214
Mathematical Aspects of Logic Programming Semantics
with the operation in question. Thus, for example, A
� denotes the allowable
c
set for c relative to the operation 8.
•
In particular, we can take C as a logic T and 8 as either disjunction or
conjunction defined on it. Indeed, the following example and the paragraph
following it show the thinking behind Definition 7.6.1, and in fact we shall take
FOU R as a running example throughout this section. Note that, throughout
this section, we take FOU R to be the set {u, f , t, b} with this given listing of
its elements, as in Chapter 1.
7.6.2 Example Consider again Belnap’s logic FOU R. Taking 8 to be disjunction ∨, the sets E and R are as follows.
(1) For u, we have n = 1, E u
∨ = {t, b}, and R u
∨ = {u}.
(2) For f , we have n = 1, E f
∨ = {u, t, b}, and R f
∨ = {f }.
(3) For t, takes the values 1 and 2,
∨ = ∅,
∨,1
n
E t
R t = {t}, and
∨,2
R t = {u, b}.
(4) For b, we have n = 1, E b
∨ = {u, t}, and R b
∨ = {b}.
Thus, for example, a countable disjunction i M s i takes value t if and only
∈
if either (i) at least one of the s i takes value t or (ii) at least one of the s i
takes value b and at least one takes value u
o
; no truth value is excluded.
Now taking 8 to be conjunction ∧, the sets E and R are as follows.
(1) For u, we have n = 1, E u
∧ = {f , b}, and R u
∧ = {u}.
(2) For f , n takes the values 1 and 2, E
∧
f = ∅,
∧,1
R f = {f }, and
,
R
∧ 2
f
= {u, b}.
(3) For t, we have n = 1, E t
∧ = {u, f , b}, and R t
∧ = {t}.
(4) For b, we have n = 1, E b
∧ = {u, f }, and R b
∧ = {b}.
In fact, Definition 7.6.1 was motivated by the problem, already mentioned,
of defining truth values of bodies of pseudo-clauses over various three-valued
logics, see [Hitzler and Seda, 1999b] and Sections 5.2.1 and 5.5 herein. The
following facts show how it works, where we take the countable set M to be
N without loss of generality. If 8 is finitely determined, then it is idempotent,
commutative, and associative, as already noted in Section 5.5. Furthermore,
g
if
i∈M s i = c, then the sequence s 1 , s 1 8 s 2 , s 1 8 s 2 8 s 3 , . . . is eventually
constant with value c. In the converse direction, suppose C is a countable set
and 8 is idempotent, commutative, and associative. Suppose further that,
for any set {s i | i ∈ M } of elements of C where M is countable, the sequence
s 1 , s 1 8s 2 , s 1 8s 2 8s 3 , . . . is eventually constant with value c. Then all products
g
in C are (well-defined and) finitely determined, where we take
i∈M s i = c
g
to define i∈M s i .
For a finitely determined binary operation 8 on C, we define the partial
Mathematical Aspects of Logic Programming Semantics
with the operation in question. Thus, for example, A
� denotes the allowable
c
set for c relative to the operation 8.
•
In particular, we can take C as a logic T and 8 as either disjunction or
conjunction defined on it. Indeed, the following example and the paragraph
following it show the thinking behind Definition 7.6.1, and in fact we shall take
FOU R as a running example throughout this section. Note that, throughout
this section, we take FOU R to be the set {u, f , t, b} with this given listing of
its elements, as in Chapter 1.
7.6.2 Example Consider again Belnap’s logic FOU R. Taking 8 to be disjunction ∨, the sets E and R are as follows.
(1) For u, we have n = 1, E u
∨ = {t, b}, and R u
∨ = {u}.
(2) For f , we have n = 1, E f
∨ = {u, t, b}, and R f
∨ = {f }.
(3) For t, takes the values 1 and 2,
∨ = ∅,
∨,1
n
E t
R t = {t}, and
∨,2
R t = {u, b}.
(4) For b, we have n = 1, E b
∨ = {u, t}, and R b
∨ = {b}.
Thus, for example, a countable disjunction i M s i takes value t if and only
∈
if either (i) at least one of the s i takes value t or (ii) at least one of the s i
takes value b and at least one takes value u
o
; no truth value is excluded.
Now taking 8 to be conjunction ∧, the sets E and R are as follows.
(1) For u, we have n = 1, E u
∧ = {f , b}, and R u
∧ = {u}.
(2) For f , n takes the values 1 and 2, E
∧
f = ∅,
∧,1
R f = {f }, and
,
R
∧ 2
f
= {u, b}.
(3) For t, we have n = 1, E t
∧ = {u, f , b}, and R t
∧ = {t}.
(4) For b, we have n = 1, E b
∧ = {u, f }, and R b
∧ = {b}.
In fact, Definition 7.6.1 was motivated by the problem, already mentioned,
of defining truth values of bodies of pseudo-clauses over various three-valued
logics, see [Hitzler and Seda, 1999b] and Sections 5.2.1 and 5.5 herein. The
following facts show how it works, where we take the countable set M to be
N without loss of generality. If 8 is finitely determined, then it is idempotent,
commutative, and associative, as already noted in Section 5.5. Furthermore,
g
if
i∈M s i = c, then the sequence s 1 , s 1 8 s 2 , s 1 8 s 2 8 s 3 , . . . is eventually
constant with value c. In the converse direction, suppose C is a countable set
and 8 is idempotent, commutative, and associative. Suppose further that,
for any set {s i | i ∈ M } of elements of C where M is countable, the sequence
s 1 , s 1 8s 2 , s 1 8s 2 8s 3 , . . . is eventually constant with value c. Then all products
g
in C are (well-defined and) finitely determined, where we take
i∈M s i = c
g
to define i∈M s i .
For a finitely determined binary operation 8 on C, we define the partial