-
Topology and the Semantics of Logic Programs
Anthony Karel Seda
Department of Mathematics, University College, Cork,
Ireland.
E-mail : [email protected]
Abstract
We introduce topologies on spaces of interpretations which
extend and gener-
alise the query and positive query topologies of Batarekh and
Subrahmanian. We
study continuity in these topologies of the immediate
consequence map associated
with any normal logic program and provide necessary and sucient
conditions for
continuity to hold. We relate these ideas to (i) computational
power of programs;
(ii) canonical programs; (iii) the declarative semantics of
denite programs; (iv)
Maher's down continuous programs; (v) the decency thesis of
Jaar, Lassez and
Maher, and (vi) compactness of sets of xed points and
models.
1 Introduction
The classical approach to the study of xed point semantics of
denite logic programs P
utilises two main facts: (1) The set I
J
L
of all interpretations of the underlying language
L based on a given preinterpretation J forms a complete lattice
under the partial
order of set inclusion. (2) The immediate consequence map T
P
dened on I
J
L
is lattice
continuous. An application is then made of a theorem, usually
attributed to Kleene,
to provide least xed points of T
P
, see [26] for a detailed account. The term classical
used here reects the fact that the underlying logic is classical
two-valued and also that
the resulting semantics is satisfactory in general so long as
only atoms are present in
clause bodies: once one permits negation symbols to be present
in clause bodies one
loses monotonicity (and hence lattice continuity) of T
P
. Nevertheless, this approach
has been extended [1] to the class of stratied programs by
adopting a dierent notion
from the usual one of ordinal powers in the case of nonmonotonic
operators.
For various reasons, including the problems caused by extended
syntax (i.e. the
presence of negation or other symbols in clause bodies), it has
been proposed that one
should work over three-valued, four-valued or even many-valued
logics and much work
has been done in this area, see [9, 10, 12, 13, 21, 22, 23, 24,
31]; in this context, even van
Emden's work [11] can be thought of as using quantitative truth
values in the interval
[0; 1]. To take just one example, in pursuing this proposal
Fitting in [12] considers
Kleene's so called strong three-valued logic. In this setting,
the notion of \basic set"
Fundamenta Informaticae 24 (4) (1995), 359-386.
1
-
dened in [12] plays the role of interpretation or valuation
(also called truth function)
and the set of all such basic sets forms a complete semilattice
but not a complete
lattice. Given a logic program P there is, moreover, an
operator
P
dened on this
set whose xed points coincide exactly with the models of a sort
of completion of P .
It is interesting to note that in general
P
is monotone (even when P is not denite)
but not continuous, a point to which we will return later.
Similar remarks apply to
[13] except that the space of interpretations there forms a
complete lattice and the
interpretations themselves take values in a bilattice. A unied
treatment of much of the
work cited above and also some which is not (such as
Quantitative Logic Programming
and Constraint Logic Programming) is to be found in [9]. What
emerges is that from
an abstract point of view semantical questions are concerned
with the following: (i)
A partially ordered set T of truth values which is perhaps a
complete partial order, a
complete semilattice or a complete lattice. (ii) The set T
D
of all interpretations dened
on the set D of all atoms in L with values in T , which itself
inherits pointwise the
underlying structure present in T (more generally at this point
the set of all structures
as dened in [9]). (iii) A monotone, weakly continuous or
continuous operator on T
D
whose pre-xed points or xed points are the models of the program
P under discussion
or its completion in some sense; these latter, i.e. the xed
points of , are found by
applying variants or generalisations of a theorem due to Tarski,
see [12, 16, 25].
Now there is another way of viewing these issues, a way familiar
in the domain
theory and denotational semantics of procedural languages but
not much considered in
logic programming, and it is as follows. Given a T
0
topological space X one may dene
a partial order on X, called the specialisation order, by x y i
x is in the closure
of fyg. Conversely, there is associated with any partially
ordered set (X;) a complete
lattice of T
0
topologies whose associated specialisation orders coincide with
, see [19,
Page 45] and [30, 37]. In fact, the collection of all sets of
the form " x = fy 2 X; x yg,
where x 2 X, forms a subbasis for the maximal element in the
complete lattice just
referred to (and sometimes called the Alexandro topology) in
which, as is well known, a
function f : X ! X is continuous i it is order preserving or
monotonic. Moreover, in
the case when X is a cpo and we replace the Alexandro topology
by the Scott topology
we have that f is partial order continuous i.e. continuous in
the lattice theoretic sense
i it is Scott continuous, again this is well known.
