Dismatching and Local Disuniﬁcation in EL (Extended Abstract)

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems.


Motivation
Unification in Description Logics was introduced in [6] as a novel inference service that can be used to detect redundancies in ontologies.For example, assume that one developer of a medical ontology defines the concept of a patient with severe head injury using the EL-concepts Patient ∃finding.(Head_injury∃severity.Severe), whereas another one represents it as Patient ∃finding.(Severe_findingInjury ∃finding_site.Head).
Formally, these expressions are not equivalent, but they are nevertheless meant to represent the same concept.They can obviously be made equivalent by treating the concept names Head_injury and Severe_finding as variables, and substituting them by Injury ∃finding_site.Head and ∃severity.Severe, respectively.In this case, we say that the concepts are unifiable, and call the substitution that makes them equivalent a unifier.In [5], we were able to show that unification in EL is NP-complete.The main idea underlying the proof of this result is to show that any solvable EL-unification problem has a local unifier, i.e., a unifier built from a polynomial number of atoms (concept names or existential restrictions), which are determined by the unification problem.This yields a brute-force NPalgorithm for unification, which guesses a local substitution and then checks (in polynomial time) whether it is a unifier.Intuitively, a unifier of two EL concepts proposes definitions for the concept names that are used as variables: in our example, we know that, if we define Head_injury as Injury ∃finding_site.Head and Severe_finding as ∃severity.Severe, then the two concepts (1) and (2) are equivalent w.r.t.these definitions.Of course, this example was constructed such that the unifier (which is local) provides sensible definitions for the concept names used as variables.In general, the existence of a unifier only says that there is a structural similarity between the two concepts.The developer that uses unification needs to inspect the unifier(s) to see whether the definitions it suggests really make sense.For example, the substitution that replaces Head_injury by Patient Injury ∃finding_site.Head and Severe_finding by Patient ∃severity.Severe is also a local unifier, which however does not make sense.Unfortunately, even small unification problems like the one in our example can have too many local unifiers for manual inspection.We propose disunification to avoid local unifiers that do not make sense.In addition to positive constraints (requiring equivalence or subsumption between concepts), a disunification problem may also contain negative constraints (preventing equivalence or subsumption between concepts).In our example, the nonsensical unifier can be avoided by adding the dissubsumption constraint to the equivalence constraint (1) ≡ ?(2).Disunification in DLs is closely related to unification and admissibility in modal logics [7,[10][11][12][13][14][15], as well as (dis)unification modulo equational theories [5,6,8,9].In the following, we shortly describe the ideas behind our work.More details can be found in [2,3].

Preliminaries
We designate certain concept names as variables, while all others are constants.An EL-concept is ground if it contains no variables.We consider (basic) disunification problems, which are conjunctions of subsumptions C ? D and dissubsumptions C ? D between concepts C, D. 1 A substitution maps each variable to a ground concept, and can be extended to concepts as usual.A substitution σ solves a disunification problem Γ if the (dis)subsumptions of Γ become valid when applying σ on both sides.We restrict σ to a finite signature of concept and role names and do not allow variables to occur in a solution-it would not make sense for the new definitions to extend the vocabulary of the ontologies under consideration, nor to define variables in terms of themselves.
In the following, we consider a flat disunification problem Γ, i.e. one that contains only (dis)subsumptions where both sides are conjunctions of flat atoms of the form A or ∃r.A, for a role name r and a concept name A. We denote by At the set of all such atoms that occur in Γ, by Var the set of variables occurring in Γ, and by At nv := At \ Var the set of non-variable atoms of Γ.Let S : Var → 2 Atnv be an assignment, i.e. a function that assigns to each variable X ∈ Var a set S X ⊆ At nv .The relation > S on Var is defined as the transitive closure of {(X, Y ) ∈ Var 2 | Y occurs in an atom of S X }.If > S is irreflexive, then S is called acyclic.In this case, we can define the substitution σ S inductively along > S as follows: if X is minimal, then σ S (X) := D∈S X D; otherwise, assume that σ S (Y ) is defined for all Y ∈ Var with X > Y , and define σ S (X) := D∈S X σ S (D).All substitutions of this form are called local.

Results
Unification in EL is local : each problem Γ can be transformed into an equivalent flat problem that has a local solution iff Γ is solvable, and hence (general) solvability of unification problems in EL is in NP [5].However, disunification in EL The decidability and complexity of general solvability of disunification problems is still open.However, we can show that each dismatching problem Γ, which is a disunification problem where one side of each dissubsumption must be ground, can be polynomially reduced to a flat problem that has a local solution iff Γ is solvable.This shows that deciding solvability of dismatching problems in EL is in NP.The main idea is to introduce auxiliary variables and flat atoms that allow us to solve the dissubsumptions using a local substitution.For example, we replace the dissubsumption ?Y from above with Y ?∃r.Z.The rule we applied here is the following: Solving Left-Ground Dissubsumptions: Condition: This rule applies to s = C1 • • • Cn ?X if X is a variable and C1, . . ., Cn are ground atoms.Action: Choose one of the following options: -Choose a constant A ∈ Σ, replace s by -Choose a role r ∈ Σ, introduce a new variable Z, replace s by X ?∃r.Z, C1 ?∃r.Z, . . ., Cn ?∃r.Z.
According to the rule, we can choose a constant or create a new existential restriction with a fresh variable, and use it in the new subsumption and dissubsumptions.In our example the left hand side of the dissubsumption ?Y is empty, hence only a subsumption is produced.
However, the brute-force NP-algorithm for checking local solvability of the resulting flat disunification problem is hardly practical.For this reason, we have extended the rule-based algorithm from [5] and the SAT reduction from [4] by additional rules and propositional clauses, respectively, to deal with dissubsumptions.The reason we extend both algorithms is that, in the case of unification, they have proved to complement each other well in first evaluations [1]: the goal-oriented algorithm needs less memory and finds minimal solutions faster, while the SAT reduction generates larger data structures, but outperforms the goal-oriented algorithm on unsolvable problems.The SAT reduction has been implemented in our prototype system UEL. 2 First experiments show that dismatching is indeed helpful for reducing the number and the size of unifiers.The runtime performance of the solver for dismatching problems is comparable to the one for pure unification problems.
is not local in this sense: consider Γ := {X ?B, A B C ? X, ∃r.X ?Y, ?Y, Y ?∃r.B} with variables X, Y and constants A, B, C. If we set σ(X) := A B C and σ(Y ) := ∃r.(A C), then σ is a solution of Γ that is not local.This is because ∃r.(A C) is not a substitution of any non-variable atom in Γ. Assume now that Γ has a local solution γ.Since γ must solve the first dissubsumption, γ(Y ) cannot be , and due to the third subsumption, none of the constants A, B, C can be a conjunct of γ(Y ).The remaining atoms ∃r.γ(X) and ∃r.B are ruled out by the last dissubsumption since both γ(X) and B are subsumed by B. This shows that Γ cannot have a local solution, although it is solvable.