Minimizing conservativity violations in ontology alignments: algorithms and evaluation

Abstract

In order to enable interoperability between ontology-based systems, ontology matching techniques have been proposed. However, when the generated mappings lead to undesired logical consequences, their usefulness may be diminished. In this paper, we present an approach to detect and minimize the violations of the so-called conservativity principle where novel subsumption entailments between named concepts in one of the input ontologies are considered as unwanted. The practical applicability of the proposed approach is experimentally demonstrated on the datasets from the Ontology Alignment Evaluation Initiative.

Appendix

This section investigates the computational complexity of diagnosis computation (“Diagnosis Computation Complexity” in section “Appendix”) and its decomposability into subproblems (“Decomposability of equivalence violations diagnosis computation” in section “Appendix”).

1.1 Diagnosis computation complexity

With the aim of proving that MAP-WFES is NP-hard, Proposition 10.1 introduces a polynomial reduction from WFES to MAP-WFES, denoted as WFES $\preceq$ MAP-WFES. The intuitive idea behind the reduction is the following. Each arc $(t,v,c)$ of the original graph is “split” into two arcs $(t,u,c)$ and $(u,v,c)$, with u a fresh node. All the nodes t, v are associated with one of the input ontology, while the fresh nodes are associated with the other. In this way, all the arcs are mappings (i.e., all of them are potentially removable, exactly as the original arcs). It is easy to see that the reduction preserves the solution weight and that a 1–1 correspondence exists between cycles in the two graphs. In addition, we remark that MAP-WFES does not break cycles traversing only vertices of one of the input ontologies. No such cycles can exist because all the arcs are mappings, as discussed above. A reduction example is given in Example 10.1, followed by the definition of the reduction in Proposition 10.1.

Fig. 11

Input graph G for WFES (left) reduced to a corresponding graph $G'$, input for MAP-WFES (right). In graph $G'$, all the round-shaped vertices belong to $V_1'$, while square-shaped vertices belong to $V_2'$. a Graph G (reduction input) b Graph $G'$ (reduction output)

Example 10.1

In Fig. 11, graphs G (left) and $G'$ (right) are shown. WFES $\preceq$ MAP-WFES coincides with $G'$. The solution to $G'$ is equal to $\Delta = \{ (c,cd,1), (b,bg,0.1), (gf,f,0.4), (a,af,0.2) \}$, with a total weight of 1.7. $D= \{ (b,g,0.1), (g,f,0.4),(a,f,0.2), (c,d,1) \}$ is the corresponding solution to the instance of the WFES problem represented by G and can be easily verified that is both minimal (having weight 1.7) and correct.

Proposition 10.1

WFES $\preceq$ MAP-WFES. A polynomial reduction from the WFES problem to the MAP-WFES problem exists. Let $G=(V,A)$ be a digraph. The reduction consists in constructing a digraph $G'=(V',A')$ such that a subset of edges, namely $\Delta \subseteq A$, is a solution to MAP-WFES iff the corresponding set of arcs, namely D, is a solution to WFES on G. The reduction is as follows:

• 1. for each $(x,y,c) \in A$, we create a fresh vertex $v_{xy}$, we add it to $V_2'$, and we create a pair of arcs $(x,v_{xy},c), (v_{xy},y,c)$ that are added to $A'$ and $\mathcal {M}$,

• 2. $V_1' = V$ and $V' = V_1' \cup V_2'$.

A set of arcs, namely $\Delta \subseteq A'$, is a solution to the MAP-WFES problem on digraph $G'$ iff the corresponding feedback edge set D is a solution to the WFES G, where $G'$ is computed from G, and for each arc of the form $(x,v_{xy},c)$ or $(v_{xy},y,c)$ in $\Delta$, we have a corresponding arc $(x,y,c)$ in D.

Proof

In order to prove the correctness of the reduction, we need to show that, if G $\preceq$ $G'$, with G an instance of the WFES problem and $G'$ an instance of the MAP-WFES problem, a set of arcs $\Delta$ is a (minimal) solution to $G'$ iff the corresponding set of arcs D is a (minimal) solution to G. As discussed in  [16], the proposed reduction is polynomial and it preserves graph connectivity and the weight of the solutions, by preserving a 1–1 correspondence between cycles of G and those of $G'$, due to the 1–1 correspondency between the arcs of A and those of $A'$.

$\Rightarrow$: If D is a solution to WFES on G, $\Delta$ is a solution to MAP-WFES on $G'$. Suppose that $\Delta$ is not a solution. This requires that either at least a cycle $\kappa '$ exists in digraph $(V',A'{\setminus }\Delta )$ or that a diagnosis $\Delta '$ exists such that $w(\Delta ') < w(\Delta )$. For the first case, given the 1–1 correspondence between cycles of G and $G'$, this implies that a corresponding cycle in G exists as well, thus contradicting that D is a solution to the instance of the WFES problem represented by G. For the latter case, given the 1–1 correspondence between arcs of G and $G'$, this implies that a solution $D'$ corresponding to $\Delta '$ exists. By the weight preservation property of the reduction, $w(D') < w(D)$ holds, contradicting that D is a (minimal) solution to the instance of the WFES problem represented by G.