It is apparent from the observations just made that the study of
T
D
in various
topologies and the continuity of therein subsumes the study of
these from the lattice-
theoretic point of view. Indeed, as suggested in [30, Page 307]
it is possible to make
ner arguments using the lattice of topologies related to a
single partial order than it
is with that partial order. Moreover, as we shall see, there are
interesting and infor-
mative topologies on T
D
not even related to the underlying partial order on T
D
, in
fact many such arise from well known function-space
constructions. Such a study as
that just envisaged raises many questions, including: Is it
possible to distinguish topo-
logically between complete partial orders, complete semilattices
and complete lattices ?
Are there syntactic criteria for determining continuity of in
various topologies ? Can
computational power of various classes of programs be related
to, say, continuity of a
suitably chosen in various topologies on T
D
? Can problems of current interest in logic
programming be translated into topological terms and vice-versa
? Can such methods
2
-
be used to study xed points of as indeed they are elsewhere in
mathematics ?
Our present purpose, therefore, is to discuss in some detail the
extent to which one
can apply the methods of general topology and analysis to the
study of the declara-
tive semantics of logic programs. We shall not be so ambitious
here as to embrace the
programme outlined above in its full generality, but will in the
main restrict ourselves
to relating these ideas to the case of two-valued logic and
current notions of logic pro-
gramming as it is described in [26]; extensions to the general
case will be considered
elsewhere. Our aims are, consequently, very much akin to those
of Batarekh and Sub-
rahmanian in their papers [2, 3, 4, 5, 6]. They have already
introduced topological
ideas into the study of logic programs; specically they have
introduced two topologies
(the query topology and the positive query topology) both of
which are naturally dened
in terms of L. It is, hence, necessary to devote a certain
amount of space to consid-
ering their results and relating our work to theirs; indeed we
intend to both extend
and complement their work. In fact, most of their studies are
conducted under a blan-
ket assumption (variously called by them the matching condition,
Henkin condition or
the reachability condition) concerning the nature of the sets D
they permit to be the
domains of preinterpretations for L. This eectively imposes the
restriction that they
work only over the Herbrand preinterpretation of L which is a
needless restriction and
one which we remove. In so doing, we relate their concepts to
our more general ones and
place theirs in our context by showing that the matching
condition provides necessary
and sucient conditions under which their topologies coincide
with certain of those
which we introduce, see Propositions 3 and 6. Thus, the main
topics we investigate are
as follows:
In Section 2 we consider topologies on spaces of
interpretations. When viewed as
function spaces it is very natural to consider these in various
product topologies.
This we do relative to arbitrary domains D, and recover the
query topology
and the positive query topology as special cases when D happens
to satisfy the
matching condition. This has several consequences: First, all
the general topology
established in [3, 4, 5, 6] follows quickly and simply from well
known facts about
product spaces and moreover these facts become available in
complete generality.
Similarly, it is immediate for the same reason that the positive
query topology
coincides with the Scott topology and that the query topology
coincides with the
Lawson topology, a topology well known in lattice theory, see
[15]. Finally, we can
actually easily display metrics generating the query topology
rather than simply
asserting that it is metrizable.
In Section 3 we briey consider the declarative semantics of
denite logic programs
and show how topological continuity of T
P
can be used as a substitute for lattice
continuity to recover the Kowalski - van Emden theorem. That it
is possible
to derive such results in this way should be no surprise given
the role of the
Scott topology in computing. Nevertheless, the approach we adopt
has some
mathematical interest for two reasons. First, it motivates the
introduction of the
greatest limit of a net of interpretations. Secondly, it brings
out the topological
interpretation of the role of lattice continuity in nding xed
points. In fact,
this turns out to be the following simple observation familiar
in certain areas
3
-
of mathematics such as dynamical systems where convergent
sequences or nets
are used to nd xed points: Suppose T : X ! X is a continuous
function
dened on a topological space X and x
0
2 X is some distinguished element of
X. Consider the \orbit" x
0
; T (x
0
); T
2
(x
0
); : : : of x
0
and suppose that the sequence
T
n
(x
0
) converges to x, say, i.e. T
n
(x
0
)! x. Then by continuity of T we have that
T (T
n
(x
0
))! T (x), and therefore T
n
(x
0
)! T (x). From this one can now deduce
that T (x) = x, i.e. that x is a xed point of T , whenever one
has unique limits.
