1. Introduction Conservative Extensions Locality-based Module Summary
Conservative Extensions and Modularity in
Ontologies
Jie Bao1
1 Iowa State University, Ames, IA
mailto:baojie@cs.iastate.edu
based on work by Bernardo Cuenca Grau, Ian Horrocks, Yevgeny Kazakov,
Ulrike Sattler Carsten Lutz, Dirk Walther, Frank Wolter and Silvio Ghilardi
Semantic Web Seminar, Spring 2008

2. Introduction Conservative Extensions Locality-based Module Summary
Outline
1 Introduction
Module and Ontology
Basic Approaches
2 Conservative Extensions
Basic Notions
Complexity Result
3 Locality-based Module
Locality: Basic Notions
Safety, Modularity and MCE
Locality

3. Introduction Conservative Extensions Locality-based Module Summary
Module and Ontology
What is an ontology module and why it is important?
Scalability Challenge
Myth: OWL is decidable thus it is guaranteed to answer a
query, e.g., a web search query
Reality: a typical web user will close a page if it does not
load in 10 seconds.
Partial Reuse Challenge
Myth: Ontologies can be reused as we referring web pages
using hyperlinks
Reality: With an OWL ontology, reuse all of it, or nothing of
it.

4. Introduction Conservative Extensions Locality-based Module Summary
Module and Ontology
What is an ontology module and why it is important?
A Module of An Ontology
is in manageable size for parse, storage and query
easy to understand, easy to maintain
has black-box behavior
has controlled interaction with other modules
thus, supports faster query and partial resuse
···

5. Introduction Conservative Extensions Locality-based Module Summary
Basic Approaches
Approaches to Support Ontology Modules
1 Modular Ontology Language: use specially designed logic
language with modular (and contextual) semantics
Distributed Description Logics (DDL)[2]
E-Connections[8]
Package-based Description Logics (P-DL)[1]
2 Design Pattern: still use the standard DL with the (global)
first order semantics, but restrict its usage to obtain
modularity
Conservative Extension (CE)[3, 10]
Locality (as an approximation to CE)[9, 7, 5, 6, 4]

6. Introduction Conservative Extensions Locality-based Module Summary
Basic Notions
Conservative Extension
Deductive Conservative Extension (DCE)
Let O and O1 ⊆ O be two L-ontologies, and S a signature over
L. We say that O is a deductive S-conservative extension of O1
w.r.t. L, if for every axiom α over L with Sig(α) ⊆ S, we have
O |= α iff O1 |= α. We say that O is a deductive conservative
extension of O1 w.r.t. L if O is a deductive S-conservative
extension of O1 w.r.t. L for S = Sig(O1 ).
Example
O1 := {C D}
O2 := {C ∃R.D, C ∀R.¬C}
S := {C, D}:

7. Introduction Conservative Extensions Locality-based Module Summary
Basic Notions
Conservative Extension
Model Conservative Extension (MCE)
Let O and O1 ⊆ O be two L-ontologies, and S a signature over
L. We say that O is a model S-conservative extension of O1 , if
for every model I of O1 , there exists a model J of O that is
obtained from I by modifying the interpretation of the
predicates in Sig(O)S while leaving the predicates in S fixed,
denoted as J |S = I|S . We say that O is a model conservative
extension of O1 if O is a model S-conservative extension of O1
for S = Sig(O1 ).
Example
O1 := {C D}
O2 := {C ∃R.D}
S := {C, D}:

8. Introduction Conservative Extensions Locality-based Module Summary
Basic Notions
Relation between DCE and MCE.
Theorem 1 [10]
If O is a model S-conservative extension of O1 , then O is a
deductive S-conservative extension of O1 , but not the converse.
Proof sketch.
1 If S-MCE(O, O1 ),then ∀ I |= O1 , ∃J |= O such that
∆I ⊆ ∆J and X I = X J for every X ∈ S. Using induction
on the structure of concepts, for every concept C,
Sig(C) ∈ S, we have that either C I = C J or
C J = C I ∪ (∆J ∆I ). Thus, if C I = ∅, then C J = ∅;
therefore, ∀J |= O s.t. C J = ∅ ⇒ ∀I |= O1 s.t. C I = ∅,
which implies S-DCE(O, O1 ).
2 S-DCE(O, O1 ) ⇒ S-MCE(O, O1 ) by example.

