Matching concept descriptions against concept patterns was introduced as a new inference task in Description Logics (DLs) almost 20 years ago, motivated by applications in the Classic system. For the DL EL, it was shown in 2000 that the matching problem is NP-complete. It then took almost 10 years before this NP-completeness result could be extended from matching to unification in EL. The next big challenge was then to further extend these results from matching and unification without a TBox to matching and unification w.r.t. a general TBox, i.e., a finite set of general concept inclusions. For unification, we could show some partial results for general TBoxes that satisfy a certain restriction on cyclic dependencies between concepts, but the general case is still open. For matching, we solve the general case in this paper: we show that matching in EL w.r.t. general TBoxes is NP-complete by introducing a goal-oriented matching algorithm that uses non-deterministic rules to transform a given matching problem into a solved form by a polynomial number of rule applications. We also investigate some tractable variants of the matching problem.
The DL E L, which offers the constructors conjunction (u), existential restriction (9r:C), and the top concept (>), has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial in E L, even in the presence of general concept inclusions (GCIs) [12]. On the other hand, though quite inexpressive, E L can be used to define biomedical ontologies, such as the large medical ontology SNOMED CT.1
Matching of concept descriptions against concept patterns is a non-standard inference task in Description Logics, which was originally motivated by applications of the Classic system [9]. In [11], Borgida and McGuinness proposed matching as a means to filter out the unimportant aspects of large concept descriptions appearing in knowledge bases of Classic. Subsequently, matching (as well as the more general problem of unification) was also proposed as a tool for detecting redundancies in knowledge bases [8] and to support the integration of ? Supported by DFG under grant BA 1122/14-2 1 see http://www.ihtsdo.org/snomed-ct/ knowledge bases by prompting possible interschema assertions to the integrator [10].
All three applications have in common that one wants to search the knowledge base for concepts having a certain (not completely specified) form. This “form” can be expressed with the help of so-called concept patterns, i.e., concept descriptions containing variables (which stand for descriptions). For example, assume that we want to find concepts that are concerned with individuals having a son and a daughter sharing some characteristic. This can be expressed by the pattern D := 9has-child:(Male u X) u 9has-child:(Female u X), where X is a variable standing for the common characteristic. The concept description C := 9has-child:(Tall u Male) u 9has-child:(Tall u Female) matches this pattern in the sense that, if we replace the variable X by the description Tall, the pattern becomes equivalent to the description. Thus, the substitution := fX 7! Tallg is a matcher modulo equivalence of the matching problem C ? D since C (D). The original paper by Borgida and McGuinness actually considered matching modulo subsumption rather than matching modulo equivalence: such a problem is of the form C v? D, and a matcher is a substitution satisfying C v (D). Obviously, any matcher modulo equivalence is also a matcher modulo subsumption, but not vice versa. For example, the substitution > := fX 7! >g is a matcher modulo subsumption of the matching problem C v? D, but it is not a matcher modulo equivalence of C ? D.
The first results on matching in DLs were concerned with sublanguages of the Classic description language, which does not allow for existential restrictions of the kind used in our example. A polynomial-time algorithm for computing matchers modulo subsumption for a rather expressive DL was introduced in [11]. The main drawback of this algorithm was that it required the concept patterns to be in structural normal form, and thus it was not able to handle arbitrary matching problems. In addition, the algorithm was incomplete, i.e., it did not always find a matcher, even if one existed. For the DL ALN , a polynomial-time algorithm for matching modulo subsumption and equivalence was presented in [6]. This algorithm is complete and it applies to arbitrary patterns. In [5], matching in DLs with existential restrictions was investigated for the first time. In particular, it was shown that in E L the matching problem (i.e., the problem of deciding whether a given matching problem has a matcher or not) is polynomial for matching modulo subsumption, but NP-complete for matching modulo equivalence.
Unification is a generalization of matching where both sides of the problem
are patterns and thus the substitution needs to be applied to both sides. In
[8] it was shown that the unification problem in the DL F L0, which offers the
constructors conjunction (u), value restriction (8r:C), and the top concept (>),
is ExpTime-complete. In contrast, unification in E L is “only” NP-complete [7].
