161
Supported Model Semantics
5.4 Consequence Operators and Fitting-Style Operators
We close this section by discussing some natural extensions of certain earlier results. These are obtained by defining a rather general semantic operator
T modelled on Fitting operators, but defined over abstract finite logics T
rather than over logics containing two, three, or four elements. We call the
resulting operators consequence operators, and an important special case of
them we call Fitting-style operators. Our main result here is a careful analysis of the continuity of these operators T in the Cantor topology Q, which
yields necessary and sufficient conditions for the continuity in Q of the singlestep operator as a special case, see Theorem 5.4.11. Once these results are
established, the aforementioned extensions we require are straightforward to
present.
Thus, let T denote a finite set {t 1 , . . . , t n } of truth values containing at
least the two distinguished values t 1 and t n , which are interpreted as being
the truth values for “false” and for “true”, respectively. We assume that we
have truth tables for the usual connectives ∨, ∧, ←, and ¬. Given a normal
logic program P , we denote the set of all (Herbrand) interpretations or valuations in this logic by I P,n ; thus, I P,n is the set of all functions I : B P → T .
If n is clear from the context, we will use the notation I P instead of I P,n ,
and we note that this usage is consistent with that already established for
n = 2, 3, and 4. As usual, any interpretation I can be extended, using the
truth tables, to give a truth value in T to any variable-free formula in the
language L underlying P . We assume throughout this section that our underlying language L contains at least one function symbol, and hence B P is
denumerable. Finally, we endow I P,n with the Cantor topology Q studied in
Chapter 3, see Theorem 3.3.1, and recall that this is the product topology of
B P copies of the discrete topology on T . We refer the reader to Theorem 3.3.4
and Proposition 3.3.9 for a summary of the properties of Q. We note that our
present assumption that B P is denumerable and that T is finite mean that Q
is second countable.
We proceed next with introducing a rather general notion of semantic operator T which subsumes many of the particular operators we have encountered
in the earlier chapters. As already noted, our main objective here is to study
11
the continuity of T in the topology Q.
5.4.1 Definition An operator T on I P is called a consequence operator for
P if for every I ∈ I P the following condition holds: for every clause A ← body
in ground(P ), where T (I)(A) = t i and I(body) = t j , say, we have that the
truth table for ← yields the truth value t n , that is, true for t i ← t j .
11 We refer the reader to [Hitzler et al., 2004] for further details concerning the material
of this section.
Supported Model Semantics
5.4 Consequence Operators and Fitting-Style Operators
We close this section by discussing some natural extensions of certain earlier results. These are obtained by defining a rather general semantic operator
T modelled on Fitting operators, but defined over abstract finite logics T
rather than over logics containing two, three, or four elements. We call the
resulting operators consequence operators, and an important special case of
them we call Fitting-style operators. Our main result here is a careful analysis of the continuity of these operators T in the Cantor topology Q, which
yields necessary and sufficient conditions for the continuity in Q of the singlestep operator as a special case, see Theorem 5.4.11. Once these results are
established, the aforementioned extensions we require are straightforward to
present.
Thus, let T denote a finite set {t 1 , . . . , t n } of truth values containing at
least the two distinguished values t 1 and t n , which are interpreted as being
the truth values for “false” and for “true”, respectively. We assume that we
have truth tables for the usual connectives ∨, ∧, ←, and ¬. Given a normal
logic program P , we denote the set of all (Herbrand) interpretations or valuations in this logic by I P,n ; thus, I P,n is the set of all functions I : B P → T .
If n is clear from the context, we will use the notation I P instead of I P,n ,
and we note that this usage is consistent with that already established for
n = 2, 3, and 4. As usual, any interpretation I can be extended, using the
truth tables, to give a truth value in T to any variable-free formula in the
language L underlying P . We assume throughout this section that our underlying language L contains at least one function symbol, and hence B P is
denumerable. Finally, we endow I P,n with the Cantor topology Q studied in
Chapter 3, see Theorem 3.3.1, and recall that this is the product topology of
B P copies of the discrete topology on T . We refer the reader to Theorem 3.3.4
and Proposition 3.3.9 for a summary of the properties of Q. We note that our
present assumption that B P is denumerable and that T is finite mean that Q
is second countable.
We proceed next with introducing a rather general notion of semantic operator T which subsumes many of the particular operators we have encountered
in the earlier chapters. As already noted, our main objective here is to study
11
the continuity of T in the topology Q.
5.4.1 Definition An operator T on I P is called a consequence operator for
P if for every I ∈ I P the following condition holds: for every clause A ← body
in ground(P ), where T (I)(A) = t i and I(body) = t j , say, we have that the
truth table for ← yields the truth value t n , that is, true for t i ← t j .
11 We refer the reader to [Hitzler et al., 2004] for further details concerning the material
of this section.