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Logics for Data and Knowledge Representation

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					           Logics for Data and Knowledge
                           Representation
    ClassL (Propositional Description Logic with Individuals)




1
Outline

 Terminology   (TBox)




 2
Terminological Axioms
Inclusion Axiom
         C⊑D (intended meaning: σ(C)⊆ (D))
Examples:
                 Master ⊑ Student,
                  Woman ⊑ Person
               Woman ⊔ Father ⊑ Person
Equivalence (Equality) Axiom
         C≡D (intended meaning σ(C)= σ(D))
Examples:
                     Student ≡ Pupil,
                 Parent ≡ Mather⊔ Father


 3
Definitions
A definition is an equality with an atomic concept on the
  left hand.

Examples

           Bachelor ≡ Student ⊓ Undergraduate
               Woman ⊑ Person ⊓ Female




 4
Terminology (TBox)


A terminology (or Tbox) is a set of a (terminological)
  axioms

Example: T is
  {Woman ⊔ Father ⊑ Person, Parent ≡ Mather⊔ Father}




 5
Outlines

 Terminology
 WorldDescriptions
 Reasoning with the TBox




 6
Satisfiability with respect to T (no ABox))


A concept P is satisfiable with respect to T, if there exists an
  interpretation I, with if I |= θ for all θ ∈ T, such that
                              I |= P.
In other words, I(P) non empty.

In this case we say also that I is a model of P




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Validity with respect to T

A (possibly empty) Tbox T of class-propositions entails
 (subsumes) a class-proposition P (written: T |= P)
 (similarly: a concept P is valid with respect to T) if forall
 interpretations I,
 with if I |= θ for all θ ∈ T, we have that I |= P.

In other words, I(P) non empty in all Interpretations.

If T |= P, then we say that P is a logical consequence of T,
   and also that T logically implies P.
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TBox reasoning
Let T be a Tbox

     Satisfiability:(with respect to T):
                              T satisfies P?
     Subsumption (with respect to T):
                              T |= P ⊑ Q?
     Equivalence (with respect to T): :
             (T|= P ≡ Q) T|= P ⊑ Q and T |= P ⊑ Q?
     Disjointness: (with respect to T):
                          T|= P ⊓ Q ⊑ ⊥?


 9
TBox reasoning

Let T be a Tbox
Satisfiability:(with respect to T):
                          T satisfies P?

A concept P is satisfiable with respect to T if there exists a model I of T
  such that I(P) is not empty. In this case we say that I is a model of P

NOTE: a property of a single model. Used to implement SAT or Eval
 (model checking)

EXAMPLE!!!
 10
TBox reasoning

Let T be a Tbox
Subsumption (with respect to T):
                  T |= P ⊑ Q (P ⊑T Q)

A concept P is subsumed by a concept Q with respect to T if I(P) is a
  subset of I(Q) for every model I of T.

NOTE: a property of all models. Used to implement Entailment and
 validity (with T empty)

EXAMPLE!!!
 11
TBox reasoning

Let T be a Tbox
Equivalence (with respect to T):
                   (T|= P ≡ Q) (P ≡T Q)
Two concepts P and Q are equivalent with respect to T if I(P) = I(Q)
  for every model I of T.

NOTE: a property of all models.

EXAMPLE!!!



 12
TBox reasoning

Let T be a Tbox
  Disjointness: (with respect to T):
                       T|= P ⊓ Q ⊑ ⊥?
Two concepts P and Q are disjoint with respect to T if I(P) intersection
  with I(Q) is empty, for every model I of T.

NOTE: a property of all models.

EXAMPLE!!!



 13
Example
 Suppose   we describe the students/listeners in LDKR
  course:

T= {Bachelor ≡ Student ⊓ Undergraduate,
  Master ≡ Student ⊓  Undergraduate,
  PhD ≡ Master ⊓ Research,
  Assistant ≡ PhD ⊓ Teach,
  Undergraduate ⊑  Teach}

T is satisfiable (build model)

 14
Example cont. Equivalence
Prove the following equivalence:
               Student ≡ Bachelor ⊔ Master
Proof:
Bachelor ⊔ Master
≡ (Student ⊓ Undergraduate) ⊔ Master
≡ (Student ⊓ Undergraduate) ⊔ (Student ⊓ 
  Undergraduate)
≡ Student ⊓(Undergraduate ⊔ Undergraduate)
≡ Student ⊓⊤
≡ Student

 15
Example cont.(Exercise)
     Let’s see the following propositions,
                         Assistant, Student
                        Bachelor, Teach
                      PhD, Master ⊓ Teach

1.    Which pairs are subsumed/supersumed?

2.    Which pairs are disjoint?



 16
Example
Suppose we describe the students/listeners in LDKR course
  in TBox as follows:
T ={Bachelor ≡ Student ⊓ Undergraduate,
  Master ≡ Student ⊓  Undergraduate,
  PhD ≡ Master ⊓ Research,
  Assistant ≡ PhD ⊓ Teach,
  Undergraduate ⊑  Teach}

Is Bachelor⊓PhD satisfiable?
Are Assistant and Bachelor disjoint?

