description logic – 2

4-Nov-05 CS6795 Semantic Web Techniques1

Description Logic – 2


History

Description Logic Handbook Chapter 1 Early knowledge representation work from 60’s

– Representing classes of objects – Abstraction Hierarchy– Properties of those objects– Constraints on the properties

Led to object oriented programming ideas– CommonLisp Object System (CLOS) – SmallTalk, C++, Java

Networks of knowledge– Semantic Networks (Quillian 1967)– Frames (Minsky 1981)


Example of Network KR

Person, Female, etc are concepts

hasChild is a property of Person

– hasChild relates Parent to Person

– Nil means infinity. A Parent is a Person with between 1 and infinity children

Large arrows are “IS-A” links– A Mother is a (specialization

of a) Parent


Networks and Logic systems were defined

First order logic can express most of the network– Predicates can be arity 1 or 2 (unary and binary)– Rules can be used for inheritance and some constraints– Mother ISA Parent

X Mother(X) Parent(X)

– All children of Parents are PersonsX,Y Parent(X) hasChild(X, Y) Person(Y)

– Expect some inferences to be made Inheritance: All properties of a superclass should also be

properties of its subclass So all children of Mothers should be Persons

X,Y Mother(X) hasChild(X, Y) Person(Y)


Network and Logic Systems have advantages/disadvantages

Networks considered more appealing and more practical than Logic

Some Network “reasoners” were ad hoc– Different reasoners give different results

Logic engines all produce the same conclusions– Clearly defined semantics

Logic engines at the time were very general – Not much known about special reasoners for these

formulas– General resolution-based theorem provers used

Generality meant reasoning procedures had high complexity


Description Logic systems

Brachman and Levesque [1984] “there is a tradeoff between the expressiveness of a representation language and the difficulty of reasoning over the representations built using that language.

– The more expressive the language, the harder the reasoning. Schmidt-Schauss and Smolka [1991 ] specialized classical

settings for deductive reasoning to the DL subsets of first-order logics,

– Using tableau calculus Best of both worlds

– Unary predicates for names of classes– Binary predicates for properties– Conclusions based on inheritance

Derived from Logic but driven by an understanding of how much reasoning power is needed.


Description Logic

This lecture from DL Handbook Chapter 2 Description Logics

– overcome the ambiguities of early semantic networks and frames

– first realized in the system KL-One [Brachman and Schmolze, 1985]

Well-studied and decidable (most DL languages) Tight coupling between theory and practice


TBox and ABox

– TBox: terminology the vocabulary of an application domain:

– Concepts: sets of individuals– Roles: binary relationships between individuals.

Examples: – Concepts: Person, Female, Mother– Role: hasChild, meaning that some person is the child of some other

– ABox: assertions about named individuals in terms of this vocabulary Example

– Elizabeth and Charles are Persons. We write this as Person(Elizabeth), and Person(Charles).

– Individuals, like “myCar”, have attributes, like “color”, and those attributes have values, like “red”. When this happens we say that red is the colorOf attribute of myCar.We write this as colorOf(myCar, red).


Architecture of a DL System


Formulas

Building blocks that allow complex descriptions of concepts and roles. – Example (we’ll look at the syntax in more detail soon.)

A Woman is a Female Person– Woman = Person u Female

A Mother is a Woman and she has a child– Mother = Woman u 9 hasChild.T

The TBox can be used to assign names to complex descriptions.

We will use the terms description and concept interchangably.


Reasoning Tasks

TBox reasoning: Determine– whether a description is satisfiable (i.e., non-contradictory) – whether description A is more general than description B

A subsumes B if every individual of concept B is also of concept A

– With Subsumption tests one can organize the concepts of a terminology into a hierarchy according to their generality.

ABox reasoning: Determine – whether its set of assertions is consistent,– whether a particular individual is an instance of a given concept

description.– A concept description can be a query, describing a set of objects

With instance tests one can retrieve the individuals that satisfy the query.

We will focus on the reasoning problem of TBox satisifiability


Model Theory

Statements in the TBox and in the ABox can be identified with formulae in first-order logic and the logical semantics is defined by those formulas


Wrapping your mind around model theory

Recall that a model provides a set of domain elements and a way to interpret each piece of syntax.

– It just tells you what is meant by what you wrote down. Example

– you can interpret Father(Bruce) and hasChild(Bruce, Mary) when I tell you who Bruce is and who Mary is and what Father means (a set of individuals) and what hasChild means (a role, a binary relation)

– Whether or not the statements are true depends on whether Bruce is a Father and on whether Bruce’s child is Mary in the world.

In model theory the weird thing we do is fix the formulas and let the interpretations vary.

