A Distributed Tableau Algorithm for Packagebased Description Logics

1
A Distributed Tableau Algorithm for Package-based
Description Logics

• Jie Bao1, Doina Caragea2 and Vasant G Honavar 1
• 1Artificial Intelligence Research Laboratory,
• Department of Computer Science,
• Iowa State University, Ames, IA 50011-1040, USA.
  
• baojie, honavar_at_cs.iastate.edu
• 2Department of Computing and Information Sciences
• Kansas State University, Manhattan, KS 66506, USA
• dcaragea_at_ksu.edu

2nd International Workshop on Context
Representation and Reasoning (CRR 2006) _at_ ECAI
2006, Aug 29, 2006, Riva del Garda, Italy
2
Dr. D. Caragea
Jie Bao
Dr. V. Honavar
3
Outline

• Requirements for reasoning with modular
  ontologies
• Package-based Description Logics (P-DL) features
  and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions

4
Modularity
5
The Need for Modular Ontologies(MO)

• Collaborative Ontology Building
• Distributed Data Management
• Large Ontology Management
• Partial Ontology Reuse

6
Reasoning with MO
Registration Office Ontology
Computer Science Dept Ontology
Bob 3304
Semantic Relations

• If GraduateOK(Jie) is consistent with the
  ontology?
• (If Jie can graduate?)

7
Reasoning with MO (2)

• Major Consideration should not require the
  integration of ontology modules.
• High communication cost
• High local memory cost
• May violate module autonomy, e.g., privacy
• Question can we do reasoning for modular
  ontologies without
• (syntactic level) an integrated ontology ?
• (semantic level) a (materialized) global tableau
  ?

8
Outline

• Requirements for reasoning with modular
  ontologies
• Package-based Description Logics (P-DL) features
  and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions

9
Package

• A package is an ontology module that captures a
  sub-domain
• Each term has a home package
• A package can import terms from other packages
• Package extension is denoted as P
• PC Package extension with only concept name
  importing
• E.g., ALCPC ALC PC

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Package Example
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Semantics of Importing

• Domain relation individual correspondence
  between local domains

• Importing establishes one-to-one domain relations
  
• Copies of individuals are shared

• Domain relations are compositionally consistent
  r13r12 O r23
• Therefore domain relations are transitively
  reusable.

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Partially Overlapping Models
?I1
?I2
x
x
CI2
CI1
x
r13
r23
CI
x
CI3
Global interpretation obtained from
local Interpretations by merging shared
individuals
?I3
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Model Projection
Global model
x
CI1
x
local models
x
CI
CI2
x
CI3
14
Outline

• Requirements for reasoning with modular
  ontologies
• Package-based Description Logics (P-DL) features
  and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions

15
Tableau Algorithm

• A tableau is a representation of a model
• Basic idea
• start with some initial facts for an ontology
• use tableau expansion rules to infer new facts,
• until no rule can be applied, or inconsistencies
  are found among those facts.
• If a clash-free fact set is found, a model of the
  ontology is constructed

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Tableau Algorithm Example
Dog(goofy)
goofy
L(goofy)Dog, Animal, eats.DogFood
Animal(goofy) ( eats.DogFood)(goofy)
eats
L(foo)DogFood
foo
eats(goofy,foo) DogFood(foo)
ABox Representation
Completion Tree Representation
Note both representations are simplified for
demostration purpose
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Federated Reasoning
Stan So they are far from us. Too Bad.
Chef Hello there, children! Where does Kyle
move to?
Chef We are in South Park, Colorado San
Francisco is in California Colorado is far from
California.
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Federated Reasoning for P-DL

• Basic strategy
• Use multiple local reasoners, each for a single
  package
• Each local reasoner creates and maintains a local
  tableau based on local knowledge
• A local reasoner may query other reasoners if its
  local knowledge is incomplete
• Global relation among tableaux is created by
  messages

(1)
(4)
(3)
(2)
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Tableau Projection
x1
x1
B1
A1
x2
x3
B2
A2
x4
x4
A3
B3
The (conceptual) global tableau
Local Reasoner for package A
Local Reasoner for package B
Shared individuals mean partially overlapped
local models
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Model Projection
Global model
x
CI1
x
local models
x
CI
CI2
x
CI3
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Tableau Expansion
Tableau Expansion for ALCPC with acyclic importing
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Communication among Local Tableaux
T1
T2