In the results for matching and unification mentioned until now, there was no
TBox involved, i.e., equivalence and subsumption was considered with respect
to the empty TBox. For unification in E L, first attempts were made to take
general TBoxes, i.e., finite sets of general concept inclusions (GCIs), into account.
However, the results obtained so far, which are again NP-completeness results,
are restricted to general TBoxes that satisfy a certain restriction on cyclic
dependencies between concepts [
For matching, we solve the general case in this paper: we show that matching in E L w.r.t. general TBoxes is NP-complete by introducing a goal-oriented matching algorithm that uses non-deterministic rules to transform a given matching problem into a solved form by a polynomial number of rule applications. The matching problems considered in this paper are actually generalizations of matching modulo equivalence and matching modulo subsumption. For the special case of matching modulo subsumption, we show that the problem is tractable also in the presence of GCIs. The same is true for the dual problem where the pattern is on the side of the subsumee rather than on the side of the subsumer.
Due to space constraints, we cannot provide complete proofs of our results.
They can be found in [
The expressiveness of a DL is determined both by the formalism for describing concepts (the concept description language) and the terminological formalism, which can be used to state additional constraints on the interpretation of concepts and roles in a so-called TBox.
The concept description language considered in this paper is called E L. Starting with a finite set NC of concept names and a finite set NR of role names, E L-concept descriptions are built from concept names using the constructors conjunction (C u D), existential restriction (9r:C for every r 2 NR), and top (>). Since in this paper we only consider E L-concept descriptions, we will sometimes dispense with the prefix E L.
On the semantic side, concept descriptions are interpreted as sets. To be more precise, an interpretation I = ( I ; I ) consists of a non-empty domain I and an interpretation function I that maps concept names to subsets of I and role names to binary relations over I . This function is inductively extended to concept descriptions as follows: >I :=
I ; (C u D)I := CI \ DI ; (9r:C)I := fx j 9y : (x; y) 2 rI ^ y 2 CI g A general concept inclusion axiom (GCI) is of the form C v D for concept descriptions C; D. An interpretation I satisfies such an axiom C v D iff CI DI : A general E L-TBox is a finite set of GCIs. An interpretation is a model of a general E L-TBox if it satisfies all its GCIs.
A concept description C is subsumed by a concept description D w.r.t. a general TBox T (written C vT D) if every model of T satisfies the GCI C v D. We say that C is equivalent to D w.r.t. T (C T D) if C vT D and D vT C. If T is empty, we also write C v D and C D instead of C vT D and C T D, respectively. As shown in [12], subsumption w.r.t. general E L-TBoxes is decidable in polynomial time.
An E L-concept description is an atom if it is an existential restriction or a concept name. The atoms of an E L-concept description C are the subdescriptions of C that are atoms, and the top-level atoms of C are the atoms occurring in the top-level conjunction of C. Obviously, any E L-concept description is the conjunction of its top-level atoms, where the empty conjunction corresponds to >. The atoms of a general E L-TBox T are the atoms of all the concept descriptions occurring in GCIs of T .
We say that a subsumption between two atoms is structural if their top-level
structure is compatible. To be more precise, following [
(a) A1 u (b) for every
s Dj .
(c) B vT 3
It is easy to see that subsumption w.r.t. ; between two atoms implies structural
subsumption w.r.t. T , which in turn implies subsumption w.r.t. T . The matching
algorithms presented below crucially depend on the following characterization
of subsumption w.r.t. general E L-TBoxes first stated in [
An E L-concept pattern C is ground if it does not contain variables, i.e., if it is a concept description. Obviously, a ground concept pattern is not modified by applying a substitution.
Definition 2. Let T be a general E L-TBox.2 An E L-matching problem w.r.t. T is a finite set = fC1 v? D1; : : : ; Cn v? Dng of subsumptions between E Lconcept patterns, where for each i; 1 i n, Ci or Di is ground. A substitution is a matcher of w.r.t. T if solves all the subsumptions in , i.e. if (C1) vT (D1); : : : ; (Cn) vT (Dn). We say that is matchable w.r.t. T if it has a matcher.
