ldk r logics for data and knowledge representation context logic originally by alessandro agostini...
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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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Outline Introduction: contexts Syntax Semantics
Local Models Contextual models
Satisfiability, validity and contextual entailment
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The notion of context The notion of “context” is used in various areas of AI,
including data and knowledge representation, NLP, and multimedia IR. However,
its meaning is frequently left to the user its use is implicit and intuitive its formalization is poor or missing
No formal definition of context was given
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example
‘Today is nice’
What do we mean by ‘nice’? Which day is ‘today’?
We cannot answer, as the proposition does not have a precise meaning or context.
Therefore, we cannot say whether the proposition is true or false.
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example (McCarthy, 1987)
‘The book is on the table’
Consider modeling the on preposition so as to draw appropriate consequences from the information expressed in the sentence
How many interpretations of it we can think? What is the right level of generality?
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example (Tarski, 1931)
‘Snow is white’
Is this proposition true? What about the color of the snow on top of Mount Etna in Sicily?
(Mount Etna is one of the most active volcanoes in the world)
Tarski made explicit the context he used to interpret it:
“I would only mention that [...] I shall be concerned exclusively with grasping the intentions which are contained in the so-called classical conception of truth.”(The first attempt to formalize a context)
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Summary These examples show that:
even a very simple proposition requires the use of some context to be interpreted
the notion of context is often unclear, undefined or left implicit.
The first attempt to formalize a context as a way to interpret (first-order) statements was introduced by Tarski (1930-31).
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Definitions of context From Webster dictionary
A context “surrounds, and gives meaning to, something else”
In linguistic
It is “the text surrounding a term in which the term is used”
In logic (Giunchiglia, 1993)
that subset of the complete state of an individual that is used for reasoning about a given goal, e.g. to formulate a query.
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Subjective Perspective (I) The “magic box” (Giunchiglia& Ghidini, 1998)
Intuitively, a context is a theory of the world which encodes (formally by using a logic called contextual logic, CxL) an individual’s subjective perspective about the world.
In this example, two observer look at the same phenomenon from two different perspectives.
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Subjective Perspective (II) The “magic box” (Giunchiglia& Ghidini, 1998)
The “magic box” represents a world with two contexts, called the local models, that encode Mr.1 and Mr. 2’s subjective view of the phenomenon.
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Syntax: alphabet and languages Alphabet of symbols
For every i∈N, with N the set of contexts, we define an alphabet Σi for a contextual language Li such that L = {Li} i∈N
Multi-Context Alphabet A multi-context alphabet is a set Σ = ∪i∈I Σi with I ⊆ N, where
each Σi is a first-order alphabet enriched by some auxiliary symbols to build contextual formulas
Family of languages From the multi-context alphabet Σ = ∪i∈I Σi we define a
family of languages L = {Li} i∈N
Each Li is the formal language used to state what is true in the context I, and it is therefore called a local language
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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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
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example Contextual Laws for ∧, ¬ and →:
i : (A∧¬B)→¬ (A→B)
i : ¬(A→B)→(A∧¬B)
Contextual Pierce’s law:
i : ((A→B)→A)→A
Contextual De Morgan’s laws: i : ¬(A∨B) ↔ (¬A∧¬B)i : ¬(A∧B) ↔ (¬A∨¬B)
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Contextual languages and theories Multi-context language
The multi-context alphabet Σ and the formation rules define a multi-context language L = {Li} i∈I
Multi-context theoryA set of closed wff’s over L is a multi-context theory
NOTE: A first order theory T is a special case of a contextual theory Ti where ψ ∈ T iff i : ψ ∈ Ti for any i ∈ I
A first-order theory Ti is called local theory
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ExampleT1 = {
1:∀x.apple(x)→Computer(x),1: apple(docPBG4pdf)
}
T2 = {2:∀x.apple(x)→Fruit(x),2:∀x.orange(x)→Fruit(x),2: apple(docRdoc),2: apple(docGtxt)
}
Same terminology, different meaning.
1
2
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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ExampleT1 = {
1:∀x.apple(x)→Computer(x),1: apple(doc1)
}
T2 = {2:∀x.Mac(x)→Computer(x),2: Mac(doc1)
}
Different terminology, same meaning.
