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 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

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

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

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

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

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

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