167
Supported Model Semantics
of T can be evaluated. Given this much and a normal logic program P , one
can then easily define a Fitting-style operator as an operator F P : I P,T → I P,T
o
which satisfies F P (I)(A) = I( C i ) for all I ∈ I P,T and all A ∈ B P . Here,
o
A ← C i is the pseudo-clause associated with A, and I P,T denotes the set
of all interpretations defined on B P taking values in T . The question then
arises of providing suitable conditions under which possibly infinite countable collections of truth values can be evaluated. This issue is taken up in
Section 7.6, where the notion of finitely determined disjunctions is given in
Definition 7.6.1 and is seen to be adequate for our present purposes. In fact, if
disjunctions are finitely determined, then disjunction is idempotent, commutative, and associative. Furthermore, the converse of this last statement holds
if T is finite.
For a collection M of subsets of a set X, we denote by σ(M ) the smallest
σ-algebra containing M , called the σ-algebra generated by M . Recall that
a function f : X → X is measurable with respect to σ(M ) if and only if
f
−1 (A) ∈ σ(M ) for each A ∈ M . If β is the subbase of a topology τ and β is
countable, then σ(β) = σ(τ ).
It turns out that Fitting-style operators are not always measurable with
respect to the σ-algebra σ(Q) generated by Q, at least if the underlying truth
set is unrestricted. However, under quite mild conditions, Fitting-style operators are always measurable, with no syntactic conditions on the program P
whatsoever, as we see next in the following result. (Note also that we make
no technical use here of the condition that t j ← t j evaluates to true for each
truth value t j ∈ T .)
5.5.1 Theorem Suppose T is a logic in which T is a countable set and
disjunctions are finitely determined. Then for any normal logic program P ,
the Fitting-style operator F P determined by P is measurable with respect to
the σ-algebra σ(Q).
As we shall see in Section 7.6, many logics of interest in logic programming
satisfy the requirement that disjunction is finitely determined. Indeed, it is
satisfied for Belnap’s logic FOU R, and hence T P , Φ P , and Ψ P are all always
measurable for any normal logic program P .
Supported Model Semantics
of T can be evaluated. Given this much and a normal logic program P , one
can then easily define a Fitting-style operator as an operator F P : I P,T → I P,T
o
which satisfies F P (I)(A) = I( C i ) for all I ∈ I P,T and all A ∈ B P . Here,
o
A ← C i is the pseudo-clause associated with A, and I P,T denotes the set
of all interpretations defined on B P taking values in T . The question then
arises of providing suitable conditions under which possibly infinite countable collections of truth values can be evaluated. This issue is taken up in
Section 7.6, where the notion of finitely determined disjunctions is given in
Definition 7.6.1 and is seen to be adequate for our present purposes. In fact, if
disjunctions are finitely determined, then disjunction is idempotent, commutative, and associative. Furthermore, the converse of this last statement holds
if T is finite.
For a collection M of subsets of a set X, we denote by σ(M ) the smallest
σ-algebra containing M , called the σ-algebra generated by M . Recall that
a function f : X → X is measurable with respect to σ(M ) if and only if
f
−1 (A) ∈ σ(M ) for each A ∈ M . If β is the subbase of a topology τ and β is
countable, then σ(β) = σ(τ ).
It turns out that Fitting-style operators are not always measurable with
respect to the σ-algebra σ(Q) generated by Q, at least if the underlying truth
set is unrestricted. However, under quite mild conditions, Fitting-style operators are always measurable, with no syntactic conditions on the program P
whatsoever, as we see next in the following result. (Note also that we make
no technical use here of the condition that t j ← t j evaluates to true for each
truth value t j ∈ T .)
5.5.1 Theorem Suppose T is a logic in which T is a countable set and
disjunctions are finitely determined. Then for any normal logic program P ,
the Fitting-style operator F P determined by P is measurable with respect to
the σ-algebra σ(Q).
As we shall see in Section 7.6, many logics of interest in logic programming
satisfy the requirement that disjunction is finitely determined. Indeed, it is
satisfied for Belnap’s logic FOU R, and hence T P , Φ P , and Ψ P are all always
measurable for any normal logic program P .