Foundations of Fuzzy Logic and Semantic Web Languages

208 Foundations of Fuzzy Logic and Semantic Web Languages

TABLE 10.8: The tableau rules for fuzzy ALC with GCIs.

(u). If (i) hC

1

u C

2

, ni ∈ L(v), (ii) {hC

1

, ni, hC

2

, ni} 6⊆ L(v), and (iii) node

v is not indirectly blocked, then add hC

1

, ni and hC

2

, ni to L(v).

(t). If (i) hC

1

t C

2

, ni ∈ L(v), (ii) {hC

1

, ni, hC

2

, ni}∩L(v) = ∅, and (iii) node

v is not indirectly blocked, then add some hC, ni ∈ {hC

1

, ni, hC

2

, ni} to

L(v).

(∀). If (i) h∀R.C, ni ∈ L(v), (ii) hR, mi ∈ L(hv, wi) with m > 1 − n, (iii)

hC, ni 6∈ L(w), and (iv) node v is not indirectly blocked, then add hC, ni

to L(w).

(∃). If (i) h∃R.C, ni ∈ L(v), (ii) there is no hR, n

1

i ∈ L(hv, wi) with hC, n

2

i ∈

L(w) such that min(n

1

, n

2

) ≥ n, and (iii) node v is not blocked, then

create a new node w, add h