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Class-Values and Truth-Values


 The intentional interpretation Ii of a proposition P determines a
  truth-value Ii(P).
 The extensional interpretation of Ie of P determines a class of
  objects Ie(P).
 What is the relation between Ii(P) and Ie(P)?




 18
PL vs. ClassL (PL, ClassL notational variants)
            PL                 ClassL
Syntax      ∧                  ⊓
            ∨                  ⊔
                              
            ⊤                  ⊤
            ⊥                  ⊥
            →                  ⊑
            ↔                  ≡
            P, Q...            P, Q...
Semantics   ∆={true, false}    ∆={o, …} (compare models)




 19
Class-Values and Truth-Values

    Intersection: Ie |=P, Ie |= Q may not imply Ie |=P⊓Q: subsumption in an extensional
     interpretation is “richer” than in an intensional interpretation (subsumption is not
     preserved by intersection)
        .. but Ie |=P⊑C, Ie |=Q⊑C always implies Ie |=P⊓Q ⊑C, namely, subsumption,
         satisfiability and validity (empty TBox) are preserved by intersection with the TBox
         axioms.


    Negation: We may have Ie |=P and Ie |= P, and Ie |= Q⊓P and Ie |= Q⊓P (satisfiability
     is preserved using two models in place of one)
        … but always not Ie |= P⊓P
        … and always not Ie |= (Q⊓P)⊓(Q⊓P)
        … and always Ie |= P⊔P, namely satisfiability, validity are preserved by negation.




    20
Class-Values and Truth-Values
P is satisfiable with respect an intensional interpretation Ii(P) if and only
   if it is satifisfiable with respect to an extensional interpretation Ie(P).

Ii(P) implies Ie(P): Build Ie(P) from Ii(P) by substituting true with U and
    false with empty set.

Ie(P) implies Ii(P): less trivial. Idea: build first a Ie’(P) which is equivalent to
   Ie(P) and which uses only U and empty set.

TO BE REFINED




 21
From TBox reasoning to PL reasoning
Let T be a Tbox, T= {θ1, …, θn }
Satisfiability:(with respect to T):
       T satisfies P? Reduces to PL satisfiability of θ1 ∧ … ∧ θn →P
Validity, entailment with respect to T:
            T |= P? Reduces to PL validity of θ1 ∧ … ∧ θn →P
Subsumption (with respect to T):
     T |= P ⊑ Q? Reduces to validity of θ1 ∧ … ∧ θn →(P →Q)
Equivalence (with respect to T): :
         T|= P ⊑ Q and T |= P ⊑ Q? Reduces to subsumption
Disjointness: (with respect to T):
          T|= P ⊓ Q ⊑ ⊥? Reduces to unsatisfiability of P ⊓ Q

NOTICE: ClassL reasoning can be implemented using DPLL
 22
Outline

 Terminology   (TBox)
 World Descriptions (ABox))
 Reasoning with TBox
 Eliminating the Tbox
 Reasoning with the Abox
 Closed vs. Open world semantics
 Properties




 23
Terminology (TBox)


Two kinds of symbols:

   base symbols (or primitive concepts), which occur only on
    the right hand side of axioms, and
   name symbols (or defined concepts) which occur on the
    left hand side of axioms

    Example:
                            A ⊑ B ⊓ (C ⊔ D)
    A defined concept; B, C, D primitive concepts

 24
Terminology (TBox)

Let A and B be atomic concepts in a terminology T.
We say that A directly uses B in T if B appears in the right-hand
 side of the defintion of A.
  Example:
                          A ⊑ B ⊓ (C ⊔ D)
  A directly uses B,C,D


We say that A uses B if B appears in the right hand side after the
 definition of A has been unfolded so that there are only
 primitive concepts in the left hand side of the definition of A
  Example:
                  {A ⊑ B ⊓ (C ⊔ D), B ⊑ (C ⊔ E)}
  A uses E, and directly uses B
 25
Terminology (TBox)

A terminology contains a cycle (is cyclic) if it contains a concept
  which uses itself. A terminilogy is acyclic otherwise


Example:
                    A ⊑ B ⊓ (C ⊔ D), B ⊑ (C ⊔ E)
  is acyclic.
                    A ⊑ B ⊓ (C ⊔ A), B ⊑ (C ⊔ E)
                   A ⊑ B ⊓ (C ⊔ D), B ⊑ (C ⊔ A)
are cyclic.