The existence of unique limits may obtain due to purely
topological features of
X, for example if X is Hausdor, or it may result from some other
feature of X.
Indeed, we encounter both situations at various points in our
work. For example,
in x3 we use the lattice structure available to dene the
aforementioned notion of
greatest limit when dealing with convergence in the topology
Q
+
dened there.
On the other hand, we use the fact that the topologyQ we dene in
x2 is Hausdor
when dealing with convergence in Q. As we shall see, the
uniqueness of greatest
limits in Q
+
turns out to be a satisfactory substitute for the Hausdor
separation
axiom, at least for the convergence questions we address in
Q
+
.
In [3, 6] several interesting conclusions are drawn from the
hypothesis that T
P
is
continuous in the query topology, where P is a (classical)
normal logic program.
These include consistency of comp(P ), the completion of P , and
criteria to ensure
that comp(P
1
) [ comp(P
2
) is consistent given that comp(P
1
) and comp(P
2
) are
consistent; both these issues are important general questions,
of course, in logic
programming. It is desirable, therefore, to have necessary,
sucient and necessary
and sucient conditions for the continuity of T
P
which are readily veried. These
we obtain in Section 4 where we consider the consequences of
these results and
establish our main conclusions. In particular, we obtain Theorem
12 of [6] as a
corollary of our results namely that in the absence of local
variables T
P
is continu-
ous in the query topology. This condition is of interest and
occurs in several places
in the literature in dierent guises, see for example [10,
x6,x7], [11, Theorem 3:1
0
],
[13, x5,x6] and [27] where it is shown that (over the Herbrand
universe) absence
of local variables characterises down-continuity up to
subsumption equivalence.
In fact, we show that in the case of the Herbrand universe down
continuity is ex-
actly the same thing as continuity relative to the query
topology. Moreover, the
observations above together with those of Blair in [8]
concerning Shepherdson's
transformation [36] of 2-register machines into logic programs
mean that the class
of programs P for which T
P
is continuous in the query topology computes all
partial recursive functions.
The equivalence noted above between down continuity of T
P
and topological conti-
nuity of T
P
, for the Herbrand universe at least, permits two further
topological interpre-
tations of issues important in logic programming. The rst is
that the programs which
Fitting [12] deemed \acceptable" and for which his map is
(partial order) continuous
is, for denite clauses, precisely the class for which is
topologically continuous, see
[17]. The other is that the programs P for which T
P
is continuous in the query topol-
ogy are canonical in the sense of [7, 18]. According to [17, 28]
the continuous programs
should not be deemed acceptable in that they are incapable of
expressing transitive
4
-
closure. We nish by observing that the result of McCarthy [29],
which we discuss in
Section 4, to the eect that programs P for which T
P
is continuous merely at the single
point T
P
# ! are canonical, provides a much larger class of canonical
programs than
the down continuous programs. Whether or not this class is
\acceptable" i.e. whether
or not those programs within it can express transitive closure
we do not know.
Acknowledgement. The author thanks two anonymous referees for
comments which
substantially improved the presentation of this paper at certain
points. In particular,
he thanks one of them for bringing [14, 20] to his attention.
These two papers use
metric space theory to establish interesting and signicant
results concerning logic pro-
gramming which are close in spirit to ours, although there is no
actual overlap between
our results and those of [14, 20].
2 Topologies on I
J
L
2.1 Notation and terminology
It will be convenient rst to establish and collect together
certain pieces of notation and
terminology which we will use throughout without further
mention. We refer to [26]
for all terms and notation relating to logic programming which
are not dened here.
Let L denote a rst order language (not containing the equality
symbol) and let
J denote a xed, but arbitrary, preinterpretation of L with
domain, dom(J), denoted
by D. Usually, in later sections, L will denote the underlying
language of a normal
logic program but for the moment L is perfectly general. Suppose
that v is a variable
assignment. Given a term t in L, let J
v
(t) 2 D denote the term assignment or denotation
of t with respect to J and v, alternatively denoted t j v when J
is understood. If
t = (t
1
; : : : ; t
n
) is a vector of terms in L, then J
v
(t), or t j v, is by denition the vector
(J
v
(t
1
); : : : ; J
v
(t
n
)). We let D
L
denote the set of all J-instances of predicate symbols in
L, thus D
L
is the set fp(d); p is a predicate symbol in L and d is a vector
of elements
of D whose length equals the arity of pg. If I is an
interpretation of L based on J , we
let I(p) denote the relation on D
n
to which I assigns the n-ary predicate symbol p. In
the usual way, we can identify I with a subset of D
L
; thus p(d) 2 I i I(p)(d) = true.
