The document summarizes data complexity results for reasoning in extensions of the EL family of description logics. It shows that instance checking is coNP-hard, and thus data intractable, for several extensions including EL∀r.⊥, EL∀r.C, EL∃¬r.C, ELC∪D, EL∃r+.C, and EL(≥kr) for k ≥ 2. The reductions are from the NP-complete 2+2SAT problem and use partitioning or covering concepts in the TBox along with a polynomial-sized ABox to encode truth assignments. Instance checking remains tractable for the data-tractable logics ELIf and extensions of DL

1. Data Complexity in EL family of Description
Logics
Adila A. Krisnadhi
(Joint work with Dr. Carsten Lutz (TU Dresden/Univ. Bremen)
2007

2. Introduction
Increased interest in lightweight description logics (DLs)
Admits tractable reasoning in large scale ontologies.
EL family: tractable ontology language; sufficient expressive power
for modeling life-science ontologies
DL-Lite family: tailored towards applications with massive amount of
instance data
Most relevant reasoning services:
instance checking: decision problem asking whether an individual is
an instance of a concept/class w.r.t. background knowledge base
(KB)
conjunctive query answering: search problem related to a generalized
form of instance checking
Relevant complexity measure: data complexity (measured w.r.t. the
size of instance data only), rather than combined complexity
(measured w.r.t. the whole input: instance data, KB schema, query)

3. Objective
Aim: investigate data complexity for the EL family
We show data (in)tractability for wide range extensions of EL in the
following. Data tractable = polynomial data complexity.
Data-intractable (coNP-hard, already for instance checking):
EL∀r .⊥ , EL∀r .C , ELC D
+
EL∃¬r .C , EL∃r .C , EL∃r ∪s.C
EL≤kr , EL≥kr for some fixed integer k ≥ 0
ELkf : EL + k-functional roles (i.e., at most k successors), k > 1
(Baader,et.al.,2005): the above DLs are at least PSpace-hard
(most are ExpTime-hard) regarding combined complexity.
Data-tractable (for conjunctive query answering):
ELI f : EL + inverse role + (1)-functional role
(Baader,et.al.,2005): Regarding combined complexity, ELI is
PSpace-hard and ELf is ExpTime-complete
(Hustadt,et.al.,2005): ELI f is data-tractable for instance checking
Data-tractability results for DL-Lite family (Calvanese, et.al., 2006)

4. EL syntax & semantics
Syntax: based on set of concept names NC and role names NR
Set of (EL)-concepts (i.e., class in OWL):
A is a concept for every A ∈ NC
is a concept and if C , D are concepts, then C D, ∃r .C are
concepts too for r ∈ NR
Semantics: Based on an interpretation I = (∆I , ·I ) where the
domain ∆I is a set of individuals and ·I maps each concept to a set
of individuals in ∆I and each role to a binary relation on ∆I .
AI ⊆ ∆I for each A ∈ NC
r I ⊆ ∆I × ∆I for each r ∈ NR
I
= ∆I
(C D)I = C I ∩ D I
(∃r .C )I = {x | ∃y : (x, y ) ∈ r I and y ∈ C I }

5. EL knowledge base (syntax)
Consists of
a TBox T : a set of concept definitions and/or concept inclusions
.
concept definition: statement otf A = C , A concept name, C
concept
concept inclusion: statement otf C D, C , D are concepts
Two kinds of TBoxes:
acyclic TBox: only allow concept definitions, LHS of all definitions
are unique, has no cyclic definition
general TBox: may also allow concept inclusion
an ABox A: a set of concept assertions and role assertions
concept assertion: expression otf A(a), A concept name, a individual
name
role assertion: expression otf r (a, b), r role name, a, b individual
names
All individual names are taken from a set NI .

