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 <A,L,S,D>, where:
A
is a set of objects which consists of the subset of basic domain
Ai0 , i  1,..., n1 , and several subsets of contexts
objects
(d )
Ai , i  1,..., nd , and
identifies
the level of the metanetwork where the context appears;
d  1,..., klev
( d ) , k  1,..., m
L
L is a set of unique names of relations k
d and
d  0,..., klev identifies the level of the metanetwork
where the relation appears;
S
(d )
(d ) (d ) (d )
is the set of relations Sr  P( Ai , Lk , Aj ), r  1,..., ld
( d )  S( d )
S
 r , and
composed in each level so that:
r
(d )
P( Ai( d ) , L(kd ) , A(j d ) ) is true when there is the relation Lk  L
( d )and A( d ) ,( A( d ) , A( d )  A)
A
between the objects i
at the
j
i
j
level d;
( d 1)
(d )
D is the set of context predicates Dr  ist ( Ai
connecting contexts of the level
level
d
( d 1)
and ist ( Ai
d+1 to the relations of the
(d )
, S r( d ) ) is true if the relation Sr
( d 1)
holds in the context Ai
, S r( d ) )
.
A Semantic Metanetwork: An Example
A''
L''1
2
L''2
S e c o n d le v e l
A''
A''
A'
A'
1
L' 2
L' 1
A'
3
1
3
A'
4
F irs t le v e l
L' 3
2
L 1
A
2
L 2
L 4
Z e ro le v e l
A
1
L 3
A
3
A Semantic Metanetwork: An Example
A''
L''1
2
L''2
S e c o n d le v e l
A''
A''
1
A'
A'
1
L' 2
L' 1
A'
3
3
A'
4
F irs t le v e l
L' 3
2
L 1
A
2
L 2
L 4
Z e ro le v e l
A


1
A
L 3
3



A  A 0 , A ' , A '' , A 0   A1 , A2 , A3  , A '  A1' , A2' , A3' , A4' , A ''  A1'' , A2'' , A3''

are the objects’ set and its three subsets accordingly to levels;





L  L0 , L' , L'' , L0   L1 , L2 , L3 , L4 , L'  L1' , L'2 , L'3 , L''  L1'' , L''2

are the set of relations’ names and its three subsets accordingly to
levels;

S  S 0 , S ' , S ''
 , S 0  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' , L'1, A2' ), S2'  P( A2' , L'2 , A3' ), S3'  P( A2' , L'3 , A4' ) ;
S1''  P( A1'' , L''1 , A2'' ), S2''  P( A2'' , L''2 , A3'' )
are the relations of all the three levels in a predicate form;
A Semantic Metanetwork: An Example
A''
L''1
2
L''2
S e c o n d le v e l
A''
A''
A'
A'
1
L' 2
L' 1
A'
3
1
3
A'
4
F irs t le v e l
L' 3
2
L 1
A
2
L 2
L 4
Z e ro le v e l
A

D  D0 , D'

,
1
L 3
A
3

D 0  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' , S 2 ), D3  ist ( A3' , S 3 ),
D4  ist ( A3' , S 4 ), D5  ist ( A4' , S 4 )
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) 
 P( Ai , SAME b , 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
Ai
Lk
Aj

Ai
~
Lk
Aj
~
P( Ai , Lk , A j )  P( A j , Lk , Ai )
~
L
where k is the new inverse relation. It is named
with unique symbol Lm and added to the set L.
For example, if
Lk
= <to_punish>, then
~
L