NOTE: NEED NICE EXAMPLE


 26
Terminology (TBox)

The expansion T’ of an acyclic terminology T is a terminology
  obtained from T by unfolding all definitions until all concepts
  occurring on the right hand side of definitions are base symbols
Example:
T is:
                   A ⊑ B ⊓ (C ⊔ D), B ⊑ (C ⊔ E)
T’ is
                A ⊑ (C ⊔ E) ⊓ (C ⊔ D), B ⊑ (C ⊔ E)


T and T’ are equivalent. Reasoning with T’ will yield the same
  results as reasoning in T



 27
Terminology (TBox)

For each concept C we define the expansion of C with respect to
  T as the concept C’ that is obtained from C by replacing each
  occurrence of a name symbol A in C by the concept D, where
  A≡D is the definition of A in T’, the expansion of T
Example: take previous Tbox (with Man defined ad being a person
  which is not a Woman)
C is:
                          Woman ⊓ Man
C’ is
              Person ⊓ Female ⊓ Person ⊓  Female


C≡TC’, C is satisfiable with respect to C’, … subsumption,
 disjointness (write precisely)
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Terminology (TBox)


The expansion of C to C’ can be costly, as in the worst case T’ is
  exponential in the size of T, and this propagates to C’


EXAMPLE: use DeMorgan laws




 29
Outlines

            (TBox)
 Terminology
 World Descriptions (ABox)




 30
ABox
The second component of the knowledge base is the world
  description, the ABox.
In a ABox, one introduces individuals, by giving them names,
  and one asserts properties about these individuals.
We denote individual names as a, b, c,…
An assertion with concept C is called concept assertion
  in the form:
                     C(a), C(b), C(c), …
  Example
                    Professor(fausto)

 31
Semantics of the ABox

We give a semantics to ABoxes by extending
 interpretations to individual names.

An interpretation I =(∆I, .I) not only maps atomic concepts
 to sets, but in addition maps each individual name a to an
 element aI ∈∆I., namely
                             I (a) = aI ∈∆I

We assume that distinct individual names denote distinct
 objects, as unique name assumption (UNA).


 32
Individuals in the TBox
Sometimes, it is convenient to allow individual names (also
  called nominals) not only in the ABox, but also in the
  description language.
The most basic one is the “set”constructor, written
                          {a1,…,an}
  Which defines a concept, without giving it a name, by
  enumerating its elements., with the semantics
                    {a1,…,an}I= {a1I,…,anI}
Example:
        StudentsFaustoClass ≡ {chen, enzo, …, zhang}

 33
Outline

 Terminology   (TBox)
 World Descriptions (ABox))
 Reasoning with TBox
 Eliminating the Tbox
 Reasoning with the Abox
 Closed vs. Open world semantics




 34
Consistency

Consistency: An Abox A is consistent with respect to a
 Tbox T if there is an interpretation I which is a model of
 both A and T.

We simply say that A is consistent if it is consistent with
 respect to the empty Tbox

Example: {Mother (Mary), Father(Mary)} is consistent but
  Not consistent with respect the family TBox
… other examples in DBs


 35
Consistency

Checking the consistency of an ABox with respect to an acyclic TBox
  can be reduced to checking an expanded ABox.

We define the expansion of an ABox A with respect to T as the ABox A’
 that is obtained from A by replacing each concept assertion C(a) with
 the assertion C’(a), with C’ the expansion of C with respect to T.

A is consistent with respect to T iff its expansion A’ is consistent
A is consistent iff A is satisfiable (in PL, under the usual translation) with
  C(a) considered as a proposition (different from C(b))

NOTE: from now on let us drop TBox (via expansion)
 36
Example
 Consider       the example of students in LDKR:
1.    Bachelor ≡ Student ⊓ Undergraduate
2.    Master ≡ Student ⊓  Undergraduate
3.    PhD ≡ Master ⊓ Research
4.    Assistant ≡ PhD ⊓ Teach
5.    Undergraduate ⊑  Teach
 Plus    that
Master(Chen), PhD(Enzo), Assistant(Rui)


 We      can conclude that:


     37
Example cont.
 Is the knowledge base consistent?
 Is α=Phd(Rui) entailed?
 Find all the instances of Undergraduate.
 Given an instance Rui, and a concept set {Student, PhD,
  Assistant} find the most specific concept C that |=C(Rui)




 38
Instance checking

Checking whether an assertion is entailed by an ABox (and
 TBox via expansion)

A |= C(a) if every interpretation which satisfies A also satisfies
  C(a).

A |= C(a) iff A conjunct with { C(a)} is inconsistent




 39
Instance retrieval

Given an ABox A and a concept C retrieve all instance a which
  satisfy C.

A |= C(a) if every interpretation which satisfies A also satisfies
  C(a).

Non optimized implementation: do instance checking for all
 instances




 40
Concept realization

Dual problem of Instance retrieval

Given an ABox A, a set of concepts and an individual a find the
  most specific concepts C such that A |= C(a)

Most specifi concept: more specific with respect the subsumption
 ordering.

Non optimized implementation: do instance checking for all
 concepts

 41
Outline

 Terminology   (TBox)
 World Descriptions (ABox))
 Reasoning with TBox
 Eliminating the Tbox
 Reasoning with the Abox
 Closed vs. Open world semantics




 42
Closed and Open world semantics

Closed world Assumption CWA (Data bases): anything which is
  not explicitly asserted is false

Open World Assumption OWA (Abox): anything which is not
 explicitly asserted (positive or negative) is unknown

DB: has/ is one model: query answering is model checking
Abox: has a set of models: query answering is satisfiability (see
 above)



 43

				
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