Hence, the setX or I
J
L
of all interpretations of L based on the preinterpretation J can
be
identied with the power set P(D
L
) of D
L
, that is to say X = I
J
L
= P(D
L
). Let 0 and 1
be distinct symbols not in L. By identifying each subset I of
D
L
with its characteristic
function
I
we can identify P(D
L
) with the set 2
D
L
of all functions from D
L
into 2,
where 2 denotes the set f0; 1g. Therefore, if we let f0; 1g
i
= f0; 1g for each i 2 D
L
,
then we can further identify P(D
L
) and 2
D
L
with the product
Q
i2D
L
f0; 1g
i
. Thus, in
terms of the framework described in the Introduction the set T
here is simply f0; 1g
partially ordered in the usual way and, as already noted, we
shall restrict ourselves to
this level of generality throughout.
An arbitrary literal, L(t) or just L, in L takes the form p(t)
or :p(t), where p(t)
is an atomic formula. Since we are working in classical two
valued logic, if I is an
interpretation of L and v is a variable assignment I j=
v
L(t) or I j= L(t), if v is
understood, means precisely the following. When L(t) is p(t), I
j=
v
L(t) i p(J
v
(t)) 2 I,
and when L(t) is :p(t), I j=
v
L(t) i p(J
v
(t)) 62 I. Suppose E = 9L(t) or E = 9(L)
5
-
denotes the existential closure of the literal L(t) and fx
1
; : : : ; x
m
g is the set of free
variables in L; such a formula E is called a query. Given a
variable assignment v and
an interpretation I of L, we write I j= E if there are elements
d
1
; : : : ; d
m
in D such that
I j=
u
L, where u = v(x
1
; : : : ; x
m
=d
1
; : : : ; d
m
) is the variable assignment v except that
x
i
is assigned to d
i
for i = 1; 2; : : : ; m. More generally, if E = 9(L
1
^ ^ L
n
), where
each L
i
is a literal L
i
(t
i
), then E is the general form of a query and we write I j= E
to mean that I j=
u
L
i
for i = 1; 2; : : : ; n, where u is as before except that fx
1
; : : : ; x
m
g
denotes the union of the sets of free variables which occur in
each of the L
i
. Of course,
the truth, or otherwise, of the statement \I j= E" is
independent of v.
It will be convenient at times, for purely technical reasons, to
slightly abuse the
notation just established in the previous paragraph and the
usual understanding of the
symbols j=, ^ and : in the following way. If J
v
(t) = d, we will sometimes write I j= p(d)
to mean exactly that p(d) 2 I or in other words that p(J
v
(t)) 2 I. Similarly, we write
I j= :p(d) to mean exactly that p(d) 62 I. Thus it is meaningful
to write I j= L(d)
or I j= L(J
v
(t)) or I j= L(t j v) and by extension to write I j= L
1
(d
1
) ^ ^ L
n
(d
n
),
this latter meaning, of course, that I j= L
i
(d
i
) for i = 1; 2; : : : ; n. Finally, if L(t) is a
literal, we will sometimes nd it convenient to write L j v for
L(t j v), where v is a
variable assignment (once again, J being understood); we call L
j v a v - instance of L.
It should be noted that :p(d) is simply a mixture of syntactic
and semantical entities
in just the same way as p(d) is such, and is a convenient device
to use in the details of
some of our later results.
2.2 The Query Topology
We begin by recalling the query topology of Batarekh and
Subrahmanian [2, 3, 4, 5,
6] and discussing certain of its limitations; in the next
subsection we show how the
diculties encountered here can be overcome. We refer to [39] as
a general reference
to matters topological, and to [32, 37, 38] as general
background for those readers
unfamiliar with the methods and use of topology in theoretical
computer science. Other
examples of the use of such methods abound in the literature on
domain theory.
Let J be a xed preinterpretation of the rst order language L and
let X denote
the set I
J
L
of all interpretations of L based on J , as described above.