L
<to_be_punished> and m
k.
Lm
=
Similarly, let Ln = <to_be_on_the_left_side_of>, then
Lq
= <to_be_on_the_right_side_of> and
In the case when
Lk
~
Lq  Ln .
is a property of an object:
~
~
P( Ai , Lk , Ai )  P( Ai , Lk , Ai ) and Lk  Lk .
The obvious property for the semantic inversion
~
~
L
operation is double inversion:
k  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
example:
Lm
and added to the set
L.
If, for
P(<Mary>, <to_love>, <Tom>) = false,
then it means the same as:
P(<Mary>, <not_to_love>, <Tom>) = true.
Thus, if
then
L k = <to_love> and Lm = <not_to_love>,
Lm  Lk .
Properties of Semantic Negation
 double negation:
 negation of inversion:
Lk  Lk ;
~
~
Lk  Lk ;
Proof:
~
~
( P( Ai , Lk , A j )  P( Ai , Lk , A j )  P( A j , Lk , Ai ) 
~
~
~
 P( A j , Lk , Ai )  P( Ai , Lk , A j ))  Lk  Lk
Semantic Operations: Semantic
Multiplication (Composition)
Lk
Ai
As
Ln
Aj

Lk
As
Ln
Aj
Ai
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(<Mary>, <to_be_married_with>, <Tom>) and
P(<Tom>, <to_have_mother>, <Diana>),
then it is also true that:
P(<Mary>, <to_have_mother-in-law>, <Diana>).
Thus, if:
L k = <to_be_married_with>, Ln = <to_have_mother>,
and Lm = <to_have_mother-in-law>,
Lm  Lk * Ln .
then
Properties of Semantic Multiplication:
 non-commutativity :
( Lk * Ln  Ln * Lk ),
Lk , Ln ( Lk  Ln  DIF  SAME b )
 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:
~ ~
~( L * L )  L
k
n
n * Lk ;
Proof:
( P( A j , Lk * Ln , Ai )  P( A j , Lk , As )  P( As , Ln , Ai ) 
~
~
 P( As , Lk , A j )  P( Ai , Ln , As ) 
~
~
 P( Ai , Ln , As )  P( As , Lk , A j ) 
~ ~
~ ~
 P( A , L * L , A ))  ~ ( L * L )  L * L .
i
n
k
j
k
n
n
k
Semantic Operations: Semantic Addition
Lk
Ai
Aj
Ln

Ai
Lk  Ln
Aj
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(<Mary>, <to_give_birth_to>, <Tom>) and
P(<Mary>, <to_take_care_of>, <Tom>), then:
P(<Mary>, <to_be_mother_of>, <Tom>) is also true.
Thus, if:
L k = <to_give_birth_to>, Ln = <to_take_care_of>, and
Lm  Lk  Ln = <to_be_mother_of>.
It is also possible to sum properties. If, for example, it
is true that P(<Tom>, <to_be_clever>, <Tom>) and
P(<Tom>, <to_be_rich>,<Tom>), then it is also true
that: P(<Tom>, <to_be_clever_and_rich>, <Tom>).
Properties of Semantic Addition
Lk  Ln  Ln  Lk ;
 commutativity:
 transitivity: Lk  ( Ln  Lm )  ( Lk  Ln )  Lm ;
 reflexivity:
Lk  Lk  Lk
;
~
~
~( L  L )  L

L
 inversion over sum:
k
n
k
n;
 distributivity (left):
Lk * ( Lm  Ln )  Lk * Lm  Lk * Ln ;
Proof:
( P( Ai , Lk * ( Lm  Ln ), A j )  P( Ai , Lk , As )  P( As , Lm  Ln , A j ) 
 P( Ai , Lk , As )  P( As , Lm , A j )  P( As , Ln , A j ) 
 P( Ai , Lk , As )  P( Ai , Lk , As ) 
 P( As , Lm , A j )  P( As , Ln , A j )  ( P( Ai , Lk , As )  P( As , Lm , A j )) 
 ( P( Ai , Lk , As )  P( As , Ln , A j )) 
 P( Ai , Lk * Lm , A j )  P( Ai , Lk * Ln , A j ) 
 P( Ai , Lk * Lm  Lk * Ln , A j )) 
 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( Ai , DIF  Lk , A j )  P( Ai , DIF , A j )  P( Ai , Lk , A j ) 

true
 P( Ai , Lk , A j ))  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
b
Properties of SAME :
b
b
~
(SAME
)

