1

Comparison and Combination ofthe Expressive Power of

Description Logics and Logic Programs

Jidi (Judy) ZhaoApril 24, 2023

2

Motivation for Extending Description Logics with Horn Logic

Rules

2By Benjamin Grosof, May, 2003

3

Examples of LP not representable in DL DL cannot represent “more than one

free variable at a time”. FriendshipBetween(?X,?Y) ← Man(?X) ∧ Woman(?Y). DLs cannot directly support n-ary

predicates Traditional expressive DLs support

transitive role axioms but they cannot derive values of properties

uncleOf (?X,?Z) ←brotherOf(?X,?Y) ∧ parentOf(?Y,?Z).

HomeWorker(?X) ←

Work(?X, ?Y) ∧ Live(?X, ?Z) ∧ Loc(?Y,?W) ∧ Loc(?Z,?W)

X

YWork

Z

Live W

Loc

Loc

4

Examples of DL not representable in LP

•Horn Logic cannot represent a (1) disjunction or (2) existential in the head.•(1) State a subclass of a complex class expression which is a disjunction. E.g.,

(Human u Adult) v (Man t Woman)•(2) State a subclass of a complex class expression which is an existential. E.g.,

Radio v 9hasPart.Tuner4

5

Differences between DLs and LPs Description Logics

Open World Assumption (OWA) May exist many models Generally no Unique Name Assumption (UNA) Classical negation

Logic Programs Closed World Assumption (CWA) Only one model Unique Name Assumption (UNA) Negation As Failure (NAF)

5

6

Semantic Web Layer Cake

URI/IRI

Data interchange:

Rules: RIF

Unifying Logic

Trust

Proof

Ontology:OWL

Crypto

RDFS

User Interface & Applications

XML

Query:

SPARQL

RDF

7

Different approaches1. approaches reducing description logics to logic programs

A. DLPB. OWL-R DL and OWL 2 RL

2. Homogeneous approachesA. OWL RulesB. SWRL

3. hybrid approaches accessing description logics through queries in logic programsA. AL-Log

8

Expressiveness of Description Logic Programs (DLP)

9

DLP comprises basic RDFS & more

by Benjamin Grosof et al.•RDFS subset of DL permits the following statements:

•Subclass, Domain, Range, Subproperty (also SameClass, SameProperty)•instance of class, instance of property

•more DL statements beyond RDFS:•Using Intersection connective (conjunction) in class descriptions•Stating that a property (or inverse) is Transitive or Symmetric•Using Disjunction or Existential in a subclass expression•Using Universal in a superclass expression

10

DLP

•Figure 1. Relationship between the fragments (profiles) of OWL 1.1•http://www.webont.org/owl/1.1/tractable.html

11

DLP mappings

12

OWL 2 RL based on Description Logic Programs

[DLP] is a syntactic profile of OWL 2 DL. allows for scalable reasoning using

rule-based technologies. trades the full expressivity of the

language for efficiency http://www.w3.org/2007/OWL/wiki/Profiles#OWL_2

_RL12

13

OWL 2 RL•achieved by restricting the use of OWL 2 constructs to certain syntactic positions.•Table 1. Syntactic Restriction on Class Expressions in SubClassOf Axioms

Subclass Expressions Superclass Expressions a class

a nominal class (OneOf)

intersection of class expressions (ObjectIntersectionOf)

union of class expressions (ObjectUnionOf)

existential quantification to a class expressions (ObjectSomeValuesFrom)

existential quantification to an individual (ObjectHasValue)

a class

intersection of classes (ObjectIntersectionOf)

universal quantification to a class expressions (ObjectAllValuesFrom)

at-most 1 cardinality restrictions (ObjectMaxCardinality 1)

existential quantification to an individual (ObjectHasValue)

14

SWRL A Semantic Web Rule Language

Combining OWL and RuleML SWRL is undecidable SWRL with the restriction of DL Safe

rules is decidable Variables in DL Safe rules bind only to

explicitly named individuals in the ontology.

14

15

AL-log [Donini et al., 1998]

Provides hybrid reasoning with representational adequacy and deductive power

An AL-log knowledge base K = (Σ, π) Σ is an ALC knowledge base, expressing

knowledge about concepts, roles and individuals. π is a constrained Datalog program

Defines an interface between DL and datalog by allowing Datalog program to “query” DL KB

15

16

Example 1

FP=Full Professor, FM=Faculty Member, NFP=Nonteaching Full Professor,

AC=Advanced Course, BC=Basic Course, TC=Teaching, CO=Course,ST=Student, TP=Topic.

17

18

Conclusion of AL-Log Defines an interface between DL and datalog by

allowing datalog program to “query” DL KB Results of DL satisfiability check used for checking

constraints in query answering AL-log does not allow relational subsystem to

deduce knowledge about the structural subsystem No roles allowed in rule bodies

AL-log extended with roles in rule body by [Rosatti, 1999]

[Eiter et al., 2004] extend the approach for more expressive DLs and more expressive LP language

19

Uncertainty extension of DL

20

Motivation for Extending Description Logics with

Uncertainty“Everything is vague to a degree you do not realize till you have tried to make it precise.”

-------Bertrand Russell British author, mathematician, & philosopher

(1872 - 1970)Nobel Prize in Literature,1950

20

21

Motivation for Extending Description Logics with

Uncertainty (Cont.) Uncertainty is an intrinsic feature of real-

world knowledge and refers to a form of deficiency or imperfection in the information.

The truth of such information is not precisely established.

People work and make decisions with imprecise data in an uncertain world.

