More on Description Logic(s)More on Description Logic(s)

Frederick MaierFrederick Maier

Note Added 10/27/03Note Added 10/27/03

So, there are a few errors that will be obvious to some:So, there are a few errors that will be obvious to some: Some of the symbols used in expressions are not in the right Some of the symbols used in expressions are not in the right

font (or even of the right type in some cases). font (or even of the right type in some cases). Instance checking is not reducible to subsumption in every Instance checking is not reducible to subsumption in every

case (e.g., see case (e.g., see thisthis). ). (The) typical means of proof is based upon satisfiability (as the (The) typical means of proof is based upon satisfiability (as the

slides on semantic tableaux indicate); I should have pointed slides on semantic tableaux indicate); I should have pointed this out more explicitly. this out more explicitly.

Again, most of the material is taken form Enrico Franconi’s Again, most of the material is taken form Enrico Franconi’s course website (I believe he’s even the originator of the DL course website (I believe he’s even the originator of the DL logo) .logo) .

I’d like to take the presentation down, as it really offers nothing that I’d like to take the presentation down, as it really offers nothing that couldn’t be found elsewhere just as readily, but I’ll wait until the end of couldn’t be found elsewhere just as readily, but I’ll wait until the end of the term. the term.

OverviewOverview

The language basicsThe language basics InterpretationsInterpretations A Family of LanguagesA Family of Languages SubsumptionSubsumption And other problemsAnd other problems ComplexityComplexity

We must understand the Syntax, We must understand the Syntax, Semantics, and Inference Semantics, and Inference Mechanisms of these languages Mechanisms of these languages if we are to use them effectively. if we are to use them effectively.

Assumption:Assumption:

OWLOWL

The language in which our ontologies The language in which our ontologies are going to be written in is likely are going to be written in is likely going to be OWL, or something like going to be OWL, or something like it. it.

And OWL is based in part on DL. And OWL is based in part on DL.

What are DL’s?What are DL’s?Key features:Key features: They are a family of Knowledge They are a family of Knowledge

Representation languages with a Representation languages with a formal semantics based largely on formal semantics based largely on FOL.FOL.

They attempt to discover “implicitly They attempt to discover “implicitly represented knowledge” using represented knowledge” using efficient inference mechanisms. efficient inference mechanisms.

The complexity of the inferences is The complexity of the inferences is an area of determined research.an area of determined research.

Basic Concepts of a DLBasic Concepts of a DL

Individuals (such as Individuals (such as johnjohn and and marymary)) Concepts (such as Concepts (such as ManMan and and

WomanWoman). ). Roles (such as Roles (such as hasChildhasChild). ).

Basic Concepts of a DLBasic Concepts of a DL

Individuals are treated exactly the same Individuals are treated exactly the same as constants in FOL. as constants in FOL.

Concepts are exactly the same as Unary Concepts are exactly the same as Unary Predicates in FOL.Predicates in FOL.

Roles are exactly the same as Binary Roles are exactly the same as Binary Predicates in FOL.Predicates in FOL.

DescriptionsDescriptions

Just Like in FOL, what we are dealing Just Like in FOL, what we are dealing with (ultimately) are sets of individuals with (ultimately) are sets of individuals and relations between individuals. and relations between individuals.

The basic unit of semantic significance The basic unit of semantic significance is the is the DescriptionDescription..

““We are We are describingdescribing sets of individuals” sets of individuals”

Defining Descriptions (Defining Descriptions (ALCALC, a typical , a typical language)language)

A description A description CC or or DD can be: can be:

AA an atomic concept. an atomic concept. **T T (top) the universal concept.(top) the universal concept. ** (bottom) the null concept (bottom) the null concept **CC a negated concept a negated concept **CC11 ∏ ∏ DD11 the intersection of concepts. the intersection of concepts. **CC11 DD11 the union of two concepts.the union of two concepts.R.CR.C (restriction) (restriction) **R.C R.C (existential quantification). (existential quantification). **

[* present in AL. Only atomic concepts can be negated. [* present in AL. Only atomic concepts can be negated. restricted restricted to to RR.T].T]

Interpretations and ModelsInterpretations and Models

Mostly, the formal semantics of a DL Mostly, the formal semantics of a DL follows FOL:follows FOL:

An individual is interpreted as an An individual is interpreted as an element from the universe of element from the universe of discourse.discourse.

A concept is interpreted as the set of A concept is interpreted as the set of elements from the universe to which elements from the universe to which the concept applies. the concept applies.

and and

and and deserve special attention. deserve special attention. Note that they only can come before a Role:Note that they only can come before a Role:

HasChild.GirlHasChild.Girl isEmployedBy.FarmerisEmployedBy.Farmer

Remember, they describe sets of Remember, they describe sets of individuals. individuals.

and and

HasChild.GirlHasChild.Girl would be interpreted would be interpreted as: as:

The setThe set { x | { x | (y)( HasChild(x,y) (y)( HasChild(x,y) Girl(y) ) Girl(y) ) }}

[Note the conditional: Am I in that set?].[Note the conditional: Am I in that set?].

and and

isEmployedBy.Farmer would be: isEmployedBy.Farmer would be:

The setThe set { x | { x | (y)( isEmployedBy(x,y) & Farmer(y) ) (y)( isEmployedBy(x,y) & Farmer(y) ) }}

A family of languagesA family of languages

The expressiveness of a description logic is The expressiveness of a description logic is determined by the operators that it uses. determined by the operators that it uses.

Add or Eliminate certain operators (e.g., Add or Eliminate certain operators (e.g., , , ), and the statements that can be ), and the statements that can be expressed are increased/reduced in number. expressed are increased/reduced in number.