Matching problems modulo equivalence and subsumption are special cases of the matching problems introduced above: – The E L-matching problem is a matching problem modulo equivalence if C v? D 2 implies D v? C 2 . This coincides with the notion of matching modulo equivalence considered in [6,5], but extended to a non-empty general TBox. – The E L-matching problem is a left-ground matching problem modulo subsumption if C v? D 2 implies that C is ground. This coincides with the notion of matching modulo subsumption considered in [6,5], but again extended to a non-empty general TBox. – The E L-matching problem is a right-ground matching problem modulo subsumption if C v? D 2 implies that D is ground. To the best of our knowledge, this notion of matching has not been investigated before. We will show in the following that the general case of matching, as introduced in Definition 2, and thus also matching modulo equivalence, is NP-complete, whereas the two notions of matching modulo subsumption are tractable, even in the presence of GCIs. 4
Matching Modulo Subsumption The case of left-ground matching problems modulo subsumption can be treated as sketched in [5] for the case without a TBox. Given a general E L-TBox T and two substitutions ; , we define vT iff (X) vT (X) for all X 2 NV : Consequently, if we define > as the substitution satisfying >(X) = > for all X 2 NV , then vT > holds for all substitutions . Since the concept constructors of E L are monotonic w.r.t. subsumption, this implies (D) vT >(D) for all concept patterns D.
Lemma 3. Let = fC1 v? D1; : : : ; Cn v? Dng be a left-ground matching problem modulo subsumption. Then has a matcher w.r.t. T iff > is a matcher of w.r.t. T . 2 Note that the GCIs in T are built using concept descriptions, and thus do not contain variables.
Proof. The “if” direction is trivial. To see the “only-if” direction, assume that is a matcher of w.r.t. T . Then we have, for all i; 1 i n, that Ci = (Ci) vT (Di) vT >(Di), which shows that > is a matcher of T . >(Ci) = w.r.t.
tu
The lemma shows that it is sufficient to test whether the substitution > is a matcher of , i.e., whether >(Ci) vT >(Di) holds for all i; 1 i n. Since in E L subsumption w.r.t. general TBoxes is decidable in polynomial time, this yields a polynomial-time algorithm for left-ground matching modulo subsumption in E L.
Theorem 4. Let be a left-ground E L-matching problem modulo subsumption and T a general E L-TBox. Then we can decide in polynomial time whether has a matcher w.r.t. T or not.
The case of right-ground matching problems modulo subsumption can be treated similarly. However, since E L does not have the bottom concept ? as a concept constructor, we cannot simply define ? as the substitution satisfying ?(X) = ? for all X 2 NV , and then show that that the right-ground matching problems modulo subsumption, , has a matcher w.r.t. T iff ? is a matcher of w.r.t. T . Instead, we need to define ? in a more complicated manner.
Given a general E L-TBox T and a right-ground matching problems modulo subsumption = fC1 v? D1; : : : ; Cn v? Dng, we use ?( ; T ) to denote the E L-concept description that is the conjunction of all the atoms of T and of D1; : : : ; Dn. We now define ?( ;T ) as the substitution satisfying ?( ;T )(X) = ?( ; T ) for all X 2 NV Lemma 5. Let = fC1 v? D1; : : : ; Cn v? Dng be a right-ground matching problem modulo subsumption. Then has a matcher w.r.t. T iff ?( ;T ) is a matcher of w.r.t. T .
Proof. The “if” direction is trivial. To see the “only-if” direction, assume that is a matcher of w.r.t. T . We need to show that this implies the ?( ;T ) is also a matcher of w.r.t. T , i.e., that it satisfies ?( ;T )(C) vT ?( ;T )(D) for every subsumption C v? D 2 .
More generally, we consider subsumptions C v? D where C is a subpattern of a pattern occurring in or T and D is an atom of T or D1; : : : ; Dn. We show the following claim: ?( ;T )(C) vT
?( ;T )(D).