9. Introduction Conservative Extensions Locality-based Module Summary
Complexity Result
Deciding DCE(O1 ∪ O2 , O1 ) in ALC.
Recall that concepts in ALC are constructed using the
grammar C|¬C|C C|∃R.C
Proof strategy: try to construct a witness concept C in the
signature Sig(O1 ) that is satisfiable w.r.t. O1 but is
unsatisfiable w.r.t. O1 ∪ O2 . If such a C is found, then not
DCE(O1 ∪ O2 , O1 ).
Theorem 2 [3]
Given two ALC TBoxes O1 and O2 , it is
2EXPTIME-complete to decide whether O1 ∪ O2 is a DCE
of O1
There are algorithms whose runtime is exponential in |O1 |,
but double exponential in |O2 |, by constructing a triple
exponential witness concepts (w.r.t. |O1 ∪ O2 |).

10. Introduction Conservative Extensions Locality-based Module Summary
Complexity Result
Deciding DCE(O1 ∪ O2 , O1 ) in ALCQI.
Recall that ALCQI allows the grammar C|¬C|C C|∃R.C|
∃R − .C| ≤ nR.C| ≤ nR − .C
Theorem 3 [10]
It is 2-EXPTIME-complete to decide DCE in ALCQI. In the
case that O1 ∪ O2 is not a DCE of O1 , there exists a witness
concept C of length at most 3-exponential in |O1 ∪ O2 |. This
bound is optimal.
Proof sketch.
Using the tree model property of ALCQI, O1 ∪ O2 is not a DCE
of O1 iff there is a tree (correspondent to a witness concept)
which is embeddable into a model of O1 but not into any model
of O1 ∪ O2

11. Introduction Conservative Extensions Locality-based Module Summary
Complexity Result
Deciding DCE(O1 ∪ O2 , O1 ) in ALCQIO.
Recall that ALCQIO allows the grammar
C|¬C|C C|∃R.C|∃R − .C| ≤ nR.C| ≤ nR − .C| o, where o
stands for nominal (concept of a single instance).
Also recall that a problem P is undecidable if a known
undecidable problem can be reduced to it.
Theorem 4 [10]
DCE in ALCQIO is undecidable.
Proof sketch.
By reducing the undecidable domino tiling problem to a
DCE problem D in ALCQIO: constructing O1 , O2 s.t. D is
solvable iff O1 ∪ O2 is not a DCE of O1 .
A solution to D (a grid of infinite plane) corresponds to a
witness concept.

12. Introduction Conservative Extensions Locality-based Module Summary
Complexity Result
Deciding MCE(O1 ∪ O2 , O1 ) in ALC.
Theorem 5 [10]
MCE in ALC is undecidable.
Proof sketch.
By a reduction from the semantic consequence problem in
modal logic. Full proof is in the TR http:
//www.csc.liv.ac.uk/~frank/publ/ijcai02.ps

13. Introduction Conservative Extensions Locality-based Module Summary
Complexity Result
Deciding DCE and MCE in EL.
Recell that EL allows the grammar |C|C C|∃R.C
Theorem 6 [11]
1 DCE in EL is decidable (ExpTime-complete).
2 MCE in EL is undecidable.
Proof sketch.
1 DCE decidability: construct C in Sig(O1 ) and D in
Sig(O1 ∪ O2 ), such that O1 ∪ O2 |= C D and C ⇒1 D
(We write C ⇒ 1D if, for all sig(O1 )-concepts E,
O1 ∪ O2 |= D E implies O1 |= C E.)
2 DCE hardness: by reduction of the two-player game Peek.
3 MCE undecidability: by reduction of halting problem for
deterministic Turing machines on the empty tape.

14. Introduction Conservative Extensions Locality-based Module Summary
Locality: Basic Notions
Both DCE and MCE are undecidable for OWL (SHOIN (D)), but. . .
There exist approximations of DCE and MCE that are
decidable.
Locality
Syntactical Locality (SynL) ⇒ Semantic Locality (SemL) ⇒
MCE ⇒ DCE
SynL is decidable in polynominal time
SemL is decidable in the same complexity of the logic for
concept satisfiability (NExpTime for OWL).

