L ogics for D ata and K nowledge R epresentation

1. Logics for Data and KnowledgeRepresentation 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 • 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) L R L R Mr.2 Mr.1 A B A B A B A B A B A B Mr.3 D D D D D D C C C C C C • 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: 8

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