6. EL KB (semantics)
In an interpretation I, each individual name a ∈ NI is mapped to a
domain individual aI ∈ ∆I
I = (∆I , ·I ) satisfies the statements/expressions:
.
A = C iff AI = C I
C D iff C I ⊆ D I
A(a) iff aI ∈ AI
r (a, b) iff (aI , b I ) ∈ r I
I satisfies (i.e., a model of) a TBox T and an ABox A iff it satisfies
all statements and expressions in them

7. Reasoning problems
Satisfiability: “given a concept C and a KB (T , A), is there a model
I of the KB that satisfies C , i.e., C I = ∅?”
Subsumption: “given concepts C , D and a KB, is C I ⊆ D I for
every model I of the KB?”
KB Consistency: “given a KB, does the KB have a model?”
Instance checking: “given an individual name a, a concept C and a
KB K, is aI ∈ C I for every model I of the KB?” (written
K |= C (a))
These reasoning tasks can (usually) be reduced to each other.
Satisfiability and KB consistency in EL is trivial; subsumption and
instance checking in EL is tractable, even with a larger language
(EL++ )
Instance checking (and conjunctive query answering) emphasize
reasoning over individuals and ABoxes, hence data complexity.

8. Conjunctive query answering & entailment
Conjunctive query: a set q of atoms otf C (v ) and r (u, v ); u, v
variables
Given an interpretation I and a mapping π that maps the set of
variables in q to ∆I , I satisfies C (v ) (resp. r (u, v )) w.r.t. π iff
π(v ) ∈ C I (resp. (π(u), π(v )) ∈ r I )
I satisfies q w.r.t. π (written I |=π q) iff I satisfies all atoms in q
w.r.t. π
I satisfies q (written I |= q) iff I |=π for some π
Given knowledge base K: K |= q iff for every model of K, we have
I |= q.
Conjunctive query entailment: given K and q, decide whether K |= q
Conjunctive query answering: given K and q, find all tuples of
individual names in K such that when variables in q are properly
substituted with the individual names, we have that K |= q
Instance checking is a special case of conjunctive query entailment
(i.e., with a single atom)
Conjunctive query answering is the search problem corresponding to
conjunctive query entailment.

9. Data intractibility results: overview
All data-intractabilty results (coNP-hardness) are for instance
checking w.r.t. acyclic TBoxes.
The matching upper bound (in coNP) for most results is obtained
from (Grimm,et.al.,2007) for SHIQ
If the TBox is empty, most cases become data tractable: since
ABoxes contain only concept names, either no query contains
complex concept constructor (thus trivially reduced to query
answering in EL which is tractable), or can be shown directly (the
ELC D case).
Exception: ELkf , k ≥ 2, instance checking is coNP-complete already
when the TBox is empty.
Some cases distinguish whether unique name assumption (UNA) is
adopted.
UNA: for two individual names a, b, a = b iff aI = b I

10. 2+2-SAT
Data intractability: coNP-hardness results by reduction from known
NP-hard problem;
(Schaerf, 1993): data complexity of instance checking for EL¬A is
coNP-hard, by reduction from the NP-complete problem 2+2-SAT
2+2-SAT: decide whether a given 2+2-formula is satisfiable
2+2-formula: a finite conjunction of 2+2-clause
2+2-clause: a propositional logic formula otf p1 ∨ p2 ∨ ¬n1 ∨ ¬n2
where each disjunct is a propositional letter or a truth constant 1, 0.
Reduction from 2+2-SAT to (negation of) instance checking for
several extensions of EL will show coNP-hardness of instance
checking for the corresponding DL.

11. General method for reduction
Let ϕ = c0 ∧ · · · ∧ cn−1 be 2+2-SAT formula containing m
propositional letters q0 , . . . , qm−1 .
Let ci = pi,1 ∨ pi,2 ∨ ¬ni,1 ∨ ¬ni,2 for all i < n.
Define a TBox T , an ABox A and a query concept C such that
Size of A depends polynomially on size of ϕ
Size of T and C are constant
Show that A, T |= C (f ) (for some individual name f ) iff ϕ is
satisfiable

12. EL¬A (Schaerf, 1993)
EL¬A = EL + atomic negation (negation only on concept names).
Semantics: (¬A)I = ∆I AI
Start from ϕ as defined earlier, define the following T , A and C :
.
T := {A = ¬A}
A := {A(1), A(0)} ∪ {c(f , c0 ), . . . , c(f , cn−1 )}∪
{p1 (ci , pi,1 ), p2 (ci , pi,2 ), n1 (ci , ni,1 ), n2 (ci , ni,2 )}
i<n
C := ∃c.(∃p1 .A ∃p2 .A ∃n1 .A ∃n2 .A)
where f , 1, 0, q0 , . . . , qm−1 , c0 , . . . , cn−1 are individual names and c,
p1 , p2 , n1 and n2 are role names. Note: size of A is polynomial in
the size of ϕ
A, T |= C (f ) iff ϕ is satisfiable. (See picture)
Idea: C expresses that ϕ is not satisfied, i.e., there’s a clause in
which the two positive literals and the two negative literals are false.
(LR): Given a model I of A, T such that f I ∈ C I , show that a
/
truth assignment that satisfies ϕ exists.
(RL): Given a truth assignment satisfying ϕ, construct a model of
A, T that does not satisfy C (f ).

