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Brief History
Description Logics
Temporal Description Logics
Conclusions
Description Logics and Time
Jie Zhang
David R. Cheriton School of Computer ScienceUniversity of Waterloo
February 27, 2006
Jie Zhang Description Logics and Time
http://find/http://goback/8/8/2019 Presentations Jiezhang Slides
2/22
Brief History
Description Logics
Temporal Description Logics
Conclusions
Outline
1 Brief History
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
2 Description LogicsDifferent Versions
Syntax and Semantics
Reasoning, Relation to DB and Extensions
3
Temporal Description LogicsDimensions of Temporal DL
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
4 Conclusions
Jie Zhang Description Logics and Time
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
Knowledge Representation
An important role in knowledge-based systemsRule-based systems
Use rules as the knowledge representation for knowledge
coded into the system
Case-based systems
Expertise is embodied in a library of past cases, rather than
being encoded in classical rules
Each case contains a description of problem + a solution or
the outcomeReasoning process used by an expert to solve the problem
is not recorded, but is implicit in the solution.
Jie Zhang Description Logics and Time
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
Knowledge Representation
Tasks of Knowledge Representation
Representing knowledge base of application domain, aka
the WorldReasoning to infer implicitly represented knowledge fromthe knowledge that is explicitly contained in the knowledgebase
Old Approaches
Logic-based formalismsNon-logic-based representation: Network-based structure
Jie Zhang Description Logics and Time
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
Logic-based Formalisms
Based on first-order logic
Knowledge is represented by first-order logic predicatecalculus, reasoning to verifying logical consequence
Why not propositional logic
Cannot access the structure of atomic sentencesExample: jack is a master student
In propositional logic: considered as a whole
In first-order logic: Master (jack), very expressive
Problems
Expressive power is too high for obtaining decidable andefficient inference problemsInference power may be too low for expressing interesting,but still decidable theories
Jie Zhang Description Logics and Time
B i f Hi
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
Non-logic-based representation
Based on graphical interfaces
Semantic networks and frames: Network-based Structure
Person
StudentEmployee
Professor
supervise
Master PhD
Jie Zhang Description Logics and Time
B i f Hi t
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
Non-logic-based representation
Advantages
Straight forwardEffectiveMore appealing
ProblemsLack precise semantic characterization
Represented semantics are ambiguous
Arcs can represent different kinds of relations
Systems developed behave differently
Reasoning on it is inefficient
Jie Zhang Description Logics and Time
Brief History
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Knowledge Representation
Logic-based Formalisms and Network-based Structure
Description Logics
Description Logics
Combination of both approaches (Hayes, 1979)
Giving semantics to network-based structures by usingfirst-order logicsA fragment of first-order logic
The Result: Description Logics
Concept: a description of a collection of individuals withcommon properties, i.e. Student
Roles: interrelationships between individuals, i.e. superviseIndividuals: constants of concepts, i.e. Jack is a constant ofMaster student
Jie Zhang Description Logics and Time
Brief History
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Different Versions
Syntax and Semantics
Reasoning, Relation to DB and Extensions
Different Versions
F L(Brachman and Levesque in 1984)
Concept conjunction, universal role quantification andlimited existential quantification
AL (Schmidt-SchauB and Smolka in 1991)Minimal language of practical interestAdds negation constructor
F L: disallows limited existential quantification
ALC: adds full negation and disjunction on F L
ALCF: adds functions on ALC, such as agreement and
disagreement, aka features
ALN: full existential quantification, number restrictions
Jie Zhang Description Logics and Time
Brief History
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Different Versions
Syntax and Semantics
Reasoning, Relation to DB and Extensions
Syntax
C,D A|C D|C D|C|R.C|R.C
A is an atomic conceptR is an atomic role
C and D are concepts
Examples
Person Student: those persons who are studentsPerson supervise.PhD: those persons all of whomsupervise PhD studentsPerson supervise.Student: those persons at least one ofwhom supervises students
Jie Zhang Description Logics and Time
Brief History
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Different Versions
Syntax and Semantics
Reasoning, Relation to DB and Extensions
Semantics
I: an interpretation function maps
Concept to a subset of I
Role to a subset of I I
Every individual to an element of I
Syntax Formal Semantics FOL semantics
A : AI I FA(x)
C D : CI DI FC(x) FD(x)
C D : CI DI FC(x) FD(x)C : I\CI FC(x)
R.C : {a I|b.(a, b) RI b CI} z.FR(x, z) Fc(z)
R.C : {a I|b.(a, b) RI} b CI} z.FR(x, z) Fc(z)
Jie Zhang Description Logics and Time
Brief HistoryDiff t V i
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Brief History
Description Logics
Temporal Description Logics
Conclusions
Different Versions
Syntax and Semantics
Reasoning, Relation to DB and Extensions
Reasoning Services
Concept satisfiability
A concept C is satisfiable if there exists a model I such thatCI
SubsumptionA concept C is subsumed by a concept D if CI DI forevery model I.
