LLogics for DData and KKnowledgeRRepresentation

Context Logic

Originally by Alessandro Agostini and Fausto GiunchigliaModified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

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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

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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

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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 R2: C L 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}

L

L

R

RC

Mr.1

Mr.2

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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 = {<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}

L L

L

R R

RC

Mr.1

Mr.2

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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.2L RC

L RC

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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 = {<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;}

L

L

R

RC

Mr.1

Mr.2L RC

L RC

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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.2L R

Mr.3A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

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Exercise: bridges 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= { 2 : I(colour) = T, …}