Higher expressiveness implies higher Higher expressiveness implies higher complexity. complexity.

The Language ALThe Language AL

A description A description CC or or DD can be: can be:

AA an atomic concept. an atomic concept.

T T (top) the universal concept.(top) the universal concept.

(bottom) the null concept (bottom) the null concept

CC a negated Atomic concepta negated Atomic concept

CC11 ∏ ∏ DD11 the intersection of concepts. the intersection of concepts.

R.CR.C (restriction) (restriction)

R.R.T T (Limited existential quantification). (Limited existential quantification).

A family of languagesA family of languages

Operation Notation

Union (U) C B

Complementation (C) C (Any Concept)

Full Existential Quantification (E) R.C

Cardinality (N) ≥ nR, ≤nR

Qualified Cardinality (Q) ≥ nR.C, ≤nR.C

Enumeration (O) {a,b,…}

Selection (F) f:C

AxiomsAxioms

We may assign names to complex We may assign names to complex descriptions: descriptions:

Bachelor ≡ Unmarried ∏ Male Bachelor ≡ Unmarried ∏ Male Or assert that one concept is Or assert that one concept is

subsumed by another:subsumed by another:

C C D D These are These are AxiomsAxioms of the system. of the system.

SubsumptionSubsumption

A concept C subsumes a concept D iffA concept C subsumes a concept D iff

I(D) I(D) I(C) I(C)

on every interpretation on every interpretation II. This means . This means the same as the assertion:the same as the assertion:

(x)(D(x) (x)(D(x) C(x)) where C(x)) where

D and C are complex statementsD and C are complex statements

The Subsumption ProblemThe Subsumption Problem

C C D D

??Determining whether one concept Determining whether one concept

logicallylogically contains another is called contains another is called the the subsumption problem.subsumption problem.

Satisfiability of a Concept or KB {C, C}C}

Instance Checking Father(john)?

Equivalence CreatureWithHeart ≡ ≡ CreatureWithKidney

Disjointness C ∏ D∏ D

Retrieval Father(X)? X = {john, robert}

Realization X(john)? X = {Father}

Other Problems:Other Problems:

ReductionReduction

These problems can be reduced to These problems can be reduced to subsumption (for languages with subsumption (for languages with negation).negation).

They can be reduced to the They can be reduced to the satisfiability problem, as well. satisfiability problem, as well.

ComplexityComplexity

The Subsumption Problem: The Subsumption Problem: It’s undecidable for reasonably It’s undecidable for reasonably

expressive languages, expressive languages, It’s non-polynomial for fairly It’s non-polynomial for fairly

restricted languages. restricted languages.

ComplexityComplexityLanguage Subsumption Instance Checking

FL- P P

AL P P

ALE NP PSPACE

ALC PSPACE PSPACE

ALCO PSPACE PSPACE

SHIQ EXPTIME EXPTIME

KL-ONE undecidable undecidable

OWL-Lite ? ?

Inference MechanismsInference Mechanisms

ALC is equivalent to LALC is equivalent to L2 2 and so, and so, theoretically, we could translate all theoretically, we could translate all the expressions of the DL into Lthe expressions of the DL into L22 and and then use resolution or some then use resolution or some algorithm as a decision procedure. algorithm as a decision procedure.

However, it is generally the case that However, it is generally the case that Tableau algorithmsTableau algorithms are are computationally less expensive.computationally less expensive.

Tableau algorithmsTableau algorithms

They work by systematically building up They work by systematically building up a tree of possible models to for a KB.a tree of possible models to for a KB.

If every branch of the tree possesses a If every branch of the tree possesses a contradiction, then the KB is contradiction, then the KB is unsatisfiable. unsatisfiable.

Tableau proofs are sound and complete Tableau proofs are sound and complete for many languages, including for many languages, including ALC.ALC.

Complexity: NotesComplexity: Notes In complexity theory the class PSPACE is the set of In complexity theory the class PSPACE is the set of

decision problems that can be solved by a Turing decision problems that can be solved by a Turing machine using a polynomial amount of memory, machine using a polynomial amount of memory, and unlimited time. and unlimited time.

In complexity theory, EXPTIME is the set of all In complexity theory, EXPTIME is the set of all decision problems solvable by a deterministic Turing decision problems solvable by a deterministic Turing machine in O(2machine in O(2p(n)p(n)) time, where p(n) is a polynomial ) time, where p(n) is a polynomial function of n. function of n.

EXPTIME is known to be a subset of EXPSPACE and a EXPTIME is known to be a subset of EXPSPACE and a superset of PSPACE, NP-complete, NP, and P. That is superset of PSPACE, NP-complete, NP, and P. That is significant because it is currently unknown which (if significant because it is currently unknown which (if any) of those four sets are equal to each other. It is any) of those four sets are equal to each other. It is known however that P is a strict subset of EXPTIMEknown however that P is a strict subset of EXPTIME

[From www.wikipedia.org][From www.wikipedia.org]

ReferencesReferences

The Description Logic Website:The Description Logic Website:http://dl.kr.org/http://dl.kr.org/

Presentations from Enrico Franconi’s DL Presentations from Enrico Franconi’s DL course*: course*:

http://www.inf.unibz.it/~franconi/dl/course/ http://www.inf.unibz.it/~franconi/dl/course/

Chapter 2 of the Description Logic Chapter 2 of the Description Logic Handbook:Handbook:

http://www.inf.unibz.it/~franconi/dl/course/dlhb/dlhb-http://www.inf.unibz.it/~franconi/dl/course/dlhb/dlhb-02.pdf02.pdf

*Upon which this presentation is mostly based.*Upon which this presentation is mostly based.