ldk r logics for data and knowledge representation classl (part 2): reasoning with a tbox 1

LLogics for DData and KKnowledge RRepresentation

ClassL (part 2): Reasoning with a TBox

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Outline Terminology (TBox) Normalization of a TBox Reasoning with the TBox

Some definitions Primitive and defined concepts Use and directly use Cyclic and acyclic terminologies Expansion of a TBox

Eliminating the TBox: Reducing to DPLL reasoning

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Terminological axioms Two new logical symbols

⊑ (subsumption), with σ ⊨ C⊑D iff σ(C)⊆ σ(D)≡ (equivalence), with σ ⊨ C≡D iff σ(C) = σ(D)

Inclusion axioms

C ⊑ D has to be read “C is subsumed by (more specific than / less general than) D”

Equality axioms

C ≡ D has to be read “C is equivalent to D”

NOTE: C ≡ D holds iff both C ⊑ D and D ⊑ C hold

NOTE: ⊑ and ≡ in CLassL are the alter ego of the → and ↔ in PL

NOTE: if C ⊑ D, we can use the symbol ⊒ to say that D ⊒ C to be read “D subsumes (less specific than / more general than) C”

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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX

Semantics: Venn diagrams to represent axioms σ(A ⊑ B)

σ(A ≡ B)

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B

A

BA


Examples of terminological axioms Inclusion Axioms (inclusions)

Master ⊑ StudentWoman ⊑ PersonWoman ⊔ Father ⊑ Person

Equality Axioms (equalities)

Student ≡ PupilParent ≡ Mother ⊔ FatherWoman ≡ Person ⊓ Female

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Definitions and specializations A definition is an equality with an atomic

concept on the left hand

A specialization is an inclusion with an atomic concept on the left hand

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Bachelor ≡ Student ⊓ UndergraduateWoman ≡ Person ⊓ FemaleParent ≡ Mother ⊔ Father

Student ⊑ Person ⊓ StudyPhD ⊑ Student ⊓ Lecturer


Terminology (TBox) A terminology (or TBox) is a set of definitions and

specializations

Terminological axioms express constraints on the concepts of the language, i.e. they limit the possible models

The TBox is the set of all the constraints on the possible models

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Woman ≡ Person ⊓ FemaleMan ≡ Person ⊓ ¬WomanStudent ⊑ Person ⊓ StudyBachelor ≡ Student ⊓ UndergraduatePhD ⊑ Student ⊓ Lecturer


Normalization of a TBox It is always possible to transform a specialization into

a definition by introducing an auxiliary symbol as follows:

If from a TBox we transform all specializations into definitions we say we have normalized the TBox

A TBox with definitions only is called a regular terminology

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Woman ⊑ Person (the specialization)

Woman ≡ Person ⊓ Female (the normalized specialization)


Reasoning with a TBox T Given two class-propositions P and Q, we want to

reason about the following relations between them:

Satisfiability w.r.t. T T ⊨ P ?

Subsumption T ⊨ P ⊑ Q? T ⊨ Q ⊑ P?

Equivalence T ⊨ P ⊑ Q and T ⊨ Q ⊑ P?

Disjointness T ⊨ P ⊓ Q ⊑ ⊥?

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Satisfiability with respect to a TBox T A concept P is satisfiable w.r.t. a terminology T, if there

exists (or for all) an interpretation I with I ⊨ θ for all θ ∈ T, and such that I ⊨ P, namely I(P) is not empty

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

In other words, the interpretation I not only satisfies P, but also complies with all the constraints in T

NOTE: Instead of σ in DL literature the symbol I (interpretation) is preferred. Therefore, from now on we use it instead of σ.

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Satisfiability with respect to a TBox T Suppose we describe the students/listeners in a course:

The TBox is satisfiable. A possible model is:

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Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T

Student Undergraduate

BachelorMaster

Research PhD

Teach

Assistant

In this model the two concepts Bachelor and Assistant are

satisfiable w.r.t. T, while the

concept Assistant ⊓

Bachelor is not.


