L ogics for D ata and K nowledge R epresentation

Logics for Data and KnowledgeRepresentation Description Logics

Outline • Overview • Syntax • Semantics • Terminology • Reasoning with Terminologies • World Description • Reasoning with World Descriptions • Tableau Algorithm

Overview • Description Logics (DLs) is a family of KR formalisms that represent the knowledge of an application domain (“the world”) by • defining the relevant concepts of the domain (i.e., its terminology), and then • using these concepts to specify the properties of objects in the domain (i.e., the world’s description).

Some History on DLs • Descended from the “structured inheritance networks” (Brachman, 1977). • Introduced to overcome the ambiguities of early semantic networks and frames. • First realized in the system Kl-One by Brachman and Schmolze (1985). First DL presented in the B & S’s paper. • Some ‘nicknames’ for DLs: • Terminological knowledge representation languages, • Concept languages, • Term subsumption languages, • Kl-One-based knowledge representation languages.

Three Basic Features Extended from ClassL • The basic syntactic building blocksare atomic concepts, atomic roles, individuals. • The expressive power of DLs is restricted to a rather small set of constructorsfor building complex concepts and relations. • Implicit knowledge about concepts and individuals can be inferred automatically with the help of specific reasoning services. Syntax Extended from ClassL to Complest Role Logical

Architecture of a KR system on DL TBox Description Language Reasoning ABox Application Programs Rules

Language (Syntax) • The first step in setting up a formal language (viz. a descriptive language) is to list the symbols, that is, the alphabet of symbols. • We denote a generic alphabet of a descriptive (or ‘description’) language: dΣ. • Similarly to any logical language we can divide symbols in dΣ in ‘descriptive’ (nonlogical) and ‘non-descriptive’ (logical).

Syntax of DL • Descriptive dΣ consists of concept names (set C), which denote sets of individuals, role names (set R), which denote binary relations between individuals, and individual names, (set I), which denote individuals. • e.g.:concept names: Room, Person, Fruitrole names: likeSkiing, hasChild, partOf, isA,..individual names: I, you, apple, Fido, ...

DL Language and Previous Languages • concept names are propositional variables • (PL/ClassL) • role names are binary predicate symbols • (FOL) • individual names are constants • (FOL)

Language (Syntax) • Non-descriptive dΣ provides concept constructors to build complex formulas, called concept descriptions and role descriptions, from atomic formulas. • e.g.: (negation), ⊓(conjunction),∀(for all), ∃(there exists)

AL-family Languages • We shall now discuss various descriptive languages from the family of AL-languages. • An AL-language (= Attributive Languages) is a minimal DL language of practical interest. • More expressive descriptive languages are usually extensions of some AL-language. • AL-languages do not deal with individuals.

AL Logical Symbols • Universal concept symbol: ⊤. • Bottom concept symbol: ⊥. • Parentheses (auxiliary symbols): (, ) • Logical constants (concept constructors): (atomic negation), ⊓ (conjunction) ∀R (for all atomic roles)∃R (there exists an atomic role) The SAME as in ClassL New!

AL Non-logical Symbols The SAME as in ClassL • Atomic concept names: A, B, ... • Concept names: C, D, ... • Atomic role names: R (generic) • NOTE: There is no logical symbol in AL for logical implication (as ‘→’ in PL and in FOL). For this, we will use the subsumption symbol ‘⊑’ instead (as in classL). New!

Defined Symbols • Similarly to ClassL, ⊤and ⊥ can be defined: • For all concept names C, • ⊥ =dfC ⊓  C • ⊤ =df ⊥ or also ⊤ =df U for U be a special coincept name denoting the Universal Concept. • We prefer to consider ⊤ and ⊥AL’ symbols.

Formation Rules for AL • Atomic Concepts: 1. A, B,..., ⊥, ⊤. • Concepts (concept descriptions): 2. All the atomic concepts3. A for A (atomic concept negation)4. C⊓D (intersection) 5. ∀R.C (value restriction)6. ∃R.⊤(limited existential quantification) • Resulting language: attributive language (AL).

Examples • Atomic concepts: Person, Female, Room, ...Atomic roles: hasChild, partOf, isIn, isA,... • Concepts: Person ⊓ Female,Person ⊓ ∃hasChild.⊤ ( ∃hasChild )Person ⊓ ∀hasChild.⊥(Not:¬∃hasChild.⊤)Person ⊓ ∀hasChild.¬Female • Question: What is the intended meaning?

Solution • Person ⊓ Female “persons that are female” • Person ⊓ ∃hasChild. ⊤ “(all those) persons that have a child” • Person ⊓ ∀hasChild. ⊥“(all those) persons without a child” • Person ⊓ ∀hasChild.Female “persons all of whose children are female”

AL’s Extensions - ALU • Extended Alphabet: Logical constants (concept constructors):⊔ (disjunction). • Extended concepts (descriptions): C ⊔ D (union) • The resulting new language (i.e. AL plus the new set of concepts) usually denoted ALU. • E.g.: Mother⊔Father describes the extension of ‘parent’.