SAME
 inversion of SAME :
;
b
b
 neutrality to semantic sum: SAME  Lk  Lk ;
 neutrality to semantic multiplication:
Lk * SAME b  SAME b * Lk  Lk ;

SAME b , for relations;
~
 annihilation: a) Lk  Lk   Lk , for properties.
~
~
b
L
*
L

L
*
L

SAME
b) k
.
k
k
k
 elimination (left): Lk  Lk * Lm  Lk * Lm .
Proof:
Lk  Lk * Lm  Lk * SAME b  Lk * Lm 
 Lk * ( SAME b  Lm )  Lk * Lm
;
 elimination (right): Lk  Lm * Lk  Lm * Lk .
Semantic Equations
1 Lx
F1( L )
2 Lx
 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
L x  Lw
(
equation
)
use notation
as a statement that
relation Lw is the solution of the equation.
Semantic Equations: Basic Properties
( F1 ( L1 )  F2 ( L2 )) Lx  Lw 
~ 1
~ 2 L x  Lw
 ( F1 ( L )  F2 ( L ))
( F1 ( L1 )  F2 ( L2 )) Lx  Lw 
 ( F1 ( L1 )  F2 ( L2 )) Lx  Lw
( F1( L1)  F2 ( L2 )) Lx  Lw 
 ( F1( L1)  Lk  F2 ( L2 )  Lk ) Lx  Lw , Lk  L
( F1( L1)  F2 ( L2 )) Lx  Lw 
 ( F1( L1) * Lk  F2 ( L2 ) * Lk ) Lx  Lw , ( Lk  DIF  L)
( F1( L1)  F2 ( L2 )) Lx  Lw 
 ( Lk * F1( L1)  Lk * F2 ( L2 )) Lx  Lw , ( Lk  DIF  L)
Semantic Equations: Solution of Basic
Equations
~
~
~
~
~
Lx  Li  Lx  Li  Lx  Li ;
Lx  Li  Lx  Li  Lx  Li ;
~
~
Lx  Li  L j  Lx  Li  Li  L j  Li 
~
~;
b
 Lx  SAME  L j  Li  Lx  L j  Li
~
~
Lx * Li  L j  Lx * Li * Li  L j * Li 
~
~;
b
 Lx * SAME  L j * Li  Lx  L j * Li
~
~
Li * Lx  L j  Li * Li * Lx  Li * L j 
~
~
b
 SAME * Lx  Li * L j  Lx  Li * L j .
Operation of Semantic Interpretation
a)
Ai
L'n
Lk
Aj

Ai
L'n
Lk
Aj
Al'
b)
L'n
Al'
Ai
Lk

L'n
Lk
Ai
P( Ai , Lk , A j )  P( Al' , L'n , Al' )  ist ( Al' , P( Ai , Lk , A j )) 
L'n
 P( Ai , Lk , A j )
L'n
where Lk
denotes the interpretation of the
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
L
 Lk .
1.
k
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
SAME
 Lk .
2.
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
(
SAME
)

L
3.
k.
This means that interpretation of the equivalence
relation between two objects inherits the
properties of context L k producing its binary
b
L
equivalent k ;
Expansion of Definitions for Semantic
Constants:
L
Lm
m
(
L

L
)

