LDK R 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 ::= | | ( {, }*) ::= ( {, }*) | = ::= | ¬ | ∧ | ∨ | → | ∀ | ∃  Contextual formulas ::= i : for each i ∈ I (also called i-formula or L i -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

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 T i  Principle of Compatibility  There is compatibility among the kinds of reasoning performed in different contexts

4 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 2: C   L   R  Compatible situations: C = { } c 1 = { I : I(L) = T, I(R) = T} c 2 = { I : I(C) = T, I(L) = F, I(R) = F} L L R RC Mr.1 Mr.2

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  Compatible situations: C = { } c 1 = { I : I(L) = T, I(R) = F; J : J(L) = F, I(R) = T} c 2 = { I : I(L) = T, I(C) = T, I(R) = T} LL L RR RC Mr.1 Mr.2

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: 1: L  R 2: (L   C   R)  (  L  C   R)  (  L   C  R) L L R RC Mr.1 Mr.2 LRC LRC

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: Intuitively, the balls must be in the same column as seen from Mr. 2 such that the first hides the second. C = { } c 1 = { I : I(L) = T, I(R) = T} c 2 = { 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;} L L R RC Mr.1 Mr.2 LRC LRC

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: LR Mr.1 Mr.2 LR Mr.3 AB C D AB C D AB C D AB C D AB C D AB C D

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