Logics for Data and Knowledge Representation

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
Outline

• Introduction contexts
• Syntax
• Semantics
• Local Models
• Contextual models
• Satisfiability, validity and contextual
  entailment
• Reasoning with beliefs and equivalence with modal
  logic

2
3
The notion of context
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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

3
4
Example
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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.

4
5
Example (McCarthy, 1987)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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?

5
6
Example (Tarski, 1931)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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)

6
7
Summary
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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).

7
8
Definitions of context
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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.

8
9
Subjective Perspective (I)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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 individuals subjective
  perspective about the world.
• In this example, two observer look at the same
  phenomenon from two different perspectives.

9
10
Subjective Perspective (II)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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. 2s subjective view of the
  phenomenon.

10
11
Syntax alphabet and languages
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Alphabet of symbols
• For every i?N, with N the set of contexts, we
  define an alphabet Si for a contextual language
  Li such that L Li i?N
• Multi-Context Alphabet
• A multi-context alphabet is a set S ?i?I Si
  with I ? N, where each Si is a first-order
  alphabet enriched by some auxiliary symbols to
  build contextual formulas
• Family of languages
• From the multi-context alphabet S ?i?I Si 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

11
12
Syntax formation rules
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• First order formulas
• lttermgt ltvariablegt ltconstantgt ltfunction
  symgt (lttermgt,lttermgt)
• ltatomic formulagt ltpredicate symgt
  (lttermgt,lttermgt)
• lttermgt lttermgt
• ltwffgt ltatomic formulagt ltwffgt ltwffgt ?
  ltwffgt ltwffgt ? ltwffgt
• ltwffgt ? ltwffgt ? ltvariablegt ltwffgt ?
  ltvariablegt ltwffgt
• Contextual formulas
• ltcwffgt i ltwffgt 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 ?)
• A contextual formula is a kind of labeled formula

12
13
Example
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Contextual Laws for ?, and ?
• i (A?B)? (A?B)
• i (A?B)?(A?B)
• Contextual Pierces law
• i ((A?B)?A)?A
• Contextual De Morgans laws i (A?B) ?
  (A?B)i (A?B) ? (A?B)

13
14
Contextual languages and theories
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Multi-context language
• The multi-context alphabet S and the formation
  rules define a multi-context language L Li
  i?I
• Multi-context theory
• A set of closed wffs over L is a multi-context
  theory
• NOTE A first order theory T is a special case
  of a contextual theory Ti where ? ? T iff i ? ?
  Ti for any i ? I
• A first-order theory Ti is called local theory

14
15
Example
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• T1 1?x.apple(x)?Computer(x),1
  apple(docPBG4pdf)
• T2 2?x.apple(x)?Fruit(x),2?x.orange(x)?Frui
  t(x),2 apple(docRdoc),2 apple(docGtxt)
• Same terminology, different meaning.

1
2
15
16
Example
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• T1 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
16
17
Local model semantics
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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

17
18
Example viewpoints (I)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• The magic box (Giunchiglia Ghidini, 1998)

• Locality Mr. 1 and Mr. 2 do not have any
  perception of the depth of the box

18
19
Example viewpoints (II)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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
19
20
Example viewpoints (III)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Compatibility there are cases in which observers
  cannot really distinguish among different
  situations

In these cases we need a third view
20
21
Viewpoints applications
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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

21
22
Local models and compatibility sequences
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Given 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 ltc0, c1, . . . , ci, . . . gt
• where ci is a subset of M(Li).
• NOTE For I 1,2, c is called a compatibility
  pair

22
23
Model and compatibility relation
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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 ltØ, Ø, . . .gt
  is not in C

23
24
Example viewpoints semantics (I)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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

24
25
Example viewpoints semantics (II)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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

25
26
Formal definition of context
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Let model C ltc0, c1, . . . , ci, . . . gt 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

26
27
Truth relation (satisfaction relation)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• The scope of the FO-logic truth relation is
  extend
• A model C satisfies an i-formula i ?
• C ? i ?
• if for all ltc0, c1, . . . , ci, . . .gt ? C and
  for all m ? ci, m ? ?
• C is a model of i ?
• i ? is true in C

27
28
Satisfiability and validity
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• An Li-formula i ? 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 G (C ? G )
  if C satisfies every formula i ? in G
• G is satisfiable if C ? i ? for some C and for
  all i ? in G
• i ? is valid (? i ? ) if C ? i ? for all C

28
29
Contextual entailment
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• A set G of formulas entails a formula i ?
  w.r.t. a model C
• (i ? is a logical consequence of G w.r.t. C)
• G ?C i ?
• if for every compatibility sequence c ? C and
  for all j?I with j?i, if cj ? Gj then for all m ?
  ci, if m ? Gi then m ? ?
• If G is empty then i ? is a tautology
• Intuitively, given a contextual model C, for any
  c ? C
• (1) distinguish between the local formulas Gi
  and the others Gj
• (2) throw away c if cj ? Gj, continue otherwise
• (3) throw away all the local models m ? ci that
  m ? Gi
• (4) the remaining models m ? ci locally satisfy ?

29
30
Example the magic box
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• 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.

30
31
Context logics reasoning with beliefs (I)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Consider two agents a and b. We can imagine that
  each agent generates a different context
  (different language, knowledge and reasoning
  capabilities). What does a believe of b? (and
  vice versa)
• aa what a believes of a
• ab what a believes of b
• ba what b believes of a

i
i 1
31
32
Context logics reasoning with beliefs (II)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• At each level we generate a different context
  that we can link using some rules
• Assume P ? Li and ? ? Li
• If P ? Li and Q ? Li then P ? Q ? Li
• If P ? Li then B(P) ? Li1 where B stands for
  believes and is a modal operator
• This generates a hierarchy of languages
• We can then generate for each path in the tree a
  different compatibility sequence
• When reasoning we add the following bridge
  rules
• i B(P) i1 P (Rdowni)
• i1 P i B(P) (Rupi)

32
33
Equivalence Modal logics Context logics (I)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Bridge rules generate an accessibility relation
  that has a direct translation in modal logic
  where B(P) is translated as ?P.
• RECALL In the canonical Kripke model K the
  following schema is valid ?(P ? Q) ? (?P ? ?Q)
• We need to prove that this property is true also
  for context logic

33
34
Equivalence Modal logics Context logics (II)
INTRODUCTION SYNTAX LOCAL MODELS
CONTEXTUAL MODELS SAT, VAL AND ENT BELIEFS

• Let us prove it for context logics with beliefs
  B(P ? Q) ? (B(P) ? ?Q).
• Let us assume both i B(P ? Q) and i B(P)
• i B(P) i B(P ? Q)
• ---------- (Rdowni) -------------- (Rdowni)
• i1 P i1 P ? Q
• -----------------------------------------------
  ----- (i1 P ?(?P ? Q) implies i1 Q)
• i1 Q
• -----------------------------------------------
  ----- (Rupi)
• i B(Q)
• -----------------------------------------------
  ----- (?i)
• i B(P) ? B(Q)
• -----------------------------------------------
  ----- (?i)
• i B(P ? Q) ? (B(P) ? B(Q))

34