# Logics for Data and Knowledge Representation

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

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Outline

 Terminology   (TBox)

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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

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

Examples

Woman ⊑ Person ⊓ Female

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Terminology (TBox)

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

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

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Outlines

 Terminology
 WorldDescriptions
 Reasoning with the TBox

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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 ⊑ ⊥?

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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!!!
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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!!!
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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!!!

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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!!!

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

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

T is satisfiable (build model)

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

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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?

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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,

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)?

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

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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.

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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

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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
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Outline

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

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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

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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
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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

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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

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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

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Outlines

(TBox)
 Terminology
 World Descriptions (ABox)

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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)

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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).

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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}

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Outline

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

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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

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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)
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Example
 Consider       the example of students in LDKR:
1.    Bachelor ≡ Student ⊓ Undergraduate
2.    Master ≡ Student ⊓  Undergraduate
3.    PhD ≡ Master ⊓ Research
4.    Assistant ≡ PhD ⊓ Teach
 Plus    that
Master(Chen), PhD(Enzo), Assistant(Rui)

 We      can conclude that:

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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)

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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

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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

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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

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Outline

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

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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)

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