(
L
4. k
n
n  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;
Lk
 b
L

SAME
Consequence:
;
k
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.
Consequence:
Lk
Lk  SAME b .
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
~( LLm )  L
k
k .
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  L k  Lk .
Transformations with contexts in the
Algebra
Addition in the context.
The result of interpretation of the sum of two notconflicting relations in a context is equal to the
semantic sum of these two relations interpreted
separately in the same context. Formally:
( Lk  Lm )
Ln
Ln
Ln
 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:
L
L
( Lk * Lm ) Ln  Lk n * Lmn , 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 notconflicting 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:
Lk m  Lr n  ( Lk  Lr ) Lm  Ln , when Lk  Lr , Lm  Ln .
L
L
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:
Lk m * Lr n  ( Lk * Lr )Lm  Ln , when Lk  Lr , Lm  Ln .
L
L
Multilevel interpretation.
The result of a relation interpretation in several
levels of contexts does not depend on the order of
interpretation:
L
( Lmn )
Lm Ln
( Lk )  Lk
, when Lm  Lk , Lm  Ln .
Equations with Contexts
Basic rules:
( F1( L1)  F2 ( L2 )) Lx  Lw 
 ( F1( L1) Lk  F2 ( L2 ) Lk ) Lx  Lw , Lk  L
( F1( L1)  F2 ( L2 )) Lx  Lw 

F1 ( L1 )
( Lk

F2 ( L2 ) Lx  Lw
Lk
)
, Lk
L
Solution of basic equations:
L
LLxi  L j  ( LLxi ) Li  L j i 

( Li Li )
Lx
L
L
Li x

L
( Li i ) Lx
L
L
 L j i  LSAME
 L j i  Lx  L j i ;
x

 Lj 
Lj
Li
L
( Li x )
Li
 SAME
Lx
Lj
 Li

Lj
Li

 Lx 
Lj
Li .
Semantic Equations: An Example
~
L1 * L2
~
~ ~
L6
( L3 * L x * L4  L5 )
~
 L1 * L2
 L2
~
~ ~
L6
( L3 * L x * L4  L5 )
 L2
 L2
~
~ ~
L6
( L3 * Lx * L4  L5 )
~
~ ~
L6
( L3 * Lx * L4  L5 )
~ ~
( L3 * L x * L4  L5 )
* L7  L8  L9 
~
* L7  L9  L8 
~
~
~
* L7  L1 * ( L9  L8 ) 
~
~
 L1 * ( L9  L8 ) * L7 
~
~ L~6
 ( L1 * ( L9  L8 ) * L7 ) 
~ ~
 L3 * Lx * L4  L5  L2
~ ~ L~6
( L1*( L9  L8 )* L7 )

~
~
~ L6
~ ~
~
(
L
*(
L

L
)*
L
1
9
8
7)
 L3 * Lx * L4  L2
 L5 
~ ~ L~6
~
~
~
~
 Lx * L4  L3 * ( L2( L1*( L9  L8 )* L7 )  L5 ) 
~
~
~ L6
~
~
~
(
L
*(
L

L
)*
L
1
9
8
7)
 Lx  L3 * ( L2
 L5 ) * L4 
 Lx  ~( L3 * ( L2
~ ~ L~6
( L1*( L9  L8 )* L7 )
~
~
 L5 ) * L4 ).
Reasoning with Contexts
Deriving
an
interpreted
(decontextualization).
knowledge
Deriving of interpreted knowledge means here
deriving the formula for knowledge interpreted using
all known levels of its context.
We define the knowledge of a relation LAi  A j between
any pairs of objects ( A j , Ak ) from the same level of a
semantic metanetwork as a semantic sum over all possible
paths between these objects ( A j , Ak ) that exist at this level
of the metanetwork.
We define the knowledge LAi 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:
LAi   LAi  Aj
j
.
The interpreted knowledge of any relation, considering all
contexts and metacontexts, is derived by the following
schema:
 interpreted knowledge  
  knowledge 
 knowl. about metacont. of n  th level
 knowl. about context  ...
.
Example of Decontextualization (1)
A''
L''1
2
L''2
S e c o n d le v e l
A''
A''
A'
A'
1
L' 2
L' 1
A'
3
1
3
A'
4
F irs t le v e l
L' 3
2
L 1
A
2
L 2
L 4
Z e ro le v e l
A
1
L 3
A
3
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 A''  L A''  A''  L A''  A''  L A''  A'' 
1
1
2
1
3
1
1
 L''1  L''1 * L''2  SAME  L''1 *( SAME  L''2 )  L''1 * L''2 ;
~
L A''  L1''  L''2 ;
2
~ ~
L A''  L2'' * L1'' .
3
Example of Decontextualization (2)
A''
2
L''1
L''2
S e c o n d le v e l
A''
A''
1
A'
A'
1
L' 2
L' 1
A'
3
3
A'
4
F irs t le v e l
L' 3
2
A
L 1
2
L 2
L 4
Z e ro le v e l
A
1
A
L 3
3
We continue at the first level and derive the
interpreted knowledge of the first level relations:
L A'  A' 
1
2
L ''
'
( L1 ) A1
L A'  A' 