13. EL∀r .⊥
Essential technique from the previous proof: A and ¬A (in the
TBox) partition the domain ∆I , i.e., every element in ∆I is either
in AI or in (¬A)I .
For EL∀r .⊥ (extension with ∀r .⊥), the partitioning concepts are
∃r . and ∀r .⊥
Semantics: (∀r .bot)I = {x | ¬∃y : (x, y ) ∈ r I }
Reduction proceeds like in EL¬A using the same ABox A and query
concept C but the following TBox:
. .
T = {A = ∃r . , A = ∀r .⊥}
Again, A, T |= C (f ) iff ϕ is satisfiable.

14. EL(≤kr ) , fixed k ≥ 1
EL( ≤ kr ): extension with (≤ kr )
No partitioning concepts; use covering concepts: ∃r . and (≤ kr )
I I
Semantics: (≤ kr ) = {x | #{y | (x, y ) ∈ r } ≤ k} where #S
denotes cardinality of the set S.
Reduction: ABox A and query concept C remains the same. TBox:
. .
T = {A = ∃r . , A = (≤ kr )}
A, T |= C (f ) iff ϕ is satisfiable
(LR): From a model I of A, T that doesn’t satisfy C (f ), define a
truth assignment s.t. t(qi ) = 1 implies qiI ∈ AI and t(qi ) = 0
I
implies qiI ∈ A . Such a truth assignment exist (due to covering),
but needs not be unique since the covering concepts need not be
disjoint. However, such a truth assignment always satisfies ϕ.
(RL): Given a truth assignment t, define a model I of A, T that
does not satisfy C (f ). Basically, the model resembles the ABox with
additional individual d, and interprets A, A and r as follows:
I
AI = {1} ∪ {qi | i < m and t(qi ) = 1} A = {∆I AI }
r I = {(e, d) | e ∈ AI }

15. EL∀r .C , EL∃¬r .C
EL∀r .C : extension with ∀r .C where C is a concept
Semantics: (∀r .C )I = {x | ∀y : (x, y ) ∈ r I → y ∈ C I }.
Reduction for EL∀r .C uses the covering concepts: ∃r . and ∀r .X
EL∃¬r .C : extension with ∃¬r .C where C is a concept
Semantics: (∃¬r .C )I = {x | ∃y : (x, y ) ∈ r I and y ∈ C I }.
/
Reduction for EL∃¬r .C uses the covering concepts: ∃r . and ∃¬r .

16. +
ELC D
, EL∃r .C
ELC D
: extension with C D; (C D)I = C I ∪ D I
Reduction: use the following TBox, ABox and query concept:
. . .
T := {V = X Y,A = X,A = Y}
A := {A(1), A(0)} ∪ {c(f , c0 ), . . . , c(f , cn−1 )} ∪ {V (qi ) | i < m}∪
{p1 (ci , pi,1 ), p2 (ci , pi,2 ), n1 (ci , ni,1 ), n2 (ci , ni,2 )}
i<n
C := ∃c.(∃p1 .A ∃p2 .A ∃n1 .A ∃n2 .A)
+
EL∃r .C : extension with ∃r + .C ;
(∃r + .C )I = {x | ∃y : (x, y ) ∈ (r I )+ ∧ y ∈ C I } where the ’+’
indicates the transitive closure of the corresponding role name.
Reduction uses the same ABox and query concept as above and
. . .
TBox T = {V = ∃r + .C , A = ∃r .C , A = ∃r .∃r + .C }
Similar reduction can also be done for EL∃r ∪s.C .