21

2222

URW3 Situation Report: uncertainty ontology

URW3

22

23

Probability, Possibility and Fuzzy logic

Probabilistic Description Logic: Statistical information e.g. John is a student with the probability 0.6

and a teacher with the probability 0.4 Fuzzy Description Logic:

Express vagueness and imprecision e.g. John is tall with the degree of truth 0.9

Possibilistic Description Logic: Particular rankings and preferences e.g. John prefers an ice cream to a beer

23

24

Probability, Possibility and Fuzzy logic (Cont.)

Previous work on uncertainty extension to DL can be classified based on (a) the generalization of classical

description logics (b) the supported forms of uncertain

knowledge (c) the underlying semantics (d) their inference problems and reasoning

algorithms.

24

25

A norm-parameterized fuzzy description logic

[Zhao, Boley, Du, 2009]

26

Fuzzy Sets Fuzzy sets and set membership is the key

to decision making when faced with uncertainty (Zadeh, 1965).

Fuzzy Logic is particularly good at handling vagueness and imprecision.

Generalize crisp sets to Fuzzy Sets (concepts).

26

27

Fuzzy values

Cheetahs run very fast. John is young. Mary is old. John is tall.

27

28

Membership Functions

28

29

Fuzzy Operations fuzzy intersection (t-norm) fuzzy union (s-norm) fuzzy set complement (negation)

29

30

A Knowledge Base (KB) <T,A>= a Tbox + an Abox

A TBox (terminology) is a finite set of fuzzy concept inclusion axioms

in FOC

fuzzy concept equivalence axioms

fuzzy DL Knowledge Bases(I)

30

31

fuzzy role inclusion axioms fuzzy role equivalence axioms

An ABox (Assertion) is a set of fuzzy assertions about individuals fuzzy concept assertions fuzzy role assertions individual inequality

fuzzy DL Knowledge Bases (II)

31

32

Semantics (I)

32

Semantics given by standard FO model theory and Fuzzy Logic

A fuzzy interpretation I is a tuple (I, •I) I is the domain (a set)•I is a mapping that maps:

Each object (individual/constant) to an element of I

Each unary predicate (classe/concept) C to a membership function of CI: I →[0,1]

Each binary predicate (propertie/role) R to a membership function of RI: I ×I →[0,1]

33

Semantics (II)

33

Concept Negation

E.g. Concept Conjunction

E.g.

34

Semantics (III)

34

Concept Disjunction

E.g.

Role Exists Restrictionin FOCexistential quantier: supremum or least upper bound

35

Semantics (IV) Role Exists Restriction E.g.

36

Semantics (V) At-least Number Restriction

in FOC

Inverse Role

37

Semantics (VI)

37

38

Reasoning Procedure

39

Probabilistic reasoning in terminological

logics(Jaeger,1994) Propositional concept language (PCL)

Syntax: Terminological axioms Probabilistic terminological axioms Probabilistic assertions

Semantics: The probability measure that interprets an individual will

be defined by Jeffrey’s rule.

A C or A C

( | )P C D p( )P a C p

40

Probabilistic reasoning in terminological

logics(Jaeger,1994) Reasoning Tasks:

(1)derive additional conditional probabilities.

(2) derive additional probabilistic assertions.

The former codifies statistical information that will be gained generally by observing a large number of individual objects and checking their membership of the various concepts.

The latter expresses a degree of belief in a specific proposition. Its value most often will be justified only by a subjective assessment of likelihood.

41

Probabilistic reasoning in terminological

logics(Jaeger,1994) Example: TBox

PTBox

PABox

42

Probabilistic reasoning in terminological

logics(Jaeger,1994) Reasoning on TBox and PTBox:

( _ | _ )( _ _ )

( _ )( _ _ )( _ | ) ( )( _ | _ ) ( _

u Flying bird Bird Antarctic birdP Flying bird Bird Antarctic bird

P Bird Antarctic birdP Flying bird Antarctic birdP Antarctic bird Bird P BirdP Antarctic bird Flying bird P Flying bir

)

( _ | ) ( )(1 ( _ | _ )) ( _ )

(1 ( _ | )) ( )( _ _ )(1 ) ( _ )

( _ )(1 ( _ |

dP Antarctic bird Bird P Bird

P Antarctic bird Flying bird P Flying birdP Antarctic bird Bird P Bird

P Antarctic bird Flying bird P Flying birdP Flying birdP Antarctic bird B

)) ( )( _ ) ( _ _ )

(1 ( _ | )) ( )( _ ) ( _ | _ ) ( _ )

(1 ( _ | )) ( )0.95 0.2*0

ird P BirdP Flying bird P Antarctic bird Flying bird

P Antarctic bird Bird P BirdP Flying bird P Flying bird Antarctic bird P Antarctic bird

P Antarctic bird Bird P Bird

.01 0.958

1 0.01

43

Probabilistic reasoning in terminological

logics(Jaeger,1994) Reasoning on KB:

According to Jeffrey’ rule,

Present a naive method for computing the probability of new knowledge

( _ )( _ )* ( _ | _ )( _ )* ( _ | _ )

0.9*0.2 0.1*0.9580.2758

P Opus Flying birdP Opus Antarctic bird u Flying bird Antarctic birdP Opus Bird Antarctic bird u Flying bird Bird Antarctic bird

44

Research Challenges in DL Extensions

Syntax and Semantics Decidability Reasoning algorithms for

possible extensions Soundness and completeness Complexity/efficiency Effective methods for

reasoning under uncertainty

44

45

Questions?