72
Mathematical Aspects of Logic Programming Semantics
Of course, a function f : D → E from a domain D to a domain E is called
Scott continuous if it is continuous in the Scott topologies on D and E. However, it is well-known that a function f between domains is Scott continuous
if and only if it is continuous in the sense of Definition 1.1.7, see Proposition A.6.4. Moreover, by virtue of Theorem 1.3.2 and Proposition A.6.5, we
have the following result.
3.2.5 Proposition Suppose that the truth set T is a domain. Then in the
Scott topology I(X, T ) is a compact T 0 topological space, but is not T 1 in
general.
Nets (and convergence classes), like sequences, are normally simple to handle, and their use makes checking continuity relatively straightforward, as we
will see later on in several places. However, we move next to consider the significance of Theorem 3.2.4 in the case of spaces I(X, T ) of valuations, where
the set (T , ≤) of truth values is a domain. Indeed, suppose that (T , ≤) is a domain and that the net (v i ) converges to v in the Scott topology on the domain
I(X, T ). According to Theorem 3.2.4, this holds if and only if for each finite
valuation u with u [ v, there is an index i 0 such that u [ v i whenever i 0 ≤ i.
In fact, when applied to the particular truth sets discussed in Section 1.3.2,
Theorem 3.2.4 gives the following result.
3.2.6 Theorem Suppose that (I i
I
interpretation.
(a) Let T denote the truth set T W O. Then, in the ordering [ t on I(X, T ), we
have that (I i ) converges to I in the Scott topology if and only if whenever
x ∈ I, eventually x ∈ I i .
(b) Let T denote the truth set T HREE. Then the following statements hold.
(i) In the ordering [ k on I(X, T ), we have that (I i ) converges to I in
the Scott topology if and only if whenever x ∈ I t , eventually x ∈ I it ,
and whenever x ∈ I f , eventually x ∈ I i f .
(ii) In the ordering [ t on I(X, T ), we have that (I i ) converges to I in
the Scott topology if and only if whenever x ∈ I t , eventually x ∈ I it ,
and whenever x ∈ I u , eventually x ∈ I iu ∪ I it .
(c) Let T denote the truth set FOU R. Then the following statements hold.
(i) In the ordering [ k on I(X, T ), we have that (I i ) converges to I
in the Scott topology if and only if whenever x ∈ I t , eventually
x ∈ I it ∪ I i b , whenever x ∈ I f , eventually x ∈ I i f ∪ I i b , and whenever
x ∈ I b , eventually x ∈ I i b .
(ii) In the ordering [ t on I(X, T ), we have that (I i ) converges to I
in the Scott topology if and only if whenever x ∈ I u , eventually
x ∈ I iu ∪ I it , whenever x ∈ I b , eventually x ∈ I i b ∪ I it , and whenever
x ∈ I t , eventually x ∈ I it .
) is a net of interpretations and that is an
Mathematical Aspects of Logic Programming Semantics
Of course, a function f : D → E from a domain D to a domain E is called
Scott continuous if it is continuous in the Scott topologies on D and E. However, it is well-known that a function f between domains is Scott continuous
if and only if it is continuous in the sense of Definition 1.1.7, see Proposition A.6.4. Moreover, by virtue of Theorem 1.3.2 and Proposition A.6.5, we
have the following result.
3.2.5 Proposition Suppose that the truth set T is a domain. Then in the
Scott topology I(X, T ) is a compact T 0 topological space, but is not T 1 in
general.
Nets (and convergence classes), like sequences, are normally simple to handle, and their use makes checking continuity relatively straightforward, as we
will see later on in several places. However, we move next to consider the significance of Theorem 3.2.4 in the case of spaces I(X, T ) of valuations, where
the set (T , ≤) of truth values is a domain. Indeed, suppose that (T , ≤) is a domain and that the net (v i ) converges to v in the Scott topology on the domain
I(X, T ). According to Theorem 3.2.4, this holds if and only if for each finite
valuation u with u [ v, there is an index i 0 such that u [ v i whenever i 0 ≤ i.
In fact, when applied to the particular truth sets discussed in Section 1.3.2,
Theorem 3.2.4 gives the following result.
3.2.6 Theorem Suppose that (I i
I
interpretation.
(a) Let T denote the truth set T W O. Then, in the ordering [ t on I(X, T ), we
have that (I i ) converges to I in the Scott topology if and only if whenever
x ∈ I, eventually x ∈ I i .
(b) Let T denote the truth set T HREE. Then the following statements hold.
(i) In the ordering [ k on I(X, T ), we have that (I i ) converges to I in
the Scott topology if and only if whenever x ∈ I t , eventually x ∈ I it ,
and whenever x ∈ I f , eventually x ∈ I i f .
(ii) In the ordering [ t on I(X, T ), we have that (I i ) converges to I in
the Scott topology if and only if whenever x ∈ I t , eventually x ∈ I it ,
and whenever x ∈ I u , eventually x ∈ I iu ∪ I it .
(c) Let T denote the truth set FOU R. Then the following statements hold.
(i) In the ordering [ k on I(X, T ), we have that (I i ) converges to I
in the Scott topology if and only if whenever x ∈ I t , eventually
x ∈ I it ∪ I i b , whenever x ∈ I f , eventually x ∈ I i f ∪ I i b , and whenever
x ∈ I b , eventually x ∈ I i b .
(ii) In the ordering [ t on I(X, T ), we have that (I i ) converges to I
in the Scott topology if and only if whenever x ∈ I u , eventually
x ∈ I iu ∪ I it , whenever x ∈ I b , eventually x ∈ I i b ∪ I it , and whenever
x ∈ I t , eventually x ∈ I it .
) is a net of interpretations and that is an