Logics for Data and Knowledge
Representation
Context Logic
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese
Syntax: formation rules

First order formulas
<term> ::= <variable> | <constant> | <function sym> (<term>{,<term>}*)
<atomic formula> ::= <predicate sym> (<term>{,<term>}*) |
<term> = <term>
<wff> ::= <atomic formula> | ¬<wff> | <wff> ∧ <wff> | <wff> ∨ <wff> |
<wff> → <wff> | ∀ <variable> <wff> | ∃ <variable> <wff>

Contextual formulas
<cwff> ::= i : <wff> for each i ∈ I (also called i-formula or Li-formula)

Using contextual formulas we turn a meta-theoretic object (the name i of a
context) into a theoretic object (an i-formula i : ψ)

A contextual formula is a kind of labeled formula
2
Local model semantics

Local model semantics (LMS)
Provide the meaning of the sentences and model reasoning as logical
consequence over a multi-context language. LMS formalizes:

Principle of Locality




Principle of Compatibility

3
We never consider all we know, but rather a very small subset of it
Modeling reasoning which uses only a subset of what reasoners actually
know about the world
The part being used while reasoning is what we call a context, i.e., a local
theory Ti
There is compatibility among the kinds of reasoning performed in
different contexts
Exercise: viewpoints
Consider a ‘magic box’ composed of 2 x 3 cells where:
 Mr.1 sees one ball on the left and one on the right
 Mr.2 sees one ball in the center

Provide the local views, contextual formulas and the compatible situations

Local views:

Contextual formulas:
1: L  R
L
R
2: C  L  R
Mr.1

L
Mr.2
C
R
Compatible situations:
C = {<c1,c2>}
c1= { I : I(L) = T, I(R) = T}
c2= { I : I(C) = T, I(L) = F, I(R) = F}
4
Exercise: viewpoints (II)
Consider a ‘magic box’ composed of 2 x 3 cells where:
 Mr.1 sees one ball either on the left or one ball on the right
 Mr.2 sees one ball all over the places

Provide the local views, contextual formulas and the compatible situations

Local views:

Contextual formulas:
1: (L  R)  (L  R)
L
R
L
R
2: L  C  R
Mr.1

L
Mr.2
C
R
Compatible situations:
C = {<c1,c2>}
c1= { I : I(L) = T, I(R) = F;
J : J(L) = F, I(R) = T}
c2= { I : I(L) = T, I(C) = T, I(R) = T}
5
Exercise: viewpoints (III)
Consider a ‘magic box’ composed of 2 x 3 cells where:
 Mr.1 sees two balls
 Mr.2 sees one ball

Provide the local views, contextual formulas and the compatible situations

Local views:
L

Contextual formulas:
1: L  R
R
Mr.1
L
C
R
Mr.2
C
R
2: (L  C   R) 
(L  C  R) 
L
6
L
C
R
(L  C  R)
Exercise: viewpoints (III) cont.
Consider a ‘magic box’ composed of 2 x 3 cells where:
 Mr.1 sees two balls
 Mr.2 sees one ball

Provide the local views, contextual formulas and the compatible situations

Local views:
L

R
Mr.1
L
C
R
L
C
R
Mr.2
L
C
R
Compatible situations:
Intuitively, the balls must be in the same
column as seen from Mr. 2 such that the
first hides the second.
C = {<c1,c2>}
c1= { I : I(L) = T, I(R) = T}
c2= { I : I(L) = T, I(C) = F, I(R) = F;
J : J(L) = F, J(C) = T, J(R) = F;
K : K(L) = F, K(C) = F, K(R) = T;}
7
Exercise: viewpoints (IV)
Consider a ‘magic box’ composed of 2 x 2 cells where:
 Mr.1 sees two balls
 Mr.2 sees two balls
 Mr.3, watching from the top, sees two balls

Provide the local views, contextual formulas and the compatible situations

Local views:
L
R
L
R
Mr.2
Mr.1
A
B
A
B
A
B
A
B
A
B
A
B
C
D
C
D
C
D
C
D
C
D
C
D
Mr.3
8
Exercise: bridges
Consider the following two classifications and determine compatibilities

color
colour
black
white
1: color  2: colour
C = {<c1,c2>}
c1= { I : I(color) = T, …}
c2= { I : I(colour) = T, …}
9