DESCRIPTION LOGICS Paula Severi University of Leicester October 12, 2009
Description Logics Outline Introduction: main principle, why the name description logic, application to semantic web. Syntax and semantics of the description logic ALC. Inference problems. Relation with predicate logic. Decidability of ALC. Relation with modal logic. Semantic tableaux. Extensions to description logic. Description logic of OWL 2 (ontology web language for the semantic web).
DESCRIPTION LOGICS Principle from Knowledge Representation (Artificial Intelligence) Knowledge should be represented by characterizing classes of objects and the relationships between them.
DESCRIPTION LOGICS Principle from Knowledge Representation (Artificial Intelligence) Knowledge should be represented by characterizing classes of objects and the relationships between them. Knowledge Description First Order representation Logic Logic Classes Concept descriptions Unary predicates Relations Role descriptions Binary predicates
DESCRIPTION LOGIC DESCRIPTION LOGIC refers to 1 concept and roles descriptions used to describe classes and relationships between them
DESCRIPTION LOGIC DESCRIPTION LOGIC refers to 1 concept and roles descriptions used to describe classes and relationships between them 2 the logic-based semantics which can be given by a translation into first-order predicate logic.
Description Logics (DLs) Description Logics (DLs) They are a family of knowledge representation languages which can be used to represent the conceptual knowledge of an application domain in a structured and formally well-understood way.
Description Logics (DLs) Description Logics (DLs) They are a family of knowledge representation languages which can be used to represent the conceptual knowledge of an application domain in a structured and formally well-understood way. Why formally well understood? They are decidable fragments of first order logic (a bit more expressive than proposition logic). Complexity usually polynomial.
Most succesfull area of application. SEMANTIC WEB SEMANTIC WEB web enriched with a logic to check consistency and do queries OWL (web ontology language) a description logic where logical constructors are written as XML-tags
From http://www.w3.org/2003/08/owlfaq.html Jim Hendler, co-chair of the W3C Web Ontology Working Group, and the W3C Communications Team says: What does the acronym OWL stand for? Actually, OWL is not a real acronym. The language started out as the Web Ontology Language but the Working Group disliked the acronym WOL. We decided to call it OWL. The Working Group became more comfortable with this decision when one of the members pointed out the following justification for this decision from the noted ontologist A.A. Milne who, in his influential book Winnie the Pooh stated of the wise character OWL: He could spell his own name WOL, and he could spell Tuesday so that you knew it wasn t Wednesday...
Description Logic ALC. Syntax and Semantics. Notation: C, D for concept descriptions and R, S for roles Semantics. Interpretation I = ( I, I) such that C I I and R I I I
Description Logic ALC. Syntax and Semantics. Notation: C, D for concept descriptions and R, S for roles Semantics. Interpretation I = ( I, I) such that C I I and R I I I Constructor Syntax Example Semantics Atomic A Human A I I concept Atomic R haschild R I I I role Intersection C D Human Male C I D I Union C D Female Male C I D I Negation C Female I /C I Exists R.C haschild.female {x y.(x, y) R I restriction y C I } Value R.C haschild.female {x y.(x, y) R I restriction y C I }
Knowledge Base. TBOX and ABOX Knowledge base TBox (terminological knowledge) is a set of inclusions C D Background knowledge ABox (assertional knowledge) is a set of assertions a : C and < a, b >: R Knowledge about individuals
Example of Knowledge Base. Knowledge base 1 TBox (background knowledge) Altarpiece Picture hasfigure.religious hasfigure.(religious Donor) hasheight. > 120 ABox (knowledge about individuals) Sistine Madonna : Altarpiece Raphael : Painter Christ : Religious < Raphael, Sistine Madonna >: haspainted < Sistine Madonna, Christ >: hasfigure < Sistine Madonna, 180 >: hasheight 1 From Ontology of Altarpieces. David Ekserdijan, Jose Fiadeiro, Paula Severi and Monika Solanski
Altarpiece: Sistine Madonna by Raphael
Open vs Closed World Assumption Closed world assumption: Databases Any statement that is not known to be true is false. Abscense of information in a database is interpreted as negative information. Open world assumption: Description Logics The truth-value of a statement is independent of whether or not it is known by any single observer or agent to be true. Absence of information in a Abox only indicates lack of knowledge.
Basic Inference Problems Let K = T, A be a knowledge base. Consistency of K. Is there a model I of K? C satisfiable w.r.t K. Is there a model of K such that C I? C K D. Does C I D I holds for all models I of K? a is an instance of C w.r.t K. Does a I C I holds for all models I of K? a, b is an instance of R w.r.t K. Does (a I, b I ) R I holds for all models I of K.?
