 L 3 3 We continue at the first level and derive the interpreted knowledge of the first level relations:

L '' '' '' ' A1 ' L1*L2 L ' '  (L1)  (L1) ; A1 A2 L '' L '' ' A1 ' A2 L ' '  (L1) *(L2)  A1 A3 '' '' ~'' '' '' '' ~'' ' L1*L2 ' L1  L2 ' ' L1*L2  L1  (L1) *(L2)  (L1 * L2) ; L '' L '' '' '' ~'' ~'' ' A1 ' A3 ' L1 *L2 ' L2 *L1 L ' '  (L1) *(L3)  (L1) *(L3) A1  A4 ; ... L '' L '' ~' A3 ' A2 L ' '  (L3) *(L2 )  A4  A3 ~'' ~'' ~'' '' ~'' ~'' '' ~' L2 *L1 ' L1  L2 ~' ' L2 *L1  L2  (L3) *(L2 )  (L3 * L2) . Example of Decontextualization (3)

L' A ' 3 2

A 2 L L L 2 4 1 Zero l evel A A 1 L 3 3 ' ' ' The knowledge about contexts A1, A2, A3 of the first level is derived as follows:

'' '' ~'' '' '' ~'' ~'' ' ' L1*L2  L1 ' ' L1*L2  L2 *L1 L '  (L1 * L2 )  (L1 * L3)  A1 '' '' ~'' ~'' ' ' ' L1*L2  L2 *L1  (L1 *(L2  L3)) ; '' '' ~'' '' ~'' ~'' ~' L1*L2 ' L1  L2 ' L2 *L1 L '  (L1)  (L2 )  (L3)  A2 '' '' ~'' ~'' ~' ' ' L1*L2  L2 *L1  (L1  L2  L3) ;

'' '' ~'' '' ~'' ~'' ~' ~' L1*L2  L1 ~' ' L2  L2 *L1 L '  (L2 * L1)  (L2 * L3)  A3 '' '' ~'' ~'' ~' ~' ' L1*L2  L2 *L1  (L2 *(L1  L3)) ;

Example of Decontextualization (4)

L' A ' 3 2

A 2 L L L 2 4 1 Zero l evel A A 1 L 3 3 Now it is possible to derive the interpreted knowledge about the relation between A1 and A3 taking all contexts and metacontexts into account:

L L L A' A' A' L  (L ) 1 *(L ) 2  (L ) 3  A1A3 1 2 3 '' '' ~ '' ~ '' ' ' ' ' ~ ' ~ ' ~ ' ' ( L1*L2  L2 *L1 ) ( L1 *L2  L1 *L3  L2 *L1  L2 *L3 )  (L1 * L2  L3) .

Reasoning with Contexts

Deriving unknown knowledge that is interpreted when the result of interpretation and the context of interpretation are known (contextualization).

This problem occurs when some knowledge has been interpreted in some context and we have all knowledge about this context and knowledge that is the result of interpretation. For example, let us suppose that your colleague, whose context you know well, has described you a situation. You use knowledge about context of this person to interpret the “real” situation. Example is more complicated if several persons describe you the same situation. In this case, the context of the situation is the semantic sum over all personal contexts.

This second reasoning problem can be solved using the following equation:

knowledge about context Lx  interpreted knowledge  .

Deriving unknown context of interpretation when the knowledge and its interpretation in this context are known (context recognition). This problem occurs when we have knowledge that has been interpreted in some unknown context and we also know what is the result of interpretation. For example let us supposed that someone sends you a message describing the situation that you know well. You compare your own knowledge with the knowledge you received. Usually you can derive your opinion about the sender of this letter. Knowledge about the source of the message, you derived, can be considered as certain context in which real situation has been interpreted and sometimes it can help you to recognize a source or at least his motivation to change the reality.

This third reasoning problem can be solved using the following equation:

 knowledge Lx  interpreted knowledge  .

Lifting (relative decontextualization).

This means deriving knowledge interpreted in some context if it is known how this knowledge was interpreted in another context. This problem is solved by successive solution of the contextualization and decontextualization problems. Let Lk is the result of interpretation of some knowledge Lx in the context Lm . The problem is to derive how this knowledge would be interpreted in the context Ln. Thus we have the following procedure of lifting:

Lm Lm (Lx  Lk )  (Lx  Lk )contextualization  Ln Ln Lm  (Lx  Lk )decontextualization

Discussion 1. How to interpret the concept of Metasemantic Network in the context of multiple experts problems ?

2. What in this case would mean the semantic interpretation operation ?

3. What in this case would mean all four reasoning problems with contexts ?