17. EL(≥kr ) without UNA, k ≥ 2
No two concepts a priori cover the domain. Hence, add more
structures in the ABox.
E.g. k = 3, reduction uses the same query concept C as before with
. .
the TBox T = {A = ∃r 4 . , A = (≥ 3r )} and ABox:
A := {A(1), A(0)} ∪ {c(f , c0 ), . . . , c(f , cn−1 )}
∪ {p1 (ci , pi,1 ), p2 (ci , pi,2 ), n1 (ci , ni,1 ), n2 (ci , ni,2 )}
i<n
∪ {r (qi , b1 ), r (qi , b2 , r (qi , b3 ), r (b1 , b2 ), r (b2 , b3 ), r (b1 , b3 )}
i<m
See picture
Either two of b1 , b2 , b3 identify the same domain element or they do
not. Hence, A and A defined in T provide the covering

18. EL(≥kr ) with UNA, k ≥ 2
E.g. k = 3, reduction uses the same query concept C as before with
. . .
the TBox T = {V = ∃r .B, A = ∃r .(A B), A = (≥ 3r )} and ABox:
A := {A(1), A(0)} ∪ {c(f , c0 ), . . . , c(f , cn−1 )}
∪ {p1 (ci , pi,1 ), p2 (ci , pi,2 ), n1 (ci , ni,1 ), n2 (ci , ni,2 )}
i<n
∪ {r (qi , b1 ), r (qi , b2 , V (qi ), A(b1 ), A(b2 )}
i<m
See picture
In all models I of the KB, there is a d s.t. (qiI , d) ∈ r I for some
i < m. Either d = bj (qi satisfies ∃r .(A B), or not (qi satisfies
(≥ 3 r )).
This reduction does not works without the UNA.

19. ELkf , k ≥ 2, without UNA
This DL is EL extended with (global) k-functionality
a role r is globally k-functional iff every element in the domain has
at most k r -successor (i.e., satisfies the GCI (≤ k r ))
E.g., k = 2, reduction can be done without TBox and the following
query concept and ABox:
A := {r (1, e), A(e), B(e), r (0, e0 ), r (e0 , e1 ), r (e1 , e2 )}
∪ {c(f , c0 ), . . . , c(f , cn−1 )}
∪ {p1 (ci , pi,1 ), p2 (ci , pi,2 ), n1 (ci , ni,1 ), n2 (ci , ni,2 )}
i<n
∪ {r (qi , b1 ), r (qi , b2 ), r (qi , b3 ), r (b1 , b2 ), A(b1 ), A(b2 ), B(b3 )}
i<m
C := ∃c.(∃p1 .∃r 3 . ∃p2 .∃r 3 . ∃n1 .∃r .(A B) ∃n2 .∃r .(A B))

20. ELkf , k ≥ 2, with UNA
With UNA and without TBoxes, instance checking (and conjunctive query
answering) is data-tractable.
Consider the input ABox as complete description of an
interpretation, check all possible matches of the query. (Taking into
account possible inconsistency in the ABox).
E.g., k = 3, reduction for instance checking with acyclic TBoxes:
.
T := {V = ∃r .B}
A := {r (1, d1 ), r (1, d2 ), r (1, d3 ), s(1, d1 ), B(d1 )}
∪ {r (0, e1 ), r (0, e2 ), r (0, e3 ), s (0, e2 ), s (0, e3 ), B(d2 ), B(d3 )}
∪ {c(f , c0 ), . . . , c(f , cn−1 )}
∪ {p1 (ci , pi,1 ), p2 (ci , pi,2 ), n1 (ci , ni,1 ), n2 (ci , ni,2 )}
i<n
∪ {V (qi ), r (qi , bi,1 ), r (qi , bi,2 ), r (qi , bi,3 ), s(qi , bi,1 ), s (qi , bi,2 ), s (qi , bi,3 )}
i<m
C := ∃c.(∃p1 .∃s .B ∃p2 .∃s .B ∃n1 .∃s.B ∃n2 .∃sB)
qi satisfies either ∃s.B or ∃s .B

21. Lower bounds result summary
EL extension w.r.t. acyclic TBoxes w.r.t. general TBoxes
EL¬A coNP-complete coNP-complete
ELC D coNP-complete coNP-complete
EL∀r .⊥ , EL∀r .C coNP-complete coNP-complete
EL(≤kr ) , k ≥ 0 coNP-complete coNP-complete
ELkf w/o UNA, k ≥ 2 coNP-complete (even w/o TBox) coNP-complete
ELkf with UNA, k ≥ 2 coNP-complete (in P w/o TBox) coNP-complete
EL(≥kr ) , k ≥ 2 coNP-complete coNP-complete
EL∃¬r .C coNP-hard coNP-hard
EL∃r ∪s.C coNP-hard coNP-hard
+
EL∃r .C coNP-hard coNP-hard