Given a query E
in L we form the subset G(E) of X dened by G(E) = fI 2 X; I j=
Eg and then form
the class G of subsets of X, where G = fG(E);E is a queryg. The
query topology is,
by denition, that topology on X generated by taking G as a
subbasis. Thus, a typical
open set in the query topology is a union of sets of the type
G(E
1
^ ^ E
n
), where
E
1
; : : : ; E
n
are queries and G(E
1
^ ^ E
n
) denotes the intersection of the sets G(E
i
),
for i = 1; 2; : : : ; n; of course, the collection B of sets of
this latter type G(E
1
^ ^E
n
)
forms a basis for the query toplogy.
As already noted in the Introduction, Batarekh and Subrahmanian
have made quite
extensive studies of this toplogy with much the same aims in
mind as we have here.
However, from a certain point of view the topology described
above is not entirely
satisfactory. In order to explain this point of view we note rst
that the set of all literals
in L is countable. Therefore the set of all queries in L is
countable. In consequence
we see that the collection B is countable and that always the
query topology is second
6
-
countable irrespective of the cardinality of the domain D of J .
This is unsatisfactory
in that it leads to a paucity of open sets in I
J
L
, see Remark 2. This in turn leads to
a failure, in general, of such desirable properties as the
Hausdor separation property
(see the next example) where there is no intrinsic reason for
this to happen. Indeed,
we shall soon dene a perfectly natural generalisation of the
query topology which is
always Hausdor. By contrast, the other topologies of interest to
us viz. the positive
query topology and our generalisation of it intrinsically lack
all separation properties
other than that of being T
0
, see x3.
Example 1.
Consider the program P as follows
p(o)
p(s(x)) p(x)
and consider the following preinterpretation J of the underlying
language L of P . We
shall (i) take the domain of J to be the set R of all real
numbers; (ii) assign the constant
symbol o to the zero element, 0, of R; (iii) assign the function
symbol s to the function
s
0
: R ! R dened by s
0
(r) = r + 1. Thus, the term assignment J(s
n
(o)) of s
n
(o) is
n, i.e. J(s
n
(o)) = n. Here D
L
= fp(d); d 2 Rg, and since L contains only the predicate
symbol p any interpretation I of L based on J corresponds to a
subset of R. In other
words, the set X or I
J
L
is naturally in one-to-one correspondence with the set of
all
subsets of R. If we let I
1
= fp(1=2)g and I
2
= fp(3=4)g for example, so that I
1
; I
2
2 X,
it is straightforward to show that I
1
and I
2
cannot be separated, see [33], and therefore
that X is not Hausdor.2
In fact, Batarekh and Subrahmanian impose a condition at all
times which they call
the matching condition. Under this condition D is forced to be
countable and in that
case the query topology is Hausdor. We discuss this condition in
more detail in the
next subsection.
2.3 The topology Q
In order to overcome some of the diculties identied in the
previous subsection, we
propose now to introduce a generalisation of the query topology
as follows.
Let p be an n-ary predicate symbol in L and let x
1
; x
2
; : : : ; x
n
be variable symbols in
L. If v denotes a variable assignment with v(x
i
) = d
i
, say, then we write v(x) = d, where
of course x denotes the vector (x
1
; x
2
; : : : ; x
n
) and d denotes the vector (d
1
; d
2
; : : : ; d
n
).
Next, we dene the set G(p(x); v), also denoted by G(p(x j v)) or
G(p(d)), to be
fI 2 X; I j= p(d)g. Likewise, we dene the set G(:p(x); v),
alternatively denoted by
G(:p(x j v)) or G(:p(d)), to be fI 2 X; I j= :p(d)g. Thus,
consistent with our earlier
notation we dene G(L(d)) to be the set fI 2 X; I j= L(d)g.
Finally, let G denote
the collection of all sets of the form G(p(d)) together with
those of the form G(:p(d))
where p(d) belongs to D
L
. We dene the atomic topology Q on X = I
J
L
to be that
topology obtained by taking the collection G as a subbasis.
Thus, a basic open set
in this topology is of the form G(L
1
(d
1
)) \ \ G(L
n
(d
n
)) which we may write on
occasions as G(L
1
(d
1
) ^ ^ L
n
(d
n
)) or G(
V
n
i=1
L
i
(d
i
)) by our abuse of notation. So,
7
-
if C denotes the collection of all sets of this latter type,
then C forms a basis for the
atomic toplogy and in this topology an arbitrary open set is a
union of elements of C.