15. Introduction Conservative Extensions Locality-based Module Summary
Locality: Basic Notions
Locality
Informally, an axiom (or an ontology) is semantically local w.r.t.
a signature S if it imposes no restrictions between the
interpretation of names in S.
Example
O1 := {∃R.C D}
S1 := {C, D}, S2 := {C, D, R}
O1 is local w.r.t. S1 , is not local w.r.t. S2 .
if O is local w.r.t. S, then S is an importing “interface" of O,
such that the “original meaning” of S from any imported
ontology will not be changed by O.

16. Introduction Conservative Extensions Locality-based Module Summary
Safety, Modularity and MCE
Safety and MCE.
Safety
Given L-ontologies O1 and O2 , we say that O2 is safe for O1
w.r.t. L if O2 ∪ O1 is a DCE of O1 w.r.t. L.
Theorem 7: MCE means Safety [7]
Let O be an L-ontology and S a signature over L such that O is
a model S-conservative extension of the empty ontology
O1 = ∅; that is, for every interpretation I there exists a model J
of O such that J |S = I|S . Then O is safe for S w.r.t. L.
Proof sketch
by showing that for any O s.t. Sig(O) ∩ Sig(O ) ⊆ S, O ∪ O is
a DCE of O w.r.t. L

17. Introduction Conservative Extensions Locality-based Module Summary
Safety, Modularity and MCE
Module
Module
Let O, O andO1 ⊆ O be L-ontologies. We say that O1 is a
module for O in O w.r.t. L, if O ∪ O is a deductive
S-conservative extension of O ∪ O1 for S = Sig(O) w.r.t. L.
S-Module
Let O and O1 ⊆ O be L-ontologies and S a signature over L.
We say that O1 is a S-module in O w.r.t. L, if for every
L-ontology O with Sig(O) ∩ Sig(O ) S, we have that O1 is a
module for O in O w.r.t. L.

18. Introduction Conservative Extensions Locality-based Module Summary
Safety, Modularity and MCE
Safety ⇒ Modularity
Theorem 8: Safety vs. Modules [7]
Let L be an ontology language, and let O, O , and O1 ⊆ O be
ontologies over L. Then:
1 O is safe for O w.r.t. L iff the empty ontology ∅ is a module
for O in O w.r.t. L.
2 If O O1 is safe for O ∪ O1 then O1 is a module for O in O
w.r.t. L.
We also has a similar theorem for S-module.

19. Introduction Conservative Extensions Locality-based Module Summary
Locality
Complexity
Recall that MCE ⇒ Safety ⇒ Modularity ⇒ DCE
Theorem
1 Given ontologies O and O over L, the problem of
determining whether O is safe for O w.r.t. L is
EXPTIME-complete for L = EL, 2-EXPTIME-complete for
L = ALC and L = ALCIQ, and undecidable for
L = ALCIQO.
2 Given ontologies O, O , andO1 ⊆ O over L, the problem of
determining whether O1 is a module for O in O is
EXPTIME-complete for L = EL, 2-EXPTIME-complete for
L = ALC and L = ALCIQ, and undecidable for
L = ALCIQO

20. Introduction Conservative Extensions Locality-based Module Summary
Locality
Semantic Locality
Semantic Locality
Let E ⊆ S. A SHIQ axiom α with Sig(α) ⊆ S is semantically
local w.r.t. E if the trivial expansion I of every E-interpretation
I to S is a model of α. A SHIQ-TBox T is semantically local
w.r.t. S if every axiom in T is semantically local w.r.t. S. T is
semantically local if it is local w.r.t. an empty S.
Example
O1 := {∃R.C D}
S1 := {C, D}, S2 := {C, D, R}
O1 is local w.r.t. S1 by setting ∃R.C = ⊥,
O1 is not local w.r.t. S2 .

21. Introduction Conservative Extensions Locality-based Module Summary
Locality
Semantic Locality
Theorem 9 [6]
Let O be a of set of semantically local ontologies, then for any
O ∈ O, O is a module of the union of any set of O.
Theorem 10 [6]
Deciding semantical locality of an SHOIQ TBox is decidable in
NExpTime.
There is a syntactical testing algorithm for semantic locality,
which can be done in polynomial time.

22. Introduction Conservative Extensions Locality-based Module Summary
Summary
DCE is undecidable for ALCQIO, MCE is decidable for EL
Decide modularity of an ontology can be reduced to MCE
Semantical Locality is an approximation of modularity,
which is decidable in NExpTime for SHOIQ

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