– We don’t always have a specific world in mind When a set of formulas is true is some world we say that the

formulas are a model of that world. – They say something accurate about it, but don’t tell you everything.

What makes this useful is that when you do some syntactic manipulation to generate new formulas from the model, we expect that the new thing we found is also true in the world


DL Semantics

Defined by standard Tarski-style interpretations: I = (I, ¢I), where

– I is the domain (a non-empty set)

– ¢I is an interpretation function that maps:Concept (class) name A ! subset AI of I

Role (property) name R ! binary relation RI over I

Individual name i ! iI element of I


The Basic Description Language AL

Syntax Semantic

>I I (universal concept)

?I ; (bottom concept)

(: A)I I \ AI (atomic negation)

(C u D)I CI u DI (intersection)

(8 R.C)I {a 2 I | 8 b.(a,b) 2 RI ! b 2 CI} (value restriction)

(9 R.>)I {a 2 I | 9 b.(a,b) 2 RI} (limited exists quantification)

R RI µ I £ I (R is an atomic role)

A AI µ I (A is an atomic concept)The sublanguage FL¡ is obtained by disallowing atomic negation; and

FL0 is obtained by disallowing limited existential quantification.

Negation can only be applied to

atomic concepts

Negation can only be applied to

atomic concepts

Only the top concept is allowed in the scope of an existential quantification

over a role

Only the top concept is allowed in the scope of an existential quantification

over a role

In predicate logic, concept C can be translated into , C(x) with a free variable x.

,8hasChild.Female(x) = 8y.hasChild(x,y) ! Female(y)


What does 8 R.C and 9 R.C mean?

hasPet

A Fido

A Fluffy

B Tabby

C Rover

C Flip

Dog

Fido

Rover

Flip

Cat

Fluffy

Tabby

A DogLover is someone whose pets are all dogs, in this case {C}DogLover = 8 hasPet.Dog

{p | 8 a, (p, a) 2 hasPet ! a 2 Dog}

Also writen more simply as

{p | hasPet(p, a) ! Dog(a) }

A DogLiker is someone who owns a dog, , in this case {A, C}DogLiker = 9 hasPet.Dog

{p | hasPet(p, a) Æ Dog(a) }


The Family of AL-Languages

AL[U][E][N][C] U - union

– (C t D) I = CI [ DI

E - full existential quantification– (9 R.C)I ={a 2 I | 9 b.(a,b) 2 RI Æ b 2 CI}

N - number restrictions– at least: (> n R)I ={a 2 I | |{ b|(a,b)2RI }| > n }– at most: (6 n R)I ={a 2 I | |{ b|(a,b)2RI }| 6 n } Person u (6 1 hasChild t (> 3 hasChild u 9 hasChild.Female))denotes those persons that have either not more than one child or at

least three children, one of which is female. C - full negation

– : CI = I \ CI


DL as Fragments of Predicate Logic

Any concept D as unary predicate with one free variable Any role R as primitive binary predicate 9 R.C corresponds to 9 y.R(x,y) Æ C(y) 8 R.C corresponds to 8 y.R(x,y) => C(y) ≥ nR corresponds to

9 y1,...,yn. R(x,y1) Æ... Æ R(x,yn) Æ 8 i<j. yi≠yj

≤ nR corresponds to

8 y1,...,yn+1. R(x, y1) Æ... Æ R(x,yn+1) => 9 i<j. yi=yj

Last two examples demonstrate advantage of variable free syntax


Knowledge Base in DL

A knowledge base K = hT , Ai comprise two components:– TBOX T introduces terminology (vocabulary of an

application domain)– ABOX A contains assertions about named

individuals in terms of this vocabulary The vocabulary consists of concepts and

roles – concepts denote sets of individuals – roles denote binary relationships between

individuals


TBox and ABox

T (TBox) is a set of axioms of the form:– C v D (concept inclusion)– C ´ D (concept equivalence)– R v S (role inclusion)

- has_parent v has_ancestor

– R ´ S (role equivalence)– R +v R (role transitivity)

- has_mother +v has_ancestor

- 8has_ancestor.human applies to all successors of has_ancestor

A (ABox) is a set of axioms of the form – x 2 D (concept instantiation)– hx, yi 2 R (role instantiation)


An Example about Family Relationships

“IS-A” relationship defines a hierarchy

over concepts

“IS-A” relationship defines a hierarchy

over concepts

a role defines a property of Parent:

hasChild.Person

a role defines a property of Parent:

hasChild.Person

These are conceptsThese are concepts

value restriction(v/r): (1, NIL)

value restriction(v/r): (1, NIL)

Examples in ABox:

hasChild(MARY, PETER)

Father(PETER)

Examples in TBox:

Woman ´ Person u Female

Mother ´ Woman u 9 hasChild.Person


atomic concepts occurring in a TBox T can be divided into two sets, name symbols NT (or defined concepts) and base symbols BT (or primitive concepts, occur only on the right-hand side)

a base interpretation for T only interprets the base symbols.