Claim: For every such subsumption C v? D, it holds that (C) vT plies (D) im
Before proving the claim, let us show that this implies that ?( ;T ) solves w.r.t. T . In fact, any subsumption in is of the form C v? E1 u : : : u Ek where C is a subpattern of a pattern occurring in , and E1; : : : ; Ek are atoms of one of the Di. In addition, a substitution solves C v? E1 u : : : u Ek w.r.t. T iff it solves all the subsumptions C v? Ei for i = 1; : : : ; k.
We prove the claim by induction on the size jCj of the left-hand side C of the subsumption C v? D. Since D is ground, (C) vT (D) implies (C) vT D. Let C = F1 u : : : u F`, where F1; : : : ; F` are atoms. We distinguish the following three cases: 1. If there is an index i 2 f1; : : : ; `g such that Fi is a variable, then ?( ;T )(Fi) v
D since D occurs as a conjunct in ?( ; T ). This implies ?( ;T )(C) vT D. 2. If there is an index i 2 f1; : : : ; `g such that Fi is ground and (Fi) vT D, then ?( ;T )(Fi) = Fi = (Fi) vT D. This again implies ?( ;T )(C) vT D. 3. Assume that the above two cases do not hold. Using Lemma 1, we can distinguish two more cases, depending on whether the first or the second condition of the lemma applies. (a) If the first condition applies, then there is an index i 2 f1; : : : ; `g such that Fi vsT D. Since Fi is neither ground nor a variable, we know that Fi is a non-ground existential restriction. Thus, Fi = 9r:F 0, D = 9r:(D1 u : : : u Dm) with D1; : : : ; Dm atoms, and (F 0) vT Di for all i 2 f1; : : : ; mg. Since F 0 is a subpattern of C, Di are atoms of D, and jF 0j < jCj, we can apply the induction hypothesis to the subsumptions F 0 v? Di. This yields ?( ;T )(F 0) vT Di for all i 2 f1; : : : ; mg, and thus ?( ;T )(C) vT D. (b) If the second condition applies, then there are atoms A1; : : : ; Ak; B of T such that A1 u u Ak vT B vT D and for each 2 f1; : : : ; kg, there is j 2 f1; : : : ; `g such that i. Fj is a concept variable and (Fj ) vT A , or ii. Fj is ground and Fj vT A , or iii. Fj = 9r:F 0, A = 9r:A0 and (F 0) vT A0.
It is sufficient to show that the subsumption relationships in 3(b)i and 3(b)iii also hold if we replace by ?( ;T ). For 3(b)i this can be shown as in 1 and for 3(b)iii as in 3a.
This completes the proof of the claim, and thus of the lemma.
Since the size of ?( ; T ) is polynomial in the size of and T , this lemma yields a polynomial-time decision procedure for right-ground matching modulo subsumption.
Theorem 6. Let be a right-ground E L-matching problem modulo subsumption and T a general E L-TBox. Then we can decide in polynomial time whether has a matcher w.r.t. T or not. tu 5
The General Case NP-hardness for the general case follows from the known NP-hardness result for matching modulo equivalence without a TBox [5]. In the following, we show that matching in E L w.r.t. general TBoxes is in NP by introducing a goaloriented matching algorithm that uses non-deterministic rules to transform a given matching problem into a solved form by a polynomial number of rule applications.
Let T be a general E L-TBox and 0 an E L-matching problem. We can assume without loss of generality that all the subsumptions C v? D in 0 are such that either C or D is non-ground. In fact, if both C and D are ground, then the following holds: – If C vT D, then 0 has a matcher w.r.t. T iff 0 n fC v? Dg has a matcher w.r.t. T .
– If C 6vT D, then 0 does not have a matcher w.r.t. T .
Consequently, we can either remove all the offending ground subsumptions without changing the solvability status of the problem, or immediately decide nonsolvability. Using the fact that C vT D1 u D2 iff C vT D1 and C vT D2, we can additionally normalize 0 such that the right-hand side of each subsumption in 0 is an atom. We call an E L-matching problem normalized if C v? D 2 0 implies that (i) either C or D is non-ground, and (ii) D is an atom.