22. ELI f : data-tractability overview
ELI f : EL extended with inverse roles and functional roles
inverse role: r − ; (r − )I = {(y , x) | (x, y ) ∈ r I }
(globally) functional role; r is (globally) functional if it satisfies the
GCI (≤ 1 r )
in ELI f , both a role r and its inverse can be declared functional.
wlog. no inverse role in ABox and query concept.
In this section, we consider general TBoxes.
(Hustadt,et.al.,2005) Data-tractability results for Horn-SHIQ
implies that instance checking for ELI f w.r.t. general TBoxes is
data-tractable. Its direct proof is in my master’s thesis.
We show the data-tractability result of conjunctive query answering
for ELI f w.r.t. general TBoxes by giving a decision procedure for
conjunctive query entailment running in polytime in the size of input
ABox.

23. Assumptions
We assume TBoxes are in normal form, i.e., GCIs are otf
A B, A1 A2 B, A ∃r .B, ∃r .A B (≤ 1 r )
Every ELI f TBox T can be converted into normal form T in
polytime by introducing fresh concept name.
For every ABox A and conjunctive query q not using any of the
concept names that occur in T but not in T , we have A, T |= q
In all atoms C (v ) in a conjunctive query q, C is a concept name
(i.e., no complex concept occurs in q)
Can easily be achieved if C is not a concept name: replace C (v ) in q
.
with A(v ) and add A = C to the TBox where A is a concept name.
Conjunctive queries are connected, i.e., for all variables u, v
appearing in q, there are atoms r (u0 , u1 ), . . . , r (un−1 , un ) in q s.t.
u = u0 and v = un
Entailment of non-connected queries can be reduced to entailment of
connected queries: if q is a non connected query, then A, T |= q iff
A, T |= q for all connected components q of q (Glimm,et.al.,2007)

24. Algorithm
Given an input TBox T , ABox A and conjunctive query q
Convert T into normal form (polytime in |T |)
If UNA is made, check consistency of A w.r.t. T (polytime in |A|
due to (Hustadt,et.al.2005)). If inconsistent, answer “yes”.
Otherwise, A must be admissible, and thus continue.
If UNA is not made, convert A to make it admissible w.r.t. T by
identifying individuals (polytime in |A|)
Construct initial canonical structure I for T and A (polytime in
|A|)
Check matches of q against the above structure (polytime in |A|).

25. Admissible ABox w.r.t. TBox
An ABox A is admissible w.r.t. a TBox T iff
the UNA is made and A is consistent w.r.t. T ; or
the UNA is not made and ( (≤ 1 r )) ∈ T implies that there are
no individual names a, b, c occurring in A with r (a, b), r (a, c) ∈ A
and b = c.
Checking whether A is admissible w.r.t. T can be done in polytime
in |A|:
if the UNA is made, checking consistency of A w.r.t. T is equivalent
to a number of instance checking w.r.t. T which is bounded in |A|
if the UNA is not made, simply identify those individual names which
are r -successors of some individual for all globally functional role r ;
this is doable in polytime in |A|

26. Canonical structure
Canonical model for A and T is the limit of the sequence of
interpretations I0 , I1 , . . . defined as follows.
Non-standard representation of interpretations is used: the function
·I maps every element d ∈ ∆I to a set of concept names d I ,
instead of every concept name A to a set of elements AI .
All interpretations satisfy
∆Ii ⊆ { a, p | a ∈ Ind(A) and p ∈ ex ∗ (T )} where
ex ∗ (T ): the set of all paths (sequence of existentials) for T with ε
the empty path
ex(T ) is the set of all existentials (concepts ∃r .A occurring in the
RHS of a GCI in T ) for T
Let Γ be a finite set of concept names. If A ∈ Γ and A ∃r .B ∈ T ,
then Γ has an ∃r .B-obligation O, where O contains
B
those concept names B in T such that there exists A ∈ Γ with
∃r − .A B ∈T
whenever r is globally functional: those concept names B such that
∃A ∈ Γ with A ∃r .B ∈ T .

