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# Data Complexity in EL Family of Description Logics

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• 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.
• 32. Referenced works F. Baader, S. Brandt, and C. Lutz. “Pushing the EL envelope”. In Proc. of the 19th Int. Joint Conf. on AI (IJCAI-05), pages 364–369. Morgan Kaufmann, 2005. (And its accompanying technical report). D. Calvanese, G. D. Giacomo, D. Lembo, M. Lenzerini, and R. Rosati. “Data complexity of query answering in description logics”. In Proc. of the 10th Int. Conf. on KR (KR06). AAAI Press, 2006. U. Hustadt, B. Motik, and U. Sattler. “Data complexity of reasoning in very expressive description logics”. In Proc. of the 19th Int. Joint Conf. on AI (IJCAI05), pages 466–471. Professional Book Center, 2005. B. Glimm, I. Horrocks, C. Lutz and U. Sattler. “Conjunctive Query Answering for the Description Logic SHIQ”. In Proc. of the 20th Int. Joint Conf. on AI (IJCAI-07). AAAI Press, 2007. A. Schaerf. “On the complexity of the instance checking problem in concept languages with existential quantiﬁcation”. Journal of Intelligent Information Systems, 2:265–278, 1993.