TBox reasoning: subsumption 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) I(Q) for every model I of T

NOTE: subsumption is a property of all models. Used to implement entailment and validity (when T empty)

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Subsumption with respect to a TBox T (1) Suppose we describe the students/listeners in a course:

T ⊨ PhD ⊑ Student

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BachelorMaster

Research PhD

Teach

Assistant


Subsumption with respect to a TBox T (2)PhD ⊑ Student

Proof:

PhD

≡ Master ⊓ Research

≡ (Student ⊓ Undergraduate) ⊓ Research

⊑ Student

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TBox reasoning: equivalence 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: equivalence is a property of all models.

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Equivalence with respect to a TBox T (1) Suppose we describe the students/listeners in a course:

T ⊨ Student Bachelor ⊔ Master

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BachelorMaster

Research PhD

Teach

Assistant


Equivalence with respect to a TBox T (2)Student ≡ Bachelor ⊔ Master

Proof:

Bachelor ⊔ Master

≡ (Student ⊓ Undergraduate) ⊔ Master

≡ (Student ⊓ Undergraduate) ⊔ (Student ⊓ Undergraduate)

≡ Student ⊓ (Undergraduate ⊔ Undergraduate)

≡ Student ⊓⊤

≡ Student

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TBox reasoning: disjointness Let T be a TBox. Disjointness (with respect to T):

T ⊨ P ⊓ Q ⊑ ⊥ (P ⊓ Q ⊑T ⊥)

Two concepts P and Q are disjoint with respect to T if their intersection is empty, I(P) I(Q) = ∅, for every model I of T.

NOTE: disjointness is a property of all models

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Disjointness with respect to a TBox T (1) Suppose we describe the students/listeners in a course:

T ⊨ Undergraduate ⊓ Assistant ⊑ ⊥

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BachelorMaster

Research PhD

Teach

Assistant


Disjointness with respect to a TBox T (2) It can be proved showing that:

T ⊨ Undergraduate ⊓ Assistant ⊑ ⊥

Proof:

Undergraduate ⊓ Assistant

⊑ Teach ⊓ Assistant

≡ Teach ⊓ PhD ⊓ Teach

≡ ⊥ ⊓ PhD

≡ ⊥

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Exercise

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Suppose we describe the students/attendees in a course:

Is Bachelor ⊓ PhD satisfiable?

NO! Consider the following propositions:

Assistant, Student, Bachelor, Teach, PhD, Master ⊓ Teach1. Which pairs are subsumed/supersumed?2. Which pairs are disjoint?



TBox: primitive and defined concepts In a TBox there are two kinds of concepts (symbols):

Primitive concepts (or base symbols), which occur only on the right hand of axioms

Defined concepts (or name symbols) which occur on the left hand of axioms

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A ⊑ B ⊓ (C ⊔ D)

B, C and D are primitive concepts. A is a defined concept

Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ TeachTeach, Student and Research are primitive concepts. Undergraduate, Bachelor, Master, PhD and Assistant are defined.


Use and direct useLet 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 of the defintion of A.

We say that A uses B in T if B appears in the right hand 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

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A ⊑ B ⊓ (C ⊔ D)

A directly uses B, C, D

A ⊑ B ⊓ (C ⊔ D) ---> A ⊑ (C ⊔ E) ⊓ (C ⊔ D)

B ⊑ C ⊔ E

A directly uses B; A uses E (because B is defined in terms of E)


Cyclic and acyclic terminologies A terminology T contains a cycle (is cyclic) if it contains a

concept which uses itself.

A terminilogy is acyclic otherwise

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Parent ≡ Father ⊔ Mother

Father ⊑ Male

Mother ⊑ Female

Male ≡ Person ⊓ Female

Is acyclic

Father ≡ Male ⊓ hasChild

hasChild ≡ Father ⊔ Mother

Is cyclic


Expansion and equivalent terminologies 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 primitive

T and T’ are equivalent when they have the same expansion. Reasoning with T’ will yield the same results as reasoning with

T. If T’ is the expansion of T then they are equivalent.

NOTE: it is possible to expand also a cyclic TBox.

In some cases some models exist even if the TBox is cyclic. These models are called fixpoints and there are some methods to find them and break the recursion (we will not see them).