27. Canonical structure (2)
Start from I0 defined as:
∆I0 = { a, ε | a ∈ Ind(A)}
r I0 = {( a, ε , b, ) | r (a, b) ∈ A}
a, ε I0 = {A ∈ NC | A, T |= A(a)}
aI0 = a, ε
Construct Ii+1 from Ii :
If exists, select a, p ∈ ∆Ii and an α = ∃r .A ∈ ex(T ) s.t. a, p has
α-obligation O, and
r is not globally functional and a, pα ∈ ∆Ii ; or
/
there is no b, p ∈ ∆Ii with ( a, p , b, p ) ∈ r Ii .
Then do the following to get Ii+1 :
add a, pα to ∆Ii
if r is a role name, add ( a, p , a, pα ) to r Ii
if r = s − , add ( a, pα , a, p ) to s Ii
set a, pα Ii to subT (O), the closure of O under subsuming concept
names w.r.t. T .
Assumption: a, p is selected s.t. |p| is minimal (thus all obligations
are eventually satisfied); set ex(T ) is well-ordered and the selected α
is minimal for the node a, p , hence constructed canonical model is
unique.

28. Important lemmas
The canonical model I for T and A is a model of T and A.
Let I be a canonical model for T and A, and J be a model for T
and A. Then there is a homomorphism h from I to J s.t.:
for all individual names a, h(aI ) = aJ ;
for all concept names A and all d ∈ ∆I , d ∈ AI implies h(d) ∈ AJ ;
for all (possibly inverse) roles r and d, e ∈ ∆I , (d, e) ∈ r I implies
(h(d), h(e)) ∈ r J
Let I be a canonical model for A and T , and q a conjunctive query.
Then A, T |= q iff I |= q.
The above lemmas show that we can decide query entailment by
looking at only the canonical model, but the problem is that the
canonical model is infinite.

29. Important lemmas (2)
Idea: if we can show that the canonical model I satisfies q iff it
satisfies q for some match π that maps all variables to elements
reachable by traveling only a bounded number of role edges from
some ABox individual, then we’re done.
Let I be a canonical model for A and T . The initial canonical
model I for A and T is obtained:
∆I = { a, p | |p| ≤ 2m + k}
AI = AI ∩ ∆I
r I = r I ∩ (∆I × ∆I
aI = aI
where m is the size of T and k is the size of q.
Lemma: I |= q iff I |= q.
Polytime (in |A|) construction of initial canonical model:
I0 can be constructed in polytime in the size of A,
obligations can computed in polytime because subsumption in ELI f
is decidable and the required checks are independent of the size of A
the number of elements of initial canonical model is polynomial in
the size of A.

30. Summary
EL extension w.r.t. acyclic TBoxes w.r.t. general TBoxes
EL¬A coNP-complete coNP-complete
ELC D coNP-complete coNP-complete
EL∀r .⊥ , EL∀r .C coNP-complete coNP-complete
EL(≤kr ) , k ≥ 0 coNP-complete coNP-complete
ELkf w/o UNA, k ≥ 2 coNP-complete (even w/o TBox) coNP-complete
ELkf with UNA, k ≥ 2 coNP-complete (in P w/o TBox) coNP-complete
EL(≥kr ) , k ≥ 2 coNP-complete coNP-complete
EL∃¬r .C coNP-hard coNP-hard
EL∃r ∪s.C coNP-hard coNP-hard
+
EL∃r .C coNP-hard coNP-hard
ELI f in P P-complete
For all considered extension, data-tractability can be shown iff the logic is convex
regarding instances, i.e., A, T |= C (a) with C = D0 · · · Dn−1 implies
A, T |= Di (a) for some i < n. (Can it be generalized?)
Subtle differences such as the UNA or local vs. global functionality can have an
impact on data-tractability.

31. Results published in ...
A. Krisnadhi. Data Complexity of Instance Checking in the EL
Family of Description Logics. Master’s thesis, Technische Universit¨t
a
Dresden, March 2007
A. Krisnadhi & C. Lutz. Data Complexity in the EL family of DLs.
In Proc.of the 20th Int. Workshop on Description Logics 2007
(DL2007), p.88–99. 2007.
A. Krisnadhi & C. Lutz. Data Complexity in the EL family of
Description Logics. In Proc. of the 14th Int. Conf. on Logic for
Programming, AI, and Reasoning (LPAR2007), vol. 4790 of LNAI,
p. 333 – 347. 2007.

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