LDK R Logics for Data and Knowledge Representation ClassL (Propositional Description Logic with Individuals) 1

Outline  Terminology (TBox) 2

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 3

Definitions A definition is an equality with an atomic concept on the left hand. Examples Bachelor ≡ Student ⊓ Undergraduate Woman ⊑ Person ⊓ Female 4

Terminology (TBox) A terminology (or Tbox) is a set of a (terminological) axioms Example: T is {Woman ⊔ Father ⊑ Person, Parent ≡ Mather⊔ Father} 5

Outlines  Terminology  World Descriptions  Reasoning with the TBox 6

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 7

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. 8

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 ⊑ ⊥ ? 9

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!!! 10

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!!! 11

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!!! 12

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!!! 13

Example  Suppose 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) 14

Example cont. Equivalence Prove the following equivalence: Student ≡ Bachelor ⊔ Master Proof: Bachelor ⊔ Master ≡ (Student ⊓ Undergraduate) ⊔ Master ≡ (Student ⊓ Undergraduate) ⊔ (Student ⊓  Undergraduate) ≡ Student ⊓ (Undergraduate ⊔  Undergraduate) ≡ Student ⊓⊤ ≡ Student 15

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? 16

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, Undergraduate ⊑  Teach} Is Bachelor⊓PhD satisfiable? Are Assistant and Bachelor disjoint? 17

Class-Values and Truth-Values  The intentional interpretation I i of a proposition P determines a truth-value I i (P).  The extensional interpretation of I e of P determines a class of objects I e (P).  What is the relation between I i (P) and I e (P)? 18

PL vs. ClassL (PL, ClassL notational variants) PLClassL Syntax ∧⊓ ∨⊔  ⊤⊤ ⊥⊥ →⊑ ↔≡ P, Q... Semantics∆={true, false} ∆={o, …} (compare models) 19

Class-Values and Truth-Values  Intersection: I e |=P, I e |= Q may not imply I e |=P ⊓ Q: subsumption in an extensional interpretation is “richer” than in an intensional interpretation (subsumption is not preserved by intersection) .. but I e |=P ⊑ C, I e |=Q ⊑ C always implies I e |=P ⊓ Q ⊑ C, namely, subsumption, satisfiability and validity (empty TBox) are preserved by intersection with the TBox axioms.  Negation: We may have I e |=P and I e |=  P, and I e |= Q ⊓ P and I e |= Q ⊓  P (satisfiability is preserved using two models in place of one)  … but always not I e |=  P ⊓ P  … and always not I e |= (Q ⊓ P) ⊓ (Q ⊓  P)  … and always I e |=  P ⊔ P, namely satisfiability, validity are preserved by negation. 20

Class-Values and Truth-Values P is satisfiable with respect an intensional interpretation I i (P) if and only if it is satifisfiable with respect to an extensional interpretation I e (P). I i (P) implies I e (P): Build I e (P) from I i (P) by substituting true with U and false with empty set. I e (P) implies I i (P): less trivial. Idea: build first a I e’ (P) which is equivalent to I e (P) and which uses only U and empty set. TO BE REFINED 21

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 22

Outline  Terminology (TBox)  World Descriptions (ABox))  Reasoning with TBox  Eliminating the Tbox  Reasoning with the Abox  Closed vs. Open world semantics  Properties 23

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 24

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 25

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 26

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 27

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≡ T C’, C is satisfiable with respect to C’, … subsumption, disjointness (write precisely) 28

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 29

Outlines  Terminology (TBox)  World Descriptions (ABox) 30

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) 31

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 a I ∈ ∆ I., namely I (a) = a I ∈ ∆ I We assume that distinct individual names denote distinct objects, as unique name assumption (UNA). 32

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 {a 1,…,a n } Which defines a concept, without giving it a name, by enumerating its elements., with the semantics {a 1,…,a n } I = {a 1 I,…,a n I } Example: StudentsFaustoClass ≡ {chen, enzo, …, zhang} 33

Outline  Terminology (TBox)  World Descriptions (ABox))  Reasoning with TBox  Eliminating the Tbox  Reasoning with the Abox  Closed vs. Open world semantics 34

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 35

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) 36

Example  Consider 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  Plus that Master(Chen), PhD(Enzo), Assistant(Rui)  We can conclude that: 37

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) 38

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 39

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 40

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 41

Outline  Terminology (TBox)  World Descriptions (ABox))  Reasoning with TBox  Eliminating the Tbox  Reasoning with the Abox  Closed vs. Open world semantics 42

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) 43