Thus, assume that 0 is a normalized E L-matching problem. Our algorithm starts with := 0, and then applies non-deterministic rules to . A nonfailing application of a rule may add subsumptions to . Note, however, that a subsumption is only added if it is not yet present. New subsumptions that are added are marked as “unsolved,” as are initially all the subsumptions of 0. A rule application may fail, which means that this attempt of solving the matching problem was not successful. A non-failing rule application marks one of the subsumptions in the matching problem as “solved.” Rules are applied until all subsumptions are marked “solved” or an attempt to apply a rule has failed.
Our definition of the rules uses a function Dec(: : : ) on subsumptions of the form C v? D, where C and D are atoms and D is not a variable. A call of Dec(C v? D) returns a (possibly empty) set of subsumptions or it fails: 1. Dec(C v? D) := fC v? Dg, if C is a variable. 2. If D1; : : : ; Dn are atoms, then Dec(9r:C0 v? 9r:(D1 u u Dn)) fails if there is an i 2 f1; : : : ; ng such that both sides of C0 v? Di are ground and C0 6vT Di. Otherwise, Dec(9r:C0 v? 9r:(D1 u u Dn)) := fC0 v? Di j 1 i n and C0 or Di is non-groundg. 3. If C = 9r:C0 and D = 9s:D0 for roles s 6= r, then Dec(C v? D) fails. 4. If C = A is a concept name and D = 9r:D0 an existential restriction, then
Dec(C v? D) fails. 5. If D = A is a concept name and C = 9r:C0 an existential restriction, then
Dec(C v? D) fails. 6. If both C and D are ground and C 6vO D then Dec(C v? D) fails, and otherwise returns ;.
Algorithm 7. Let 0 be a normalized E L-matching problem. Starting with := 0, apply the rules of Figure 1 and Figure 2 exhaustively in the following order: Eager Solving – variable on the right:
Action: – If there is some subsumption of the form X v? D 2 then the rule application fails.
– Otherwise, mark C v? X as “solved.” Eager Solving – variable on the left:
Action: – If there is some subsumption of the form C v? X 2
then the rule application fails. – Otherwise, mark X v? D as “solved." where X 2 NV . where X 2 NV .
such that C 6vT D, such that C 6vT D, (1) Eager rule application: If an eager rule from Figure 1 applies, apply it and if it fails, stop and return “failure.” (2) Non-deterministic rule application: If no eager rule is applicable, let s be an unsolved subsumption in . Choose one of the non-deterministic rules of Figure 2, and apply it to s. If this rule application fails, then stop and return “failure.” If no more rule applies and the algorithm has not stopped returning “failure,” then return “success.”
In (2), the choice which unsolved subsumption to consider next is don’t care non-deterministic. However, choosing which rule to apply to the chosen subsumption is don’t know non-deterministic. Additionally, the application of a non-deterministic rules may require don’t know non-deterministic choices to be made. If a non-deterministic rule is applied to a subsumption s, then neither its left-hand side nor its right-hand side is a variable. In fact, a subsumption that has a variable on one of its sides is solved by one of the eager rules, which have precedence over the non-deterministic rules.
It is easy to see that the subsumptions added by the non-deterministic rules satisfy the normalization conditions (i) and (ii), and thus all the sets generated during a run of the algorithm are normalized E L-matching problems. The next lemma states an important property ensured by the presence of the eager rules. Lemma 8. If is a matching problem generated during a non-failing run of the algorithm, and both C v? X 2 and X v? D 2 are solved, then C vT D. Proof. Obviously, one of the two subsumptions was solved after the other. This means that, when it was solved by the application of an eager rule, the other one was already present. Since we consider a non-failing run, the application of the eager rule did not fail, which yields C vT D. tu Mutation :
Condition: This rule applies to s = C1 u u Cn v? D 2 .
Action: Its application chooses an index i 2 f1; : : : ; ng and calls Dec(Ci v? D). If this call does not fail, then it adds the returned subsumptions to , and marks s as solved. If Dec(Ci v? D) fails, it returns “failure.” Condition: This rule applies to s = C1 u u Cn v? D in .