Other: instance checking, retrieval, realization
Expressivity verses DecidabilityTrade-off (Firstly argued by Brachman and Levesque)The more expressive, the harder the reasoning will be
Jie Zhang Description Logics and Time
Brief HistoryDiff t V i
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y
Description Logics
Temporal Description Logics
Conclusions
Different Versions
Syntax and Semantics
Reasoning, Relation to DB and Extensions
Relation to DB and Possible Extensions
DL for Database ManagementStrongly related
Knowledge-based Systems contain both database systems
and description logics knowledge representation systems
Used to address many issues in DBM
Check concept consistency to verify at design time whether
an entity can have at least one instance
Query optimization and/or information integration
Possible Extensions of DL
Adding representational featuresAdding reasoning servicesTemporal extensions: handling time
Jie Zhang Description Logics and Time
Brief HistoryDimensions of Temporal DL
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y
Description Logics
Temporal Description Logics
Conclusions
Dimensions of Temporal DL
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
Dimensions of Temporal DL
Differ on notation of time
Point-based notion of timeInterval-based notion of time
Differ on ways of adding notion of time
Implicitly: temporal information only implicit in languageExplicitly: an explict notion of time is adopted
In an explicit representation of timeExternal point of view
Same individual can have different snapshots in different
moments describing various states of the individual at thesetimes
Internal point of view
differnt states of an individual are seen as different individual
components
Jie Zhang Description Logics and Time
Brief HistoryDimensions of Temporal DL
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Description Logics
Temporal Description Logics
Conclusions
Dimensions of Temporal DL
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
Pointed-based Temporal DL
ALCT (Schild): ALC + time
Syntax
C,D CUD| (C until D) (until)
CSD| (C since D) (since)C| sometime in the future
C| always in the future
C| sometime in the past
C| always in the past
Define connectives using Since and Until
C UC C SC
C C C C
Jie Zhang Description Logics and Time
Brief HistoryDimensions of Temporal DL
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Description Logics
Temporal Description Logics
Conclusions
Dimensions of Temporal DL
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
Pointed-based Temporal DL
Semantics
T = (P, t)DI(v)(x)w.(t < w < v) CI(w)(x)}
(CSD)I(t) = {x I|v.(v > t)DI(v)(x)w.(v < w < t) CI(w)(x)}
Example
PhD Person (StudentUProfessor)
Jie Zhang Description Logics and Time
Brief History
D i i L iDimensions of Temporal DL
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Description Logics
Temporal Description Logics
Conclusions
Dimensions of Temporal DL
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
Interval-based Temporal DL
Firstly proposed by Schmiedel
Based on Allens interval relationships
Jie Zhang Description Logics and Time
Brief History
D i ti L iDimensions of Temporal DL
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Description Logics
Temporal Description Logics
Conclusions
p
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
Interval-based Temporal DL
Semantics
An Interval of T, T: [t1, t2] {t P|t1 t t2}M T, I: I is a function associating to each
i= [t1, t2] T a standard non-temporal ALC interpretation(C)I(i) = {x I|j.(j, i) CI(j)(x)}
([]C)I(i) = {x I|j.(j, i) CI(j)(x)}
Example
PhD Person (Student < mi > Professor)
Jie Zhang Description Logics and Time
Brief History
Description LogicsDimensions of Temporal DL
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Description Logics
Temporal Description Logics
Conclusions
p
Pointed-based and Interval-based Temporal DL
Decidability and Complexity
Decidability and Complexity
Concept satisfiability
A concept is satisfiable if there exists M and t such thatM, t |= (C ), which means that there exists aninterpretation I such that CI(t) for some t
DecidabilityThere exists a computational process that solves theproblem in a finite number of steps
No temporal roles for ALCT
ALCT with global roles is undecidable in any unbounded
linear order
Complexity
concept satisfiability in ALCT without global roles isPSPACE complete
Jie Zhang Description Logics and Time
Brief History
Description Logics
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Description Logics
Temporal Description Logics
Conclusions
Summary
Introduction to the history of description logics
Syntax and semantics of DL and other related issues:
reasoning, relation to DB management, and possible
extensions
Syntax and semantics of temporal DL, including
point-based and interval-based temporal DL, and the
decidability and complexity issues
Jie Zhang Description Logics and Time
Brief HistoryDescription Logics
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Description Logics
Temporal Description Logics
Conclusions
References
Enrico Franconis lecture notes: Foundations of first order logic.
Alessandro Artale and Enrico Franconi. Introducing temporal description logics.1999.
Alessandro Artale and Enrico Franconi. A survey of temporal extensions ofdescription logics. Annals of Mathematics and Artificial Intelligence. 2000.
Alessandro Artale and Enrico Franconi. Temporal description logics. 2000.
F. Baader and W. Nutt. Basic description logics. 2002.
Jan Chomicki and David Toman. Temporal logic in information systems. 1998.
Francesco M. Donini, Maurizio Lenzerini, Daniele Nardi, and Andrea Schaerf.Reasoning in description logics. 1996.
J. Kelly. The Essence of Logic. Prentice Hall, 1997.
H. J. Levesque and R. J. Brachman. Expressiveness and tractability inknowledge representation and reasoning. 1987.
D. Nardi and R. J. Brachman. An introduction to description logics. 2002.
Jie Zhang Description Logics and Time
Brief HistoryDescription Logics
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Description Logics
Temporal Description Logics
Conclusions
Thank You!
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