1
2
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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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
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example: viewpoints (I) The “magic box” (Giunchiglia& Ghidini, 1998)
Locality: Mr. 1 and Mr. 2 do not have any perception of the depth of the box
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example: viewpoints (II) Compatibility: there are some compatible situations we
can list
Compatible pairs can be described: “if Mr.1 sees at least one ball then Mr.2 sees at least one ball”
NOTE: each of the configurations depicted here is what we call a local model (see next slides), i.e. one of the possible situations in a context
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example: viewpoints (III) Compatibility: there are cases in which observers cannot
really distinguish among different situations
In these cases we need a third view
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Viewpoints: applications An application where partial views matter is data
integration in federations of relational databases
Each federated database can be represented as a context
The federation of databases is a set of views of an ideal (global) database which is impossible, too complex or even not worth to reconstruct completely
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Local models and compatibility sequencesGiven a family of languages L = {Li} i∈I
Local model We denote with M(Li) the set of all models for Li
An element m ∈ M(Li) is a local model
Compatibility sequence A compatibility sequence for L is an infinite sequence
c = <c0, c1, . . . , ci, . . . > where ci is a subset of M(Li).
NOTE: For I = {1,2}, c is called a compatibility pair
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Model and compatibility relation Intuitively, local models describe what is locally true while compatibility sequences
put together local models which are mutually compatible consistently with the situation we are modeling
A compatibility relation (for L) is a set C of compatibility sequences.
A model (for L) is a non-empty compatibility relation C such that the sequence <∅, ∅, . . .> is not in C
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example: viewpoints semantics (I) Languages
We need languages L1 and L2 describing the views of Mr.1 and Mr.2.
With L1 we describe that a ball can be on the left or on the right
With L2 we describe that a ball can be on the left, in the center, or on the right.
No other constrains are specified
L1 = {Bl Br}
L2 = {Bl Br Bc}
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example: viewpoints semantics (II) Local models
We construct all the possible situations (models) for L1 and L2
This leads to the definition of four situations (models) for L1 and eight situations (models) for L2
Compatibility pairs We construct all the
compatibility pairs
Compatibility relation The collection of all the
compatibility pairs
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Formal definition of contextLet model C = {<c0, c1, . . . , ci, . . . >” be given.
A context is any ci, i.e. the set of local models m ∈ M(Li) allowed by C within any particular compatibility sequence c in C
Given c, a context captures exactly locally true facts given the constraints posed by the local models of the other contexts in the same compatibility sequence, as allowed by c
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Truth relation (satisfaction relation)The scope of the FO-logic truth relation is extend
A model C satisfies an i-formula i : ψ
C ⊨ i : ψ
if for all <c0, c1, . . . , ci, . . .> ∈ C and for all m ∈ ci, m ⊨ ψ
C is a model of i : ψ i : ψ is true in C
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Satisfiability and validity An Li-formula i : ψ is satisfied by a model C if all the
local models in each context ci satisfy it
A model C satisfies a set of formulas Γ (C ⊨ Γ ) if C satisfies every formula i : ψ in Γ
Γ is satisfiable if C ⊨ i : ψ for some C and for all i : ψ in Γ
i : ψ is valid (⊨ i : ψ ) if C ⊨ i : ψ for all C
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Contextual entailment A set Γ of formulas entails a formula i : ψ w.r.t. a model C
(i : ψ is a logical consequence of Γ w.r.t. C)
Γ ⊨C i : ψ
if for every compatibility sequence c ∈ C and for all j∈I with jǂi, if cj ⊨ Γj then for all m ∈ ci, if m ⊨ Γi then m ⊨ ψ
If Γ is empty then i : ψ is a tautology
Intuitively, given a contextual model C, for any c ∈ C
(1) distinguish between the local formulas Γi and the others Γj
(2) throw away c if cj ⊮ Γj, continue otherwise
(3) throw away all the local models m’ ∈ ci that m’ ⊮ Γi
(4) the remaining models m ∈ ci locally satisfy ψ
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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Example: the “magic box” If Mr. 1 sees a ball on the left
and Mr. 2 does not see any ball on the right, then Mr. 2 sees a ball on the left or in the center.
(1) In the premises, local conditions are in blue, the others in red
(2) Throw away all the compatibility sequences that do not satisfy the red condition
(3) Throw away all the local models that do not satisfy the blue one
(4) Notice how the remaining local models for Mr. 2 satisfy the condition in green.
INTRODUCTION :: SYNTAX :: LOCAL MODELS :: CONTEXTUAL MODELS :: SATISFIABILITY, VALIDITY AND ENTAILMENT
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