Foundations of Semantic Web Technologies

OWL Formal Semantics 199

Consider a knowledge base H = {R v S

−

, S v S

−

1

, S

1

v S

2

} which consists

only of a role hierarchy. Then H

K

= H ∪ {R

−

v S, S

−

v S

1

, S

−

1

v S

−

2

}.

Furthermore, R is a subrole of S

−

, S

1

, and S

2

. S is a subrole of S

−

1

and S

−

2

;

S

1

is a subrole of S

2

; R

−

is a subrole of S, S

−

1

and S

−

2

, etc.

Now consider the knowledge base

K = {R

−

(a, b), R(b, c), S(c, d), S

−

(d, e), S

1

(e, f ), R v S},

which, apart from R v S, can be depicted by the following diagram.

a

R

−

//

b

R

//

c

S

//

d

S

−

//

e

S

1

//

f

Then a is an R

−

-predecessor of b, and hence also an S

−

-predecessor of b.

At the same time, c is an R-successor of b, and hence also an S-successor

of b. We thus have that the ...