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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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 :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Subsumption with respect to a TBox T (1) Suppose we describe the students/listeners in a course:
T ⊨ PhD ⊑ Student
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Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T
Student Undergraduate
BachelorMaster
Research PhD
Teach
Assistant
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Subsumption with respect to a TBox T (2)PhD ⊑ Student
Proof:
PhD
≡ Master ⊓ Research
≡ (Student ⊓ Undergraduate) ⊓ Research
⊑ Student
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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T
Student Undergraduate
BachelorMaster
Research PhD
Teach
Assistant
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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 :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Disjointness with respect to a TBox T (1) Suppose we describe the students/listeners in a course:
T ⊨ Undergraduate ⊓ Assistant ⊑ ⊥
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Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T
Student Undergraduate
BachelorMaster
Research PhD
Teach
Assistant
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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?
Undergraduate ⊑ TeachBachelor ≡ Student ⊓ UndergraduateMaster ≡ Student ⊓ UndergraduatePhD ≡ Master ⊓ ResearchAssistant ≡ PhD ⊓ Teach TBox T
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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.
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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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TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Conversion in PL
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