Example of Knowledge Base. INCONSISTENT KNOWLEDGE BASE TBox (background knowledge) Altarpiece Picture hasfigure.religious hasfigure.(religious Donor) hasheight.> 120 ABox (knowledge about individuals) Sistine Madonna : Altarpiece Christ : Religious < Sistine Madonna, Christ >: hasfigure < Sistine Madonna, 100 >: hasheight
Example of Knowledge Base. INCONSISENT KNOWLEDGE BASE TBox (background knowledge) Altarpiece Picture hasfigure.religious hasfigure.(religious Donor) Figure = Religious Donor Other Religious Donor = Religious Other = Donor Other = ABox (knowledge about individuals) Sistine Madonna : Altarpiece Christ : Religious Paula : Other < Sistine Madonna, Christ >: hasfigure < Sistine Madonna, Paula >: hasfigure
Example of Knowledge Base. IS THIS KNOWLEDGE BASE CONSISTENT??? TBox (background knowledge) Altarpiece Picture hasfigure.religious hasfigure.(religious Donor) ABox (knowledge about individuals) Sistine Madonna : Altarpiece No information about religious figures of Sistine Madonna
Example of Knowledge Base. THIS KNOWLEDGE BASE IS CONSISTENT TBox (background knowledge) Altarpiece Picture hasfigure.religious hasfigure.(religious Donor) ABox (knowledge about individuals) Sistine Madonna : Altarpiece Justification of Consistency. Construction of Model. Define I as the following extension of the term model. I = {Sistine Madonna, figure1} figure1 Religious I Sistine Madonna, figure1 hasfigure I
Example of Knowledge Base. THIS KNOWLEDGE BASE IS INCONSISTENT TBox (background knowledge) Altarpiece Religious = Picture hasfigure.religious hasfigure.(religious Donor) ABox (knowledge about individuals) Sistine Madonna : Altarpiece Justification of Inconsistency. Suppose that exists a model of the above knowledge base. Since SistineMadonna I Altarpiece I, there exists fig1 such that fig1 Religious I. This contradicts the fact that Religious I =.
Unique Name Assumption (UNA) Unique Name Assumption in Description Logic Different names always refer to different entities, i.e. if a and b are different individual names then a I b I. OWL does not assume UNA It provides explicit constructs owl:sameas and owl:differentfrom.
DL and Predicate Logic Translation of ALC into first order logic atomic concept unary predicates role names binary predicates concepts formulas with one free variables Translation of concepts π x (A) π x (C D) π x ( R.C) π x ( R.C). = A(x) = π x (C) π x (D) = y.(r(x, y) π y (C)) = y.(r(x, y) π y (C)) Translation of TBox and ABox π(t ) = C D ( x.π x(c) π x (D)) π(a) = a:c A π x(c)[x/a] (a,b):r C R(a, b)
DL and Predicate Logic Theorem The reasoning problems for ALC are all decidable. Proof. The above translation to Predicate Logic uses only two variables x and y. The two variable fragment of first order logic is decidable.
Semantic Tableaux Most widely used technique to solve the reasoning problems is the tableux based approach. Algorithm to decide consistency 1 Transform the concepts to negation normal form, i.e. negation is only applied to concept names. 2 Start with the ABox A as initial interpretation. We will extend this interpretation and get a model using semantic tableaux. 3 We build a tree (actually a forest): a labelled node for each individual x where L(x) = {C x : X} and an edge from x to y if < x, y >: R.
Complexity of ALC Theorem The reasoning problems for ALC are PSPACE-complete.
Description Logics. More Constructors. Constructor Syntax Semantics Number n R {x #({y (x, y) R I }) n} restrictions n R {x #({y (x, y) R I }) n} Inverse role R 1 {(x, y) (y, x) R I } Transitive role R transitive closure of R I Role R S {(x, z) (x, y) R I composition (y, z) R I } Concrete domain D D Concrete roles U U I I D Existential U 1,... U n.p {x y 1,..., y n. predicate x U 1 y 1... x U n y n restriction (y1,..., y n ) P }
Description Logics. More Constructors. Constructor Syntax Example Number n R 3 haschild restrictions n R 1 hasmother Inverse role R 1 haschild 1 Transitive role R haschild Role composition R S haschild hasage Concrete domain D Int Concrete roles U hasage Existential U 1,... U n.p hasheight, haswidth. > predicate restriction
Description Logics. More Constructors. Constructor Syntax Example Number n R 3 haschild restrictions n R 1 hasmother Inverse role R 1 haschild 1 Transitive role R haschild Role composition R S haschild hasage Concrete domain D Int Concrete roles U hasage Existential U 1,... U n.p hasheight, haswidth. > predicate restriction All these constructors form part of OWL 2 (last version of web ontology language). Yet more constructors investigated in literature: union role, intersection role, n-ary relations, etc.
DL and Modal Logic ALC-concepts are syntactic variants of multi-modal logic K m. Translation of concepts. f(a) f(c D) f( R i.c) f( R i.c). = A = f(c) f(d) = i (f(c)) = i (f(c)) Translation of TBoxes π(t ) = C D ( f(c) f(d) ABoxes do not have direct correspondance with modal logic, but they can be seen as special cases of nominals (propositional variables that hold in only one world).