• Membership m(y,C)
• Reporting r(y,C)
• Clash bottom(y)
• Model top(y)

y
y
C?
Query if y is an instance of C
C(y)
y
y
C
Notify that y is an instance of C
y
y

X
Notify that y has local inconsistency
y
y

Notify that no more rule can be applied locally
on y
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ALCPC Expansion Example (2)

• P1 1A 1B
• P2 1B 2C
• P3 2C 3D
• Query if A D (from the point of view of P3)
• (it is not answerable by either DDL nor
  E-Connection in their current forms)
• Reasoning if A D is not true, then there
  will be clash. Hence, it must be true

L3(x)A??D, ?C?D A,?C, ?D
Transitive Subsumption Propagation
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ALCPC Expansion Example (3)
Detect Inter-module Unsatisfiability
T2
T1
r(x,B) r(x,?F)
x
x
L2(x)P,?P?B, ?P??F,B,?F
L1(x)B,?F,?B?F,F
?(x)
2P is unsatisfiable
T1
T2
r(z,A)
Reasoning from Local Point of View
y
y
L1(x)A, ?A?C,C
L2(y)A,?A??R.B, ?B?(A??C), ?R.B, ?B
(x)
?
P
r(z,A) r(z,?C)
L2(z)B,?A??R.B, ?B?(A??C), ?R.B, A??C, A, ?C
L1(z)A, ?C, ?A?C, C
z
z
?(x)
1A is unsatisfiable witnessed by P2 is
satisfiable witnessed by P1
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Soundness
(a) Augmenting
A is consistent iff A is consistent
a
a
A
A
(b) Searching
a
a
A is consistent iff A is consistent or A is
consistent
A
or
or
A
ß
ß
A
(c) Reporting
(A,B) is consistent iff (A,B) is consistent
a
A
A
a
B
B
send
infer
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Completeness
P-DL model can be constructed from a distributed
Tableau
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Outline

• Requirements for reasoning with modular
  ontologies
• Package-based Description Logics (P-DL) features
  and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions

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Other Tableau Projections
x1
x1
x2
x3
x3
x2
x3
x5
x4
x4
x5
Distributed Description Logics (DDL) Serafini
and Tamilin 2004, 2005
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Other Tableau Projections (2)
x1
A1
x2
x3
x1
A1
A3
A2
E
x4
x4
x2
E
x3
B1
B1
A3
A2
x5
x6
x5
x6
B2
B3
B2
B3
E-Connections Grau 2005
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Ongoing Work

• Working with cyclic importing

x1
x1
x1
A1,B1
B1
A1
x2
x3
x2
x3
B2
B2
A2
A2
x4
x4
x4
A3
B3
B4
B4
A3,B3
A3
PA
PB
B1
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Ongoing Work (2)

• Asynchronous reasoning
• local reasoners dont need to wait after a
  reporting message
• Thus they can concurrently search on different
  branches for a possible global tableau.
• Working with OWL
• Support SHOIQ(D)
• Implementation based on Pellet

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References

• P-DL
• J. Bao, D. Caragea, and V. Honavar. Towards
  collaborative environments for ontology
  construction and sharing. In International
  Symposium on Collaborative Technologies and
  Systems (CTS 2006). 2006.
• J. Bao, D. Caragea, and V. Honavar. Modular
  ontologies - a formal investigation of semantics
  and expressivity. 2006. In the Asian Semantic Web
  Conference (ASWC), LNCS 4185, pp. 616631, 2006.
• J. Bao, D. Caragea, and V. Honavar. On the
  Semantics of Linking and Importing in Modular
  Ontologies. accepted by the International
  Semantic Web Conference (ISWC) 2006. (In Press)
• J. Bao, D. Caragea, and V. Honavar. A
  tableau-based federated reasoning algorithm for
  modular ontologies. Submitted to 2006
  IEEE/WIC/ACM International Conference on Web
  Intelligence, 2006 (under reviewing)
• Related work
• L. Serafini and A. Tamilin. Local tableaux for
  reasoning in distributed description logics. In
  Description Logics Workshop 2004, CEUR-WS Vol
  104, 2004.
• L. Serafini and A. Tamilin. Drago Distributed
  reasoning architecture for the semantic web. In
  ESWC, pages 361-376, 2005.
• B. C. Grau. Combination and Integration of
  Ontologies on the Semantic Web. PhD thesis, Dpto.
  de Informatica, Universitat de Valencia, Spain,
  2005.

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• Thanks !

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Reasoning by Model Construction