AL’s Extensions - ALE • Extended Alphabet: Logical constants (concept constructors):∃R (there exists an arbitrary role) • Extended concepts (descriptions): ∃R.C (full existential quantification) • The resulting new language (i.e. AL plus the new set of concepts) usually denoted ALE. • E.g.: ∃hasChild.Female describes the extension of ‘those parents who has at least a daughter’.

AL’s Extensions - ALN • Extended Alphabet: Logical constants (concept constructors):≥n, ≤n for all n ∈ N (at-least/at-most n) • Extended concepts (descriptions): ≥nR (at-least number restriction)≤nR (at-most number restriction) • The resulting new language (i.e. AL plus the new set of concepts) usually denoted ALN. • E.g.: ≥2 hasChild describes the extension ‘those parents who has at least 2 children’.

AL’s Extensions - ALC • Extended Alphabet: Logical constants (concept constructors):  (general negation) • Extended concepts (descriptions):  C (full concept negation) • The resulting new language (i.e. AL plus the new set of concepts) usually denoted ALC. (C stands for “Complement”). • E.g.: (≥2 hasChild ) describes the extension ‘those parents who has at most 1 children’.

AL’s Extensions (Summary) • Extending AL by any subsets of the above constructors yields a particular DL language. • Each language is denoted by a string of the form AL[U][E][N][C], where a letter in the name stands for the presence of the corresponding constructor. Notation: AL*. • ALC as the most important in many aspects.(We’ll see that ALU ⊆ ALC and ALE ⊆ ALC.)

AL’s Sub-languages • By eliminating some of the syntactical symbols, we get some sub-languages of AL. • The most important sub-language obtained by elimination in the AL family is ClassL. • Historically, another important sub-language is named as the Frame LanguageFL0.

From AL to ClassL • Elimination (w.r.t. ALUEC!): • Name Symbols: atomic role names R • Concept constructors: ∀R, ∃R • Descriptions: ∀R.C, ∃R.C • The new language is a description language without roles which is ClassL. • NOTE: So far, we are considering DL without individuals and ABox.

AL’s Contractions: FL- • Elimination (w.r.t AL’s alphabet): • Symbols: Universal and bottom symbols ⊤,⊥ • Concept constructors: ¬ (atomic negation) • Descriptions: ⊤, ⊥ ,  A (atomic negation) • The resulting new language (i.e. AL without negation) is denoted: FL-.

AL’s Contractions: FL0 • Elimination(w.r.t FL-’s alphabet): • Concept constructors:∃R (there exists an atomic role) • Descriptions: ∃R.⊤(limited existential quantification) • The resulting new language (i.e. FL- without the limited existential quantification) is denoted: FL0. • FL = Frame Language (for historical reasons)

Summary • DLs are a family oflogic-based KR formalisms to describe a domain in terms of - concepts - roles - individuals (“grounding”). • Strictly speaking we do not need DLs to represent concepts and roles, but the variable-free syntax of DLs is much more concise! (That’s good for automation!) • ClassL is ALC without Role. This will be discussed later.

Semantics • The elements of the description languages in AL-family (AL*) are plain strings of symbols • The meaning which is intended to be attached to concept, role, and individual names form an informal interpretation of the given AL* language’s expressions. No (formal) meaning.

Recall: Interpretation (∆, I) • Def. An interpretation of a language L is a pair I = (∆,I), where:- ∆ (domain) is a non-empty set of objects - I(interpretation function) is a mapping from L to ∆. • NOTE: sometimes, the mapping ‘I’ is written as ‘⋅I’ but of the same function. • E.g.: I(Person) ={p|p is a person} is written as PersonI ={p|p is a person}.

AL* Interpretation (∆,I) • I(⊥) =∅ and I(⊤) = ∆ (full domain, “Universe”) • For every concept name A of L, I(A) ⊆ ∆. • I(¬C) = ∆ \ I(C). • I(C⊓D) = I(C) ∩ I(D); I(C ⊔ D) = I(C) ∪ I(D). • For every role name R of L, I(R) ⊆ ∆ × ∆. • I(∀R.C) = {a ∈ ∆ | for all b, if (a,b)∈I(R) then b∈I(C)}. • I(∃R. ⊤) = {a ∈ ∆ | exists b s.t. (a,b) ∈ I(R)}. • I(∃R.C) = {a ∈ ∆ | exists b s.t. (a,b) ∈ I(R), b ∈ I(C)}. • I(≥nR) = {a ∈ ∆ | |{b | (a, b) ∈ I(R)}| ≥ n}. • I(≤nR) = {a ∈ ∆ | |{b | (a, b) ∈ I(R)}| ≤ n}. The SAME as in ClassL

Interpretation of Existential Quantifier • I(∃R.C)= {a ∈ ∆ | exists b s.t. (a,b) ∈ I(R), b ∈ I(C)} • Those that have some value in C with role R. • Remark: a ∈ ∆ I(C) b a (a,b)∈I(R)

Example • “Those who have a daughter.” • I(∃R.C) = I(∃hasChild.Female) = {a ∈ ∆ | exists b s.t. (a,b) ∈ I(hasChild), b ∈ I(Female)} I(Female) betty hasChild anna Anna has a child betty and betty is a female, then Anna is an individual of the concept ∃hasChild.Female.