Name Symbol

Name Symbols vs. Base Symbols

Base symbols


Unfolding Name Symbols

Acyclic TBox can be unfolded or expanded by eliminating all defined name symbols from the right-hand-side with only base symbols:


Satisfiability: whether the assertions in an TBox and ABox has a model

Subsumption: whether one description is more general than another one

Equivalence: whether two classes denote same set

Instantiation: check if an individual is an instance of class C

Retrieval: retrieve a set of individuals that instantiate C

Problems all reducible to satisfiability.

Reasoning Services

subsumes(C,D) : sat (: C u D) subsumes(C,D) : sat (: C u D)

equiv(C,D) subsumes(C, D) Æ subsumes(D, C) equiv(C,D) subsumes(C, D) Æ subsumes(D, C)


Concept Satisfiability

The concepts woman, mother, parent are satisfiable how about :woman u mother?

The conjunct :woman u mother can never be satisfied


ABox Satisfiability

All other inference services can be reduced to asat(A) where A is the axioms in TBox and ABox– instance checking:

instance?(a, C, A) ´ :asat (A [ {a: :C})– concept satisfiability:

sat(C) ´ asat{a:C}– concept subsumption:

subsumes(C, D) ´ :sat(:CuD) ´ :asat({a::CuD}) Open world assumption

– A = {andrew:male, (charles, andrew): has_child}– Does instance?(charles, 8has_child.male, A) hold?


Open World Assumption

Can we prove that instance?(charles, 8has_child.male, A) holds?

No! Although the ABox contains only knowledge about one

male child, it is always assumed that represented information is incomplete.

To make this statement hold, we could add:– charles:8has_child.male or – assert that information about a 2nd child will not be added in

the future: charles: 9 6 1has_child (not possible in ALC as we need number restrictions)


Reasoner: The Tableau Algorithm

contains a set of completion rules operating on constraint sets or tableaux

clash triggers proof procedure:

– transform unfold the TBox– Transform all concepts into negation normal form (i.e.

negation only occurs only in front of concept names): : (C u D) ! :C t :D : 9R.C ! 8 R.:C

– apply completion rules in arbitrary order as long as possible

stops when a clash is found terminates if no completion rule is applicable

anymore satisfiable iff a clash-free tableau can be derived


Completion Rules for the Logic ALC

Clash Trigger

{a: A, a: : A} µ A

Conjunction Rule

if 1. a: C u D 2 A, and

2. {a:C, a:D} * A

then A’= A [ {a:C, a:D}

Disjunction Rule

if 1. a: C t D 2 A, and

2. {a:C, a:D} Å A = ?

then A’= A [ {a:C} or

A’= A [ {a:D}

Role Exists Restriction Rule

if 1. a: 9R.C 2 A, and

2. :9b 2 O: {(a,b):R, b:C} µ A

then A’= A [ {(a,b):R, b:C} with b fresh in A

Role Value Restriction Rule

if 1. a: 8 R.C 2 A, and

2. 9b 2 O: (a,b):R 2 A and

3. {b:C} A

then A’= A [ {b:C}


Proof for Concept Satisfiability

Does the concept woman subsume mother? Is the concept :woman u mother unsatisfiable? Application of completion rules:

:woman u mother is unsatisfiable) the concept woman subsumes the concept mother


DL for the Semantic Web

Web Ontology Language (OWL): W3C Recommendation on 10 Feb 2004

builds on RDF and RDF Schema and adds more vocabulary for describing properties and classesExtends existing Web standards

has three increasingly-expressive sublanguages: OWL Lite (based on DL SHIF (D)) , OWL DL (based on DL SHOIN(D)), and OWL Full (OWL DL + RDF)

benefits from many years of DL research– Well defined semantics– Formal properties well understood (complexity, decidability)– Known reasoning algorithms– Implemented systems (highly optimised)

– Example: Ontology of Books


OWL Class Constructor


OWL Axioms

Axioms (mostly) reducible to inclusion (v)– e.g. C ´ D iff both C v D and D v C


Concrete Abstract Syntax

Easier to read than OWL, much closer to logic ALCAS.pl

– Prolog implementation of ALC reasoner using CAS– Currently only does TBox consistency

Working on this…

– Shows actions at various steps.– See http://owl.man.ac.uk/2003/concrete/latest/

description logic – 2

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