( L1'
1
L1'' * L''2
)
3
L
' A1''
( L1)
~
' L1''  L''2
* ( L2 )
L ''
'
L A '  A '  ( L1 ) A1
1
4
L ''
'
*( L3 ) A3

' L1'' *L''2
( L1 )
;
L ''
'
* ( L2 ) A2


~
'
' L1'' * L''2  L1''
( L1 * L2 )
;
' L1'' * L''2
 ( L1 )
~ ~
' L2'' * L1''
*( L3 )
...
L ''
~' L A3''
'
L A'  A'  ( L3 )
* ( L2 ) A2 
4
3
~
~ ~
~' L~2'' * L~1''
~'
' L1''  L''2
' L2'' * L1''  L''2
 ( L3 )
* ( L2 )
 ( L3 * L2 )
.
;
Example of Decontextualization (3)
A''
L''1
2
L''2
S e c o n d le v e l
A''
A''
1
A'
A'
1
L' 2
L' 1
A'
3
3
A'
4
F irs t le v e l
L' 3
2
L 1
A
L 2
2
L 4
Z e ro le v e l
A
1
A
L 3
3
' , A' , A'
A
The knowledge about contexts 1 2 3 of the
first level is derived as follows:
~
'
' L1'' * L''2  L1''
L A'  ( L1 * L2 )
1
 ( L1' * ( L'2

~ ~
'
' L1'' * L''2  L2'' * L1''
 ( L1 * L3 )

~ ~
' L1'' * L''2  L2'' * L1''
L3 ))
;
~
~ ~
~' L1'' * L''2
' L1''  L''2
' L2'' * L1''
L A '  ( L1 )
 ( L2 )
 ( L3 )

2
~ ~
~'
'
' L1'' * L''2  L2'' * L1''
 ( L1  L2  L3 )
;
~ ~
~' ~' L1'' * L''2  L~1''
~'
' L''2  L2'' * L1''
L A'  ( L2 * L1)
 ( L2 * L3 )

3
~ ~
~'
~'
' L1'' * L''2  L2'' * L1''
 ( L2 * ( L1  L3 ))
;
Example of Decontextualization (4)
A''
L''1
2
L''2
S e c o n d le v e l
A''
A''
1
A'
A'
1
L' 2
L' 1
A'
3
3
A'
4
F irs t le v e l
L' 3
2
L 1
A
2
L 2
L 4
Z e ro le v e l
A
1
L 3
A
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 '
A
LA1  A3  ( L1 ) 1 * ( L2 )
 ( L1 * L2  L3 )
L '
A
2  (L )
3
L '
A
3 
'' '' ~ '' ~ ''
~ ~ ~
( L1' * L'2  L1' * L'3  L2 ' * L1'  L2 ' * L'3 ) ( L1* L2  L2 * L1 )
.
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:
Lx knowledge about context   interpreted knowledge  .
Reasoning with Contexts
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  .
Reasoning with Contexts
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:
( LLx m
 Lk )  ( Lx 

( LLx n

LmLn
Lk
Lm
Lk )contextualization 
)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 ?