1. Logics for Data and Knowledge Representation
Context Logic
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. 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

3. 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
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
Principle of Compatibility
There is compatibility among the kinds of reasoning performed in different contexts 3

4. Exercise: viewpoints Mr.1 Mr.2
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
2: C  L  R L R
Mr.1
Compatible situations: L C R
C = {<c1,c2>}
c1= { I : I(L) = T, I(R) = T}
c2= { I : I(C) = T, I(L) = F, I(R) = F}
Mr.2 4

5. 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)
2: L  C  R L R L R
Mr.1
Compatible situations: L C R
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}
Mr.2 5

6. 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:
Contextual formulas: L R
1: L  R
2: (L  C   R) 
(L  C  R) 
(L  C  R)
Mr.1 L C R L C R
Mr.2 L C R 6

7. 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:
Compatible situations: L R
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;}
Mr.1 L C R L C R
Mr.2 L C R 7

8. 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
Mr.1
Mr.2 L R
Mr.3 A B C D 8

9. Exercise: bridges color black colour white
Consider the following two classifications and determine compatibilities
color
black
colour
white
1: color  2: colour
C = {<c1,c2>}
c1= { I : I(color) = T, …}
c2= { I : I(colour) = T, …} 9