Remark 1 1) In forming C from G, we take all nite intersections
of elements of G.
As usual, we allow empty intersections i.e. the intersection of
the empty family, which
in this case is all of I
J
L
. It is useful notation to let
denote the empty conjunction
L
1
(d
1
)^ ^L
n
(d
n
) of literals with n = 0. Thus, G(
) is the intersection of the empty
family and therefore G(
) = I
J
L
, and hence I j=
for all I 2 I
J
L
.
2) In general, I j= p
1
(d
1
)^ ^ p
n
(d
n
)^:q
1
(e
1
)^ ^ :q
m
(e
m
) i p
i
(d
i
) 2 I and
q
j
(e
j
) 62 I for i = 1; 2; : : : ; n; j = 1; 2; : : : ; m. In
particular, this is so when I = i
n = 0, and when I = D
L
i m = 0.
We present next some of the basic separation properties of the
atomic topology in
Proposition 1. These facts are the analogues for Q of similar
results established by
Batarekh and Subrahmanian [3, 5] for the query topology. We omit
the direct proofs
which are straightforward; in any case the results follow
immediately from Theorem 1
below.
Proposition 1 Let J be an arbitrary preinterpretation of a rst
order language L.
Then
a) The set G(L(d)) is both open and closed.
b) Every basic open set in Q is closed.
c) The topology Q is regular and Hausdor.
It is apparent from this result and Example 1 that the query and
atomic topologies
are usually dierent, and we now proceed to demonstrate the true
connection between
them. In the next subsection, we establish results that give us
a very clear picture of
these topologies.
Proposition 2 The atomic topology is ner than the query
topology.
Proof Let E = 9(L
1
(t
1
) ^ ^ L
n
(t
n
)) be an arbitrary query in L and suppose
fx
1
; : : : ; x
m
g is the union of the sets of free variables which occur in each
of the literals
in E. Consider the basic open set G(E) = fI 2 X; I j= Eg in the
query topol-
ogy and let v be a variable assignment. Then an interpretation I
belongs to G(E)
i there are elements d
1
; : : : ; d
m
of D such that I j=
u
L
i
(t
i
) for i = 1; 2; : : : ; n, where
u = v(x
1
; : : : ; x
m
=d
1
; : : : ; d
m
). But then I 2 G(
V
n
i=1
L
i
(d
i
)), where d
i
= J
u
(t
i
) for
i = 1; 2; : : : ; n. Conversely, if I 2 G(
V
n
i=1
L
i
(d
i
)), then I 2 G(E). It is now evident
that G(E) is a union of certain basic open sets in the atomic
topology, this union being
taken over a collection of variable assignments u. Therefore,
G(E) is open in the atomic
topology as required. 2
Remark 2 It is apparent from this result, or rather its proof,
that the root of the
problem with the query topology is the presence of the
existential quantier in a query.
This forces one to take unions as above with the result that
there are not enough open
sets.
8
-
According to [3, 5] a preinterpretation J matches the language L
i every element d
in the domainD of J is the denotation of some variable free term
in L. For example, if J
is the Herbrand preinterpretation of L, then J matches L.
Batarekh and Subrahmanian
impose the condition at all times that J matches L and refer
also to this condition as
the Henkin condition or the reachability condition. However,
this is a severe restriction
to make since it forces the domain D of J to be countable. We
will have no need to
impose the matching condition at any time. Nevertheless, it is
under this condition
that we obtain coincidence of the two topologies we have been
discussing, as we show
in the next result, and for this reason it is of some interest
to us.
Proposition 3 The query topology coincides with Q i J satises
the matching con-
dition. In particular, these topologies coincide for the
Herbrand preinterpretation.
Proof Suppose rst that J satises the matching condition. By
Proposition 2 it suces
to show that each of the sets G(p(d)) and G(:p(d)) is open in
the query topology, for
then the two topologies coincide. So let p(d) be an arbitrary
element of D
L
. Then d is
a vector of elements of D and, by the matching condition, there
is a vector t of variable
free terms in L such that J(t) = d. Now consider the query E =
9(p(t)). Since t is
variable free, E is simply the ground atom p(t). Thus, G(E) =
G(p(t)) = G(p(d)) and
it follows that G(p(d)) is open in the query topology, as
required. A similar argument
shows that G(:p(d)) is open in the query topology. This
establishes suciency of the
condition.
