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Mathematical Aspects of Logic Programming Semantics
I t , takes value u on I u , and, by default, takes value f on X \(I t ∪I u ). Precisely
the same remarks apply to triples I = (I t , I f , I b ) and to triples I = (I t , I u , I b )
in relation to valuations v : X → FOU R.
This passage between mappings and tuples of subsets will often be made
without explicit mention. However, as noted in Remark 1.3.1, we will, in the
main, use the term valuation to refer to mappings and the term interpretation
to refer to tuples of sets, and it will be convenient to employ the following
terminology.
1.3.5 Definition A valuation or interpretation taking values in T W O,
T HREE, or FOU R will be called two-valued , three-valued , or four-valued ,
respectively.
The identification above, of valuations with tuples of sets, carries the pointwise ordering of valuations over to the “pointwise” ordering of interpretations,
and we employ exactly the same notation for the orderings in the corresponding cases. We obtain the following result, whose proof is straightforward and
will be omitted. (There is the possibility of confusion here unless one remembers that coordinate positions in the tuples are labelled with truth values and
that the truth value not present is the default value. Thus, for example, in
the case of three-valued valuations, the two coordinate positions are either
ordered with t and f in that order or ordered with t and u in that order,
and similarly for four-valued valuations. The only way to avoid this minor
irritation is to use pairs of sets to represent two-valued valuations, triples of
sets to represent three-valued valuations, and quadruples of sets to represent
four-valued valuations. However, this is not customarily done.)
1.3.6 Theorem The following statements hold in relation to interpretations
on X.
(a) If I and K are two-valued interpretations, then I [ t K if and only if
I ⊆ K as subsets of X. The bottom element for the set of two-valued
interpretations is given by the empty set, ∅.
(b) If I and K are three-valued interpretations, then I [ k K if and only
if I t ⊆ K t and I f ⊆ K f . Also, I [ t K if and only if I t ⊆ K t and
K f ⊆ I f . In both orderings, the bottom element for the set of three-valued
interpretations is given by the appropriate pair (∅, ∅).
(c) If I and K are four-valued interpretations, then I [ k K if and only if
I t ⊆ K t ∪ K b , I f ⊆ K f ∪ K b , and I b ⊆ K b . Also, I [ t K if and only
if I t ⊆ K t , I u ⊆ K u ∪ K t , and I b ⊆ K b ∪ K t . In both orderings, the
bottom element for the set of four-valued interpretations is given by the
appropriate triple (∅, ∅, ∅).
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Mathematical Aspects of Logic Programming Semantics
I t , takes value u on I u , and, by default, takes value f on X \(I t ∪I u ). Precisely
the same remarks apply to triples I = (I t , I f , I b ) and to triples I = (I t , I u , I b )
in relation to valuations v : X → FOU R.
This passage between mappings and tuples of subsets will often be made
without explicit mention. However, as noted in Remark 1.3.1, we will, in the
main, use the term valuation to refer to mappings and the term interpretation
to refer to tuples of sets, and it will be convenient to employ the following
terminology.
1.3.5 Definition A valuation or interpretation taking values in T W O,
T HREE, or FOU R will be called two-valued , three-valued , or four-valued ,
respectively.
The identification above, of valuations with tuples of sets, carries the pointwise ordering of valuations over to the “pointwise” ordering of interpretations,
and we employ exactly the same notation for the orderings in the corresponding cases. We obtain the following result, whose proof is straightforward and
will be omitted. (There is the possibility of confusion here unless one remembers that coordinate positions in the tuples are labelled with truth values and
that the truth value not present is the default value. Thus, for example, in
the case of three-valued valuations, the two coordinate positions are either
ordered with t and f in that order or ordered with t and u in that order,
and similarly for four-valued valuations. The only way to avoid this minor
irritation is to use pairs of sets to represent two-valued valuations, triples of
sets to represent three-valued valuations, and quadruples of sets to represent
four-valued valuations. However, this is not customarily done.)
1.3.6 Theorem The following statements hold in relation to interpretations
on X.
(a) If I and K are two-valued interpretations, then I [ t K if and only if
I ⊆ K as subsets of X. The bottom element for the set of two-valued
interpretations is given by the empty set, ∅.
(b) If I and K are three-valued interpretations, then I [ k K if and only
if I t ⊆ K t and I f ⊆ K f . Also, I [ t K if and only if I t ⊆ K t and
K f ⊆ I f . In both orderings, the bottom element for the set of three-valued
interpretations is given by the appropriate pair (∅, ∅).
(c) If I and K are four-valued interpretations, then I [ k K if and only if
I t ⊆ K t ∪ K b , I f ⊆ K f ∪ K b , and I b ⊆ K b . Also, I [ t K if and only
if I t ⊆ K t , I u ⊆ K u ∪ K t , and I b ⊆ K b ∪ K t . In both orderings, the
bottom element for the set of four-valued interpretations is given by the
appropriate triple (∅, ∅, ∅).
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