Action: Its application tries to choose atoms A1; : : : ; Ak; B of T such that A1 u u Ak vT B holds. If this is not possible, then it returns “failure.” Otherwise, it performs the following two steps: – Choose for each 2 f1; : : : ; kg an i 2 f1; : : : ; ng and call Dec(Ci v? A ). If this call does not fail, it adds the returned subsumptions to . Otherwise, if Dec(Ci v? A ) fails, the rule returns “failure.” – If it has not failed before and Dec(B v? D) does not fail, it adds the returned subsumptions to . Otherwise, if Dec(B v? D) fails, it returns "failure". If these steps did not fail, then the rule marks s as solved.
Any run of the algorithm terminates after a polynomial number of steps. The main reason for this is that there are only polynomially many subsumptions that can occur in the matching problems generated during a run.
Lemma 9. Let be a matching problem generated during a run of Algorithm 7. Then any subsumption occurring in is of one of the following forms: 1. A subsumption contained in the original input matching problem 0. 2. A subsumption of the form C v? D where C; D are subpatterns of concept patterns occurring in 0. 3. A subsumption of the form C v? A or A v? C where A is an atom of T and C is a subpattern of a concept pattern occurring in 0.
Since any rule application either fails while trying to solve an unsolved subsumption (in which case the algorithm stops immediately) or actually solves an unsolved subsumption, there can be only polynomially many rule applications during a run. In addition, it is easy to see that each rule application can be realized in polynomial time, with a polynomial number of possible non-deterministic choices. This shows that Algorithm 7 is indeed an NP-algorithm. It remains to show that it is sound and complete.
To show soundness, assume that is a matching problem obtained after termination of a non-failing run of the algorithm. Since the run terminated without failure, all the subsumptions in are solved. We use the subsumptions of the form X v? C 2 to define a substitution . Note that the fact that is a normalized E L-matching problem implies that C is a ground pattern, i.e., a concept description. For each variable X 2 NV , we define
SX := fC j X v? C 2 g; and denote the conjunction of all the elements of SX as uSX , where the empty conjunction is >. The substitution is now defined as
(X) := uSX for all X 2 NV : Lemma 10.
is a matcher of
w.r.t. T .
Since the input matching problem 0 is contained in , this shows that is a matcher also of 0 w.r.t. T . This completes the proof of soundness.
Regarding completeness, we can use a given matcher of 0 w.r.t. T to guide the application of the non-deterministic rules such that a non-failing run is generated.
Lemma 11. Let be a matcher of 0 w.r.t. T . Then there is a non-failing and terminating run of Algorithm 7 producing a matching problem such that is a matcher of w.r.t. T .
This lemma provides the final step towards showing that Algorithm 7 is an NP-decision procedure for matching w.r.t. general TBoxes in E L. Theorem 12. The problem of deciding whether a given E L-matching problem has a matcher w.r.t. a given general E L-TBox or not is NP-complete. 6
Conclusion We have extended the known results for matching in E L [5] to the case where subsumption and equivalence is considered w.r.t. a non-empty general TBox, i.e., a non-empty set of GCIs. For the DL F L0, matching without GCIs is polynomial, and this remains true even in the extension ALN of F L0. It would be interesting to see how one can solve matching problems w.r.t. general TBoxes in these DLs. Since already subsumption in F L0 w.r.t. general TBoxes is ExpTime-complete [4], the complexity of solving such matching problems is at least ExpTime-hard. Another interesting open problem is unification in E L w.r.t. general TBoxes. 3. Baader, F., Borgwardt, S., Morawska, B.: A goal-oriented algorithm for unification in ELHR+ w.r.t. cycle-restricted ontologies. In: Thielscher, M., Zhang, D. (eds.) Pro. of 25th Australasian Joint Conf. on Artificial Intelligence (AI’12). Lecture Notes in Artificial Intelligence, vol. 7691, pp. 493–504. Springer-Verlag (2012) 4. Baader, F., Brandt, S., Lutz, C.: Pushing the EL envelope. In: Kaelbling, L.P., Saffiotti, A. (eds.) Proc. of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005). pp. 364–369. Morgan Kaufmann, Los Altos, Edinburgh (UK) (2005) 5. Baader, F., Küsters, R.: Matching in description logics with existential restrictions.
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