3. We can average the local functionalities, as proposed in [15]:

$${f}{u}{n}{\left({r}\right)}={a}{v}{g}_{{x}}\ {f}{u}{n}{\left({r},{x}\right)}={a}{v}{g}_{{x}}{\left({\frac{{1}}{{\#{y}:{r}{\left({x},{y}\right)}}}}\right)}$$ $$={\frac{{{1}}}{{\#{x}\exists{y}:{r}{\left({x},{y}\right)}}}}\sum_{{x}}{\frac{{{1}}}{{\#{y}:{r}{\left({x},{y}\right)}}}}$$

However, the local functionalities are ratios, so that the arithmetic mean is less appropriate.

4. We can average the local functionalities not by the arithmetic mean, but by the harmonic mean instead

$$fun(r)=HM_x fun(r,x)=HM_x\Big(\frac{1}{\#y:r(x,y)}\Big)$$ $$=\frac{\#x\exists y:r(x,y)}{\sum_x\#y:r(x,y)}=\frac{\#x\exists y:r(x,y)}{\#x,y:r(x,y)}.$$

5. We may say that the global functionality is the number of first arguments per relationship instance:

$$f u n(r)={\frac{\#x\exists y:r(x,y)}{\#x,y:r(x,y)}}$$

This notion is equivalent to the harmonic mean.

With these considerations in mind, we chose the harmonic mean for the definition of the global functionality.

## B Probabilistic Modeling Of First-Order Formulas

In Section 4, we presented our probabilistic model of ontology alignment based on descriptions as first-order sentences of our rules, such as Equation (3), reproduced here:
∃r, y : r(*x, y*) ∧ (∀y
′
: r(x
′
, y′) ⇒ y6≡y
′
) ∧ fun(r**) is high =**⇒ x6≡x We derive from these equations probability assessments, such as Equation (4),
by assuming mutual independence of all distinct elements of **our models (instance equivalence, functionality, relationship inclusion, etc.). This assumption**
is of course not true in practice but it allows us to approximate efficiently the probability of the consequence of our alignment rules in a canonical manner.