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T T’

A ⊑ B ⊓ (C ⊔ D) A ⊑ (C ⊔ E) ⊓ (C ⊔ D)

B ⊑ (C ⊔ E) B ⊑ (C ⊔ E)


Expansion requires normalization To expand a terminology we should first normalize it.

Otherwise, if we use a specialization to expand a definition, the definition reduces to a specialization

From now on we deal with regular terminologies only

(see next slide for the regular version of the terminology T above)

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T

Parent ≡ Father ⊔ MotherFather ⊑ MaleMother ⊑ FemaleMale ≡ Person ⊓ Female

T’

Parent ⊑ (Person ⊓ Female) ⊔ FemaleFather ⊑ Person ⊓ Female Mother ⊑ FemaleMale ≡ Person ⊓ Female


Concept expansion 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

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TParent ≡ Mother ⊔ FatherFather ≡ Male ⊓ hasChildMother ≡ Female ⊓ hasChildMale ≡ Person ⊓ Female

The expansion of Parent w.r.t. T is: (Female ⊓ hasChild) ⊔ (Person ⊓ Female ⊓ hasChild)


Eliminating the TBox using expansionAssume C’ expansion of C w.r.t. T.

For all σ satisfying all the axioms in T we have: T ⊨ C iff σ ⊨ C’ (Satisfiability) T ⊨ C ⊑ D iff σ ⊨ C’ ⊑ D’ (Subsumption,

Equivalence) T ⊨ C ⊓ D ⊑ ⊥ iff σ ⊨ C’ ⊓ D’ ⊑ ⊥ (Disjointness)

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TPerson ≡ Male ⊓ FemaleMale ≡ Person ⊓ Female

Is Person satisfiable? NO!

The expansion of Person w.r.t. T is: (Person ⊓ Female) ⊓ Femalewhich is equivalent to ⊥ and therefore unsatisfiable


Expansion, final considerations As we are going to see, expanding a TBox is an

important step to reason on it.

The expansion of T to T’ or C to C’ can be costly:

In the worst case T’ is exponential in the size of T, and this propagates to single concepts.

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Reduction to subsumption and unsatisfiability Reduction to subsumption. Given two concepts C and D,

C is unsatisfiable ⇔ C ⊑ ⊥ C ≡ D ⇔ C ⊑ D and D ⊑ C C ⊥ D ⇔ C ⊓ D ⊑ ⊥

Reduction to unsatisfiability. Given two concepts C and D, C ⊑ D ⇔ C ⊓ D is unsatisfiable C ≡ D ⇔ both (C ⊓ D) and (C ⊓ D) are unsatisfiable C ⊥ D ⇔ C ⊓ D is unsatisfiable

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PL and ClassL: table of the symbols

PL ClassL

Syntax ∧ ⊓

∨ ⊔

⊤ ⊤

⊥ ⊥

→ ⊑

↔ ≡

P, Q... P, Q...

Semantics ∆={true, false} ∆={o, …} (compare models)

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RECALL: A proposition P is true iff it is satisfiable


PL and ClassL are notational variants Theorem: P is satisfiable w.r.t. an intensional

interpretation ν if and only if P is satifisfiable w.r.t. an extensional interpretation σ

(already seen in ClassL without the TBox)

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INTRODUCTION :: SYNTAX :: SEMANTICS :: PL AND CLASSL :: CLASSL REASONINGTBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX

Eliminating the TBox: the algorithm With acyclic TBoxes T it is always possible to reduce reasoning

problems w.r.t. T to problems without T. See for instance the algorithm for subsumption (all the others can be reduced to it).

Input: a TBox T, the two concepts C and D Output: true if C ⊑ D holds or false otherwise

boolean function IsSubsumedBy(T, C, D) {

T’ = Normalize(T);

C’ = Expand(C, T’);

D’ = Expand(D, T’);

C’ = RewriteInPL(C’);

D’ = RewriteInPL(D’);

return DPLL(C’ → D’);

}

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Normalization

Expansion, TBox elimination

DPLL Reasoning


Conversion in PL

ldk r logics for data and knowledge representation classl (part 2): reasoning with a tbox 1

Documents

fausto giunchiglia

tboxfirst version

tboxa tbox

rui zhangterminology

alessandro agostini

d c holdnote

possible modelsthe tbox

c d holds iff