Framework of representing alignment correspondences 119

proposes to operationalize it, i.e., to define operations that can be readily implemented in alignment management tools and that take into account the confidence measures attributed to relations.

If we consider two knowledge resources (AandB), several matching tools can be applied and each on of them generates an alignment between A and B. In the following section, we will introduce some operators to aggregate these alignments in order to generate a unique alignment betweenA andB.

The definition of these operators implies a definition of a clear semantics for interpreting alignments.

Relations between alignments can be of any kind and any type. In order to combine or aggregate these relations we classify them by category (logical relation such as the relations defined byA5, semantic relations used to repre-sent associations between terminological entities such as the relations defined within the SKOS formalism or other types of relations such as translation links and other associations).

The semantics of these relations is different, which makes it rather impos-sible to combine them or to use them within the same level of expressivity.

The approach that we will define in the following sections is applicable for each category of alignment relations taken individually but not for the whole mixed set of alignment relations. In order to consider and combine heteroge-neous alignments we transform (translate) the alignment relations from one formalism to another and apply the approach on a coherent formalism (for example skos:exactM atch can be transformed to=).

Since the majority of alignment tools assign a confidence measure to each alignment correspondence (for an alignment relation), the

interpreta-tion of this measure leads to the nointerpreta-tion of uncertainty. Probability, possi-bility and fuzzy based approaches define different theories for representing and interpreting uncertainty. Many research studies compared the grounds of these theories and tried to represent their common criteria [Gaines 1978]

in order to establish a proper comparison between fuzzy and probabilistic theories. [Gal et al.2005] states that a probabilistic approach encodes in-complete knowledge as “probabilities about events” where fuzzy approaches

“model the intrinsic imprecision of features”.

Some other studies compared the expressivity of these theories and their computational efficiency. [Drakopoulos 1995] demonstrated that probabilis-tic models are more expressive than fuzzy sets but less efficient since they carry too much information, which makes it very difficult to process cor-related events. For these reasons we will base our choice of interpreting confidence measures within correspondences as fuzzy measures (fuzzy set theory) or belief (Dempster-Shafer theory).

7.2.2 Interpretation of correspondences using fuzzy set the-ory

Fuzzy set theory (we use alsoFSas short form of this expression) was defined by [Zadeh 1965] as an approach for representing the belonging of an element to a set from a fuzzy angle by assigning to each element a membership degree. This theory represents the same operators as classical set theory, which satisfy the same properties as classical set operators. We define two dimensions of possible interpretations of alignments using this theory (view the alignment from two different point of views):

1. from the first point of view, an alignment is described as a collection of fuzzy relations between two sets of entities where the membership function associates values to a couple of elements from both sets (binary function). This interpretation of an alignment makes it possible to use the default fuzzy composition operator to compose alignments (see figure 7.1);

2. from a specific point of view on a couple of entities from different resources, the alignment relation between these entities is defined as the reference context, which makes it interpreted as a fuzzy set by itself.

Each alignment correspondence between the same couple of entities is represented as an element of this fuzzy set having a membership degree (unary function). The interpretation of an alignment on this dimension is used for defining aggregation operators of alignments (see figure 7.2).

7.2.2.1 Interpretation of alignments as sets of fuzzy relations LetΓ be the set of individual alignment relations of a certain category (log-ical, semantic or other). LetΘbe the power set of Γ.

An alignmentA is a set of fuzzy relations defined over pairs of entities from R₁ ×R₂. For each pair of entities e₁ ∈ R₁, e₂ ∈ R₂ and for each relation r_i ∈ Θ the alignment A provides a membership function for the correspondence:

e₁, e₂,{(r_i, µ^r_Aⁱ(e₁, e₂))}

where:

µ^r_Aⁱ ∈[0,1], r_i ∈Θ

(rm,^μAr(e1, e2)) (rs,^μAr(e1, e2))

(rj,^μAr(e1, e2))

Resource 1

Resource 2

Alignment A (ri, ^μAri ((x1, y1))

Θ=2^Γ xn

x1

y1 yn

(ri, ^μAri ((xk, y1))

(ri, ^μAri ((xn, y1))

Figure 7.1: Illustrating alignment relations as fuzzy relations

Since multiple relations can be represented within a correspondence of an alignment, then each relation between both entities is interpreted as a fuzzy relation.

7.2.2.2 Interpretation of alignment relations as fuzzy sets

As represented in the figure 7.2 a correspondence between two entities (es, et) of different resources (R₁, R₂) can be interpreted as a fuzzy membership function that assigns a membership value to each relation between both entities that belongs to the set of relations that are true between es andet. Let Γ be the set of individual alignment relations of a certain category (logical, semantic or other). Let Θbe the power set of Γ. Let X be the set

of relations that are true betweenes andet.

An alignment Abetween R1 and R2 contains an element C=he_s, et, Ri whereR ⊂X is the set of relations between es and et as couples (relation, membership):

R={(r, µ_A(r))}

where:

µA∈[0,1], r∈Θ

^es e_t

e_s -- R(true) -- ^et

Resource 1 Resource 2

Alignments: 1 - 2

A B C

(ri ,μAes-et(ri))

(ri ,μBes-et(rj))

(rk ,μCes-et(rk))

Figure 7.2: Illustrating alignment relations as fuzzy sets

The above-defined fuzzy set interpretation imposes the following condi-tion on the weights associated to the relacondi-tions of a correspondence within an alignmentA:

Condition 1 If r_i ⇒ r_j then in any correspondence he_s, et,[(r1, µA(r1)), . . .]i, we must have µA(ri)≤µA(rj).

In order to respect this condition we apply the following normalization that filters the correspondences that are in conflict with the condition 1.

Normalizing confidence measures

If we have two alignment relations within a correspondence he_s, e_t,[(r₁, µ_A(r₁)),(r₂, µ_A(r₂))]i where r₁ ⊆ r₂ and µ_A(r₁) ≥ µ_A(r₂), thenr2 is removed from the list of relations since the first one entails it.