comparison and combination of the expressive power of description logics and logic programs
DESCRIPTION
Comparison and Combination of the Expressive Power of Description Logics and Logic Programs. Jidi (Judy) Zhao October 9, 2014. Motivation for Extending Description Logics with Horn Logic Rules. By Benjamin Grosof, May, 2003. 2. Examples of LP not representable in DL. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/1.jpg)
1
Comparison and Combination ofthe Expressive Power of
Description Logics and Logic Programs
Jidi (Judy) ZhaoApril 24, 2023
![Page 2: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/2.jpg)
2
Motivation for Extending Description Logics with Horn Logic
Rules
2By Benjamin Grosof, May, 2003
![Page 3: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/3.jpg)
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
![Page 4: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/4.jpg)
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
![Page 5: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/5.jpg)
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
![Page 6: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/6.jpg)
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
![Page 7: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/7.jpg)
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
![Page 8: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/8.jpg)
8
Expressiveness of Description Logic Programs (DLP)
![Page 9: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/9.jpg)
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
![Page 10: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/10.jpg)
10
DLP
•Figure 1. Relationship between the fragments (profiles) of OWL 1.1•http://www.webont.org/owl/1.1/tractable.html
![Page 11: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/11.jpg)
11
DLP mappings
![Page 12: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/12.jpg)
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
![Page 13: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/13.jpg)
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)
![Page 14: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/14.jpg)
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
![Page 15: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/15.jpg)
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
![Page 16: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/16.jpg)
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.
![Page 17: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/17.jpg)
17
![Page 18: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/18.jpg)
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
![Page 19: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/19.jpg)
19
Uncertainty extension of DL
![Page 20: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/20.jpg)
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
![Page 21: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/21.jpg)
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
![Page 22: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/22.jpg)
2222
URW3 Situation Report: uncertainty ontology
URW3
22
![Page 23: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/23.jpg)
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
![Page 24: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/24.jpg)
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
![Page 25: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/25.jpg)
25
A norm-parameterized fuzzy description logic
[Zhao, Boley, Du, 2009]
![Page 26: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/26.jpg)
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
![Page 27: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/27.jpg)
27
Fuzzy values
Cheetahs run very fast. John is young. Mary is old. John is tall.
27
![Page 28: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/28.jpg)
28
Membership Functions
28
![Page 29: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/29.jpg)
29
Fuzzy Operations fuzzy intersection (t-norm) fuzzy union (s-norm) fuzzy set complement (negation)
29
![Page 30: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/30.jpg)
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
![Page 31: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/31.jpg)
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
![Page 32: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/32.jpg)
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]
![Page 33: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/33.jpg)
33
Semantics (II)
33
Concept Negation
E.g. Concept Conjunction
E.g.
![Page 34: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/34.jpg)
34
Semantics (III)
34
Concept Disjunction
E.g.
Role Exists Restrictionin FOCexistential quantier: supremum or least upper bound
![Page 35: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/35.jpg)
35
Semantics (IV) Role Exists Restriction E.g.
![Page 36: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/36.jpg)
36
Semantics (V) At-least Number Restriction
in FOC
Inverse Role
![Page 37: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/37.jpg)
37
Semantics (VI)
37
![Page 38: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/38.jpg)
38
Reasoning Procedure
![Page 39: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/39.jpg)
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
![Page 40: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/40.jpg)
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.
![Page 41: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/41.jpg)
41
Probabilistic reasoning in terminological
logics(Jaeger,1994) Example: TBox
PTBox
PABox
![Page 42: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/42.jpg)
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
![Page 43: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/43.jpg)
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
![Page 44: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/44.jpg)
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
![Page 45: Comparison and Combination of the Expressive Power of Description Logics and Logic Programs](https://reader035.vdocument.in/reader035/viewer/2022062816/56813e73550346895da888e0/html5/thumbnails/45.jpg)
45
Questions?