Conversely, suppose that the matching condition is not satised
and let
D
L
be
dened to be that subset of D
L
whose elements have arguments each of which is the
denotation of a variable free term in L. Then by hypothesis
D
L
6= D
L
and, since the
topology Q is Hausdor, it follows that
D
L
and D
L
can be separated by basic open
sets. Thus, there is a basic neighbourhood of D
L
in Q which does not contain
D
L
. Now
suppose that G(E) is a subbasic neighbourhood of D
L
in the query topology. Thus,
E = 9(L
1
^ ^ L
n
) is a query and D
L
j= E. Therefore, by Remark 1, all the literals
L
i
which occur in E are positive. Now assign all the variables
which occur in E, if
any, to elements of D which are the denotations of variable free
terms in L. Then for
this variable assignment J(L
i
) 2
D
L
for i = 1; 2; : : : ; n and so
D
L
j= E or, in other
words,
D
L
2 G(E). Therefore every subbasic, and hence basic, neighbourhood
of D
L
in
the query topology contains
D
L
and so these points of X cannot be separated therein.
Therefore, the query and atomic topologies do not coincide.2
Corollary 1 If J satises the matching condition, then the class
of all G(E) such that
E is a ground query forms a countable subbasis (in fact, basis)
for the query topology.
2.4 Convergence in the topology Q
Our next task is to establish basic net convergence results in
X. These results will allow
us to display the topology Q as a product topology. This in turn
permits us to obtain
all the point-set topology we have need of concerning X. Our
approach avoids a long
detour into lattice theory which is the direction taken in
[6].
9
-
Let fI
; 2 g, or just I
, denote a net of interpretations, I
, in X = I
J
L
. The
following is a key result.
Proposition 4 A net I
converges to I in Q i every element of I is eventually an
element of I
and every element of the complement of I is eventually an
element of the
complement of I
.
Remark 3 1) The condition stated in the proposition \ A, say, is
eventually an element
of I
" means that there exists
0
such that A 2 I
for all
0
.
2) The condition in the statement of Proposition 4 is to be
interpreted in the obvious
way if either I = or I = D
L
. Thus, if I = , only the second half of the condition
applies (and then the conclusion is valid), whereas if I = D
L
, only the rst half applies
(and again the conclusion is valid).
Proof of Proposition 4 Suppose I
! I and A = p(d) 2 I. Put U = G(p(d)), then U
is a neighbourhood of I and so I
eventually belongs to U . Hence there exists
0
such
that
0
implies I
2 U i.e. I
j= p(d) and so p(d) 2 I
. Thus, A 2 I
whenever
0
, and so A is eventually an element of I
. Now suppose that A 62 I, where again
A = p(d). Then I j= :p(d). Put U = G(:p(d), so that I 2 U .
Hence I
2 U eventually
i.e. there exists
0
such that
0
implies I
2 G(:p(d)). Thus, I
j= :p(d) or, in
other words, p(d) 62 I
. Therefore, A 62 I
whenever
0
as required.
Conversely, suppose that the \eventual" conditions hold relative
to I
and I. Let U
be a neighbourhood of I; without loss of generality we can
suppose that U is a basic
neighbourhood of I of the form U = G(
V
n
i=1
p
i
(d
i
) ^
V
m
j=1
:q
j
(d
0
j
)). Then we have that
p
1
(d
1
); : : : ; p
n
(d
n
) 2 I and q
1
(d
0
1
); : : : ; q
m
(d
0
m
) 62 I:
By hypothesis and the directedness of (;), we can choose
0
so large (after n +m
steps) that whenever
0
we have
p
1
(d
1
); : : : ; p
n
(d
n
) 2 I
and q
1
(d
0
1
); : : : ; q
m
(d
0
m
) 62 I
:
Thus, I
2 U whenever
0
and so I
! I in Q as required. 2
Let us now use the identication described in x2:1 to carry the
topology Q over to
2
D
L
; thus a net
I
!
I
i I
! I. Suppose in fact that
I
!
I
and A 2 D
L
.
If A 2 I, then
I
(A) = 1. But since I
! I, we have, by Proposition 4, that A is
eventually in I
i.e. eventually
I
(A) = 1 also. On the other hand, if A 62 I, then
I
(A) = 0. But again using the fact that I
! I and Proposition 4, we see that
eventually
I
(A) = 0 also. Thus, for each A 2 D
L
, we see that
I
(A)!
