1. Logics for Data and Knowledge Representation
ClassL (part 2): Reasoning with a TBox

2. 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

3. Terminological axioms
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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” 3

4. Semantics: Venn diagrams to represent axioms
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
σ(A ⊑ B)
σ(A ≡ B) B A A B 4

5. Examples of terminological axioms
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Inclusion Axioms (inclusions)
Master ⊑ Student
Woman ⊑ Person
Woman ⊔ Father ⊑ Person
Equality Axioms (equalities)
Student ≡ Pupil
Parent ≡ Mother ⊔ Father
Woman ≡ Person ⊓ Female

6. Definitions and specializations
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
Bachelor ≡ Student ⊓ Undergraduate
Woman ≡ Person ⊓ Female
Parent ≡ Mother ⊔ Father
Student ⊑ Person ⊓ Study
PhD ⊑ Student ⊓ Lecturer

7. Terminology (TBox)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE 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
Woman ≡ Person ⊓ Female
Man ≡ Person ⊓ ¬Woman
Student ⊑ Person ⊓ Study
Bachelor ≡ Student ⊓ Undergraduate
PhD ⊑ Student ⊓ Lecturer 7

8. Normalization of a TBox
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE 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
Woman ⊑ Person (the specialization)
Woman ≡ Person ⊓ Female (the normalized specialization) 8

9. Reasoning with a TBox T
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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 ⊑ ⊥? 9

10. Satisfiability with respect to a TBox T
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
A concept P is satisfiable w.r.t. a terminology T, if there exists 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.
ADD EXAMPLE

11. Satisfiability with respect to a TBox T
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Suppose we describe the students/listeners in a course:
The TBox is satisfiable. A possible model is:
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T
ADD EXAMPLE
Student
Undergraduate
Bachelor
Master
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. 11

12. TBox reasoning: subsumption
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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)

13. Subsumption with respect to a TBox T (1)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Suppose we describe the students/listeners in a course:
T ⊨ PhD ⊑ Student
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T
ADD EXAMPLE
Student
Undergraduate
Bachelor
Master
Research
PhD
Teach
Assistant 13

14. Subsumption with respect to a TBox T (2)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
PhD ⊑ Student Proof: PhD ≡ Master ⊓ Research ≡ (Student ⊓  Undergraduate) ⊓ Research ⊑ Student
ADD EXAMPLE 14

15. TBox reasoning: equivalence
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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.

16. Equivalence with respect to a TBox T (1)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Suppose we describe the students/listeners in a course:
T ⊨ Student  Bachelor ⊔ Master
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T
ADD EXAMPLE
Student
Undergraduate
Bachelor
Master
Research
PhD
Teach
Assistant 16

17. Equivalence with respect to a TBox T (2)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Student ≡ Bachelor ⊔ Master Proof: Bachelor ⊔ Master ≡ (Student ⊓ Undergraduate) ⊔ Master ≡ (Student ⊓ Undergraduate) ⊔ (Student ⊓  Undergraduate) ≡ Student ⊓ (Undergraduate ⊔ Undergraduate) ≡ Student ⊓⊤ ≡ Student
ADD EXAMPLE 17

18. TBox reasoning: disjointness
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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

19. Disjointness with respect to a TBox T (1)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Suppose we describe the students/listeners in a course:
T ⊨ Undergraduate ⊓ Assistant ⊑ ⊥
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T
ADD EXAMPLE
Student
Undergraduate
Bachelor
Master
Research
PhD
Teach
Assistant 19

20. Disjointness with respect to a TBox T (2)
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
It can be proved showing that:
T ⊨ Undergraduate ⊓ Assistant ⊑ ⊥
Proof:
Undergraduate ⊓ Assistant
⊑  Teach ⊓ Assistant
≡  Teach ⊓ PhD ⊓ Teach
≡ ⊥ ⊓ PhD
≡ ⊥
ADD EXAMPLE 20

21. Assistant, Student, Bachelor, Teach, PhD, Master ⊓ Teach
Exercise
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Suppose we describe the students/attendees in a course:
Is Bachelor ⊓ PhD satisfiable?
NO!
Consider the following propositions:
Assistant, Student, Bachelor, Teach, PhD, Master ⊓ Teach
Which pairs are subsumed/supersumed?
Which pairs are disjoint?
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
TBox T

22. TBox: primitive and defined concepts
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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
A ⊑ B ⊓ (C ⊔ D)
B, C and D are primitive concepts. A is a defined concept
Undergraduate ⊑  Teach
Bachelor ≡ Student ⊓ Undergraduate
Master ≡ Student ⊓  Undergraduate
PhD ≡ Master ⊓ Research
Assistant ≡ PhD ⊓ Teach
Teach, Student and Research are primitive concepts.
Undergraduate, Bachelor, Master, PhD and Assistant are defined. 22

23. Use and direct use Let A and B be atomic concepts in a terminology T.
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE 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 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
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) 23

24. Cyclic and acyclic terminologies
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
A terminology T contains a cycle (is cyclic) if it contains a concept which uses itself.
A terminilogy is acyclic otherwise
Father ≡ Male ⊓ hasChild
hasChild ≡ Father ⊔ Mother
Is cyclic
Parent ≡ Father ⊔ Mother
Father ⊑ Male
Mother ⊑ Female
Male ≡ Person ⊓  Female
Is acyclic 24

25. Expansion and equivalent terminologies
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE 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 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).
T T’
A ⊑ B ⊓ (C ⊔ D) A ⊑ (C ⊔ E) ⊓ (C ⊔ D)
B ⊑ (C ⊔ E) B ⊑ (C ⊔ E) 25

26. Expansion requires normalization
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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) T
Parent ≡ Father ⊔ Mother
Father ⊑ Male
Mother ⊑ Female
Male ≡ Person ⊓  Female T’
Parent ⊑ (Person ⊓  Female) ⊔ Female
Father ⊑ Person ⊓  Female
Mother ⊑ Female
Male ≡ Person ⊓  Female 26

27. Concept expansion
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE 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 T
Parent ≡ Mother ⊔ Father
Father ≡ Male ⊓ hasChild
Mother ≡ Female ⊓ hasChild
Male ≡ Person ⊓  Female
The expansion of Parent w.r.t. T is:
(Female ⊓ hasChild) ⊔ (Person ⊓  Female ⊓ hasChild) 27

28. Expansion, final considerations
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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. 28

29. PL and ClassL: table of the symbols
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX PL
ClassL
Syntax ∧ ⊓ ∨ ⊔  ⊤ ⊥ → ⊑ ↔ ≡
P, Q...
Semantics
∆={true, false}
∆={o, …} (compare models)
RECALL: A proposition P is true iff it is satisfiable 29

30. PL and ClassL are notational variants
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
INTRODUCTION :: SYNTAX :: SEMANTICS :: PL AND CLASSL :: CLASSL REASONING
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)
MODIFIED 30

31. Reduction to subsumption and unsatisfiability
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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 31

32. Eliminating the TBox using expansion
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
Assume 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) T
Person ≡ Male ⊓ Female
Male ≡ Person ⊓  Female
Is Person satisfiable? NO!
The expansion of Person w.r.t. T is: (Person ⊓  Female) ⊓ Female
which is equivalent to ⊥ and therefore unsatisfiable
CHANGED AND MOVED 32

33. Eliminating the TBox: the algorithm
TBOX :: NORMALIZATION :: REASONING WITH A TBOX :: DEFINITIONS :: ELIMINATING THE TBOX
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’); }
Normalization
Expansion, TBox elimination
Conversion in PL
DPLL Reasoning 33