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LLogics for DData and KKnowledge RRepresentation
ClassL (Propositional Description Logic with Individuals)
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Outline
Terminology (TBox)
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Terminological AxiomsInclusion 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
Bachelor ≡ Student ⊓ UndergraduateWoman ⊑ 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
TerminologyWorld DescriptionsReasoning 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 reasoningLet 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 TboxSatisfiability:(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 TboxEquivalence (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 TboxDisjointness: (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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ExampleSuppose 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)
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Example cont. EquivalenceProve the following equivalence:
Student ≡ Bachelor ⊔ MasterProof:Bachelor ⊔ Master≡ (Student ⊓ Undergraduate) ⊔ Master≡ (Student ⊓ Undergraduate) ⊔ (Student ⊓
Undergraduate)≡ Student ⊓(Undergraduate ⊔ Undergraduate) ≡ Student ⊓⊤≡ Student
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Example cont.(Exercise) Let’s see the following propositions,
Assistant, StudentBachelor, Teach
PhD, Master ⊓ Teach
1. Which pairs are subsumed/supersumed?
2. Which pairs are disjoint?
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ExampleSuppose 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)?
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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-ValuesP 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 reasoningLet 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 TBoxEliminating the TboxReasoning with the AboxClosed vs. Open world semanticsProperties
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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
Terminology (TBox)World Descriptions (ABox)
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ABoxThe 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), …
ExampleProfessor(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,…,an
I}
Example: StudentsFaustoClass ≡ {chen, enzo, …, zhang}
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Outline
Terminology (TBox)World Descriptions (ABox))Reasoning with TBoxEliminating the TboxReasoning with the AboxClosed 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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ExampleConsider 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
6.Plus that Master(Chen), PhD(Enzo), Assistant(Rui)
1.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 TBoxEliminating the TboxReasoning with the AboxClosed 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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