$\Leftarrow$: If $\Delta$ is a solution to MAP-WFES on $G'$, D is a solution to WFES on G. Suppose that D is not. This requires that either a cycle $\kappa$ of G exists in digraph $(V,A {\setminus } D)$ or that a solution $D'$ exists such that $w(D') < w(D)$. The first case requires the existence of a cycle $\kappa '$ of $G'$ (corresponding to $\kappa$) that is left unbroken. In turn, this either violates that $\Delta$ is a solution, or that, for some $i \in \{1,2\}$, $\kappa '$ exclusively traverses elements of $V'_i$. This situation is excluded by construction of $G'$, because no arcs between vertices of the same subset $V'_i$ of $V'$ exist. For the latter case, this implies that a diagnosis $\Delta '$ corresponding to the (minimal) solution $D'$ does not exist. This requires that at least an element of $D'$ cannot belong to a diagnosis (i.e., it cannot be removed). By construction of $G'$, we have that $\mathcal {M} = A'$. Given that only elements of $A' {\setminus } \mathcal {M}$ cannot belong to a diagnosis for the MAP-WFES problem, this results in a contradiction, thus proving the correctness of the reduction. $\square$

From the results of Proposition 10.1, it follows that MAP-WFES is NP-hard, as detailed in Proposition 10.2.

Proposition 10.2

MAP-WFES is NP-hard.

Proof

The proof follows from the polynomial reduction from the WFES problem, that is NP-hard  [16], to MAP-WFES. $\square$

1.2 Decomposability of equivalence violations diagnosis computation

Proposition 10.3 relates unsafe cycles and problematic SCCs, showing that each unsafe cycle results in a problematic SCC.

Proposition 10.3

A SCC is problematic iff it (totally) contains at least one unsafe cycle.

Proof

$\Rightarrow$: Consider a problematic SCC S, with projections $\Pi _1$ and $\Pi _2$ on the input ontologies. From problematic SCC definition (Definition 4.6), at least one of the projections of S, say $\Pi _1$, is not a local SCC. We therefore also know that $\Pi _1$ is not a SCC; otherwise, it would also be a local SCC. Suppose that $\Pi _1$ is a subset of a SCC $\Pi '$. This implies that all the elements of $\Pi '$ belongs to S as well, and therefore, $\Pi '$ and $\Pi _1$ are identical, but this contradicts the assumption that $\Pi _1$ is not a SCC.

$\Leftarrow :$ from the definition of cycle and SCC, each cycle $\kappa$ is contained in a SCC S (i.e., $\kappa \subseteq S$). Let $\kappa$ be an unsafe cycle, and let also $\kappa _1$ (resp. $\kappa _2$) be the subset of vertices of $\kappa$ belonging to an input ontology ${\mathcal {O}}_1$ (resp. ${\mathcal {O}}_2$). By definition of unsafe cycle (Definition 4.3), at least one of this subsets, say $\kappa _1$, is not contained in any local SCC. But given that $\kappa \subseteq S$, $\kappa _1 \subseteq \Pi _{{\mathcal {O}}_1}(S)$ holds. Therefore, $\Pi _{{\mathcal {O}}_1}(S)$ is not contained in any local SCC either and, by Definition 4.6, S is a problematic SCC. $\square$

Proposition 10.3 guarantees completeness for a detection technique for violations to the conservativity principle on a graph representation of an aligned ontology, based on problematic SCCs. Given that all the violations result in unsafe cycles and that they totally belong to a single problematic SCC, completeness for a repair technique breaking all the unsafe cycles follows.

Notice also that a (unsafe) cycle always belongs to one and only one (problematic) SCC (as expressed by Proposition 10.4), while a problematic SCC may contain more than one cycle. Therefore, a technique detecting problematic SCCs may be more efficient than one directly addressing unsafe cycles.

Proposition 10.4

Safe cycle never traverse multiple SCCs of the same input ontology.

Proof

By Definition 4.3, all the vertices belonging to a projection $\Pi$ of a safe cycle $\kappa ^s$ need to be traversed by a cycle $\kappa '$ in the input ontology these vertices belong to. The claim is that cycle $\kappa '$ identifies either a SCC of the aligned ontology or a subset of a SCC. Assume that vertices of $\Pi$ belong to at least two SCCs $S_1, S_2$, that is, $\Pi \cap S_1 \ne \emptyset$ and $\Pi \cap S_2 \ne \emptyset$. Being traversed by a cycle, all the vertices of $\Pi$ are mutually reachable. Then, from transitivity of reachability, it follows that all the vertices in $S_1 \cup S_2$ are mutually reachable. This contradicts the hypothesis that $S_1$ and $S_2$ are two distinct SCCs, thus proving the proposition. Such argument be can straightforwardly generalized to more than two SCCs. $\square$

Proposition 10.5 proves the correctness of our approach and the optimality of the computed (global) diagnosis.

Proposition 10.5

Computing a (global) diagnosis for a graph G, representing an aligned ontology, can be reduced to computing the (local) diagnoses for the problematic SCCs of G. The (minimal) global diagnosis is the union of the (minimal) local diagnoses.

Proof

From Proposition 10.3, it follows that: (i) all and only problematic SCCs contain unsafe cycles, (ii) an unsafe cycle does not traverse vertices of more than one SCC (i.e., the unsafe cycles of distinct SCCs are totally disjoint). From (i), completeness follows (it is sufficient to compute a diagnosis for each problematic SCCs to remove all the unsafe cycles in the aligned ontology). (ii) ensures the independence of SCCs, and this guarantees minimality and correctness for local diagnoses computed in isolation. Finally, it is immediate to see that the minimality property is preserved by the union of local diagnoses, and this concludes the proposition. $\square$

Proposition 10.5 thus guarantees that the global diagnosis computed as the union of the diagnoses of the problematic SCCs is both minimal and correct (that is, it breaks all the unsafe cycles).