Interpretation of Value Restriction • I(∀R.C) = {a ∈ ∆ | for all b, if (a,b)∈I(R) then b∈I(C)} • Those that have only value in C with role R. • Remark: a ∈ ∆ I(C) b b' b'' a if (a,b) ∈I(R) if (a,b') ∈I(R) if (a,b'') ∈I(R)

Example • “Those who have only daughter(s)” • I(∀R.C) = I(∀hasChild.Female)= {a ∈ ∆ | for all b, if (a,b)∈I(R) then b∈I(C)} I(Female) betty hasChild bianca babara if anna has a child betty, then betty is a female. anna if anna has a child bianca, then bianca is a female. if anna has a child babara, then babara is a female. if ...

Interpretation of Number Restriction • I(≥nR) = {a∈∆ | |{b | (a, b) ∈ I(R)}|≥ n} • Those that have relation R to at leastn individuals. ∆ b b' … a |{b | (a, b) ∈ I(R)}| ≥ n

Example • “Those who has at least 2 children” • I(≥nR) = I(≥2hasChild)={a ∈ ∆ | |{ b | (a, b) ∈ I(R)}|≥n } ∆ betty hasChild bianca babara anna Anna’s children {betty, bianca, babara,...} are ≥ 2. Anna is one individual of those who has at least 2 children.

Interpretation of Number Restriction Cont. • I(≤nR) = {a ∈ ∆ | |{b | (a, b) ∈ I(R)}| ≤n } • Those that have relation R to at most n individuals. ∆ b b' … a |{b | (a,b) ∈ I(R)}| ≤n

Example • “Those who as at most 2 children” • I(≤nR) = I(≤2hasChild)={a∈∆ | |{b | (a, b) ∈ I(R)}|≤n } ∆ betty hasChild anna Anna has only one child Betty, so Anna belongs to the concept of those who has at most 2 children.

Simple Exercises • Verify the following equivalences hold for all interpretations (∆,I): • I(¬(C ⊓ D)) = I(¬C ⊔ ¬D) • I(¬(C ⊔ D)) = I(¬C ⊓ ¬D) • I(¬∀R.C) = I(∃R.¬C) • I(¬∃R.C) = I(∀R.¬C) • E.g.: I(¬∃R.C)= {a ∈ ∆ | not exists b s.t. (a,b) ∈ I(R), b ∈ I(C)} = {a ∈ ∆ | ¬ ∃b. R(a,b)∧C(b)} = {a ∈ ∆ | ∀b. ¬ (R(a,b)∧C(b))} = {a ∈ ∆ | ∀b. R(a,b)→¬C(b)} = I(∀R.¬C)

Terminological Axioms Inclusion Axiom C⊑D e.g.: Master ⊑≥18Pass.Exam⊓≥1Finish.Project, Professor⊑∃Give.Lecture⊓∃Organize.(Exam⊔Experiment) Equivalence (Equality) Axiom C≡D e.g.: PhD ≡ Postgraduate⊓≥3Publish.Paper, Parent ≡ Man⊓∃hasChild.⊤ Is the TBox definition the SAME here as in ClassL? R⊑S R≡S

Terminology (TBox) A terminology (or TBox) is a set of a (terminological) axioms. e.g.: T is { PhD≡ Postgraduate⊓≥3Publish.Paper, Parent ≡ Person⊓∃hasChild.Person, hasGrandChild⊑hasChild } NOTE: the punctuations mean logical conjunctions. .

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 ⊑ ⊥? NOTE: they are the SAME as in ClassL with more concepts built by the role constructors.

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 Role R is called Role assertion (in contrast to Concept assertion C(a)in ClassL) as: R(a, b), R(b, c), … e.g. Study(Tin, ldkr) NOTE: the other part of the ABox is the Concept assertions the SAME as in Ground ClassL. The SAME as in ClassL

Semantics of the ABox • We give a semantics to ABox by extending interpretations to individual names. • An interpretation I =(∆, ⋅I) not only maps atomic concepts to sets, but in addition maps each individual name a to an element aI∈∆., namely I (C(a)) = aI∈CI, I(R(a, b))=(aI, bI)∈RI We still assume unique name assumption as in ClassL. ‘set’ constructor is the SAME as in ClassL, which allows individual description in TBox. The SAME as in ClassL

L ogics for D ata and K nowledge R epresentation