I
(A) in the
discrete topology on 2. So in fact,
I
converges pointwise to
I
. Clearly, the above
arguments reverse and so
I
!
I
in the image of Q under the identications being
considered i
I
converges to
I
pointwise. Thus, the topology on 2
D
L
is the topology
of pointwise convergence. Therefore, the atomic topology Q on X
= P(D
L
) carries
over to the product topology on
Q
i2D
L
f0; 1g
i
, where each set f0; 1g
i
is endowed with
the discrete topology. Many results follow from this central
fact as we now see.
10
-
Theorem 1 Let J denote an arbitrary preinterpretation of L, let
X = I
J
L
and let D
denote the domain of J as usual. Then
a) In the topology Q, X is a totally disconnected compact
Hausdor space.
b) Q is metrizable i D is countable.
c) Q is second countable i D is countable.
d) In the topology Q, X is a Cantor space. In fact, when J is
the Herbrand preinter-
pretation, and is innite, the space X of all Herbrand
interpretations is homeomorphic
to the usual Cantor set in the closed unit interval within the
real line.
Proof All these results follow from results concerning product
spaces, see Willard [39],
as indicated. For a) see Page 211 and other well known results
such as Tychono's
theorem. For b) see Page 161, Theorem 22.3. For c) see Page 108,
Theorem 16.2.
Finally, for d) see Page 121 Art. 17.9 and Page 217, Cor.
30.5.2
It is well known, see [15, Chapter 111], that the Lawson
topology of the complete
lattice 2
D
L
coincides with the product topology on 2
D
L
when 2 is endowed with the
discrete topology. We therefore immediately obtain, as another
consequence of our
observations above, the following result rst established in [6]
under the matching con-
dition.
Theorem 2 For an arbitrary preinterpretation J the topology Q on
I
J
L
coincides with
the Lawson topology.
We nish this section by describing metrics which generate Q in
the case that the
domain D of J is countable.
Suppose that D is countable and, as usual, form the set D
L
= fp(d); p is a predicate
symbol and d is a vector of elements of Dg. In this case it is
clear that the set D
L
is
countable, although there is no obvious way of listing the
elements of D
L
in a sequence.
Let A = p(d) denote a typical element of D
L
and choose some xed listing fA
i
g of the
elements of D
L
, where i = 1; 2; 3; : : :.
Now in general, if (M
i
;
i
)
i=1;2;3;:::
is a sequence of metric spaces, where each
i
is
bounded by 1, and we form the product M =
Q
1
i=1
M
i
, then the product topology on
M is generated by the metric , where
(;
0
) =
X
1
i=1
i
(
i
;
0
i
)
2
i
in which expression = (
1
;
2
; : : :) and
0
= (
0
1
;
0
2
; : : :) are elements of M . These
facts are contained in the proof of [39, Theorem 22.3, Page
161].
As before, let 2 = f0; 1g endowed with the discrete topology and
let 2
A
i
= 2
i
= 2
for i = 1; 2; 3; : : :. Now dene the metric
i
on 2
i
for each i by
i
(0; 0) = 0 =
i
(1; 1) and
i
(1; 0) = 1 =
i
(0; 1):
Then
i
generates the discrete topology on 2
i
and the metric described above generates
the product topology on
Q
1
i=1
2
i
, and hence the atomic topology on P(D
L
). This metric
has an intuitive description when applied to elements of P(D
L
) as we now show.
11
-
Consider the identications above of P(D
L
), 2
D
L
and
Q
1
i=1
2
i
. These equate I with
I
which in turn is equated with the sequence I = (
1
;
2
; : : :), where
n
= 1 i A
n
2 I
and
n
= 0 i A
n
62 I. Thus, if I and I
0
are two interpretations and I and I
0
the
corresponding sequences, then
(I; I
0
) = (I; I
0
) =
X
1
i=1
i
(
i
;
0
i
)
2
i
:
Consider now the symmetric dierence I 4 I
0
of I and I
0
. An element A
i
belongs to
this symmetric dierence i either A
i
2 I n I
0
or A
i
2 I
0
n I. In other words, i either
i
= 1 and
0
i
= 0 or
i
= 0 and
0
i
= 1 which is the case i
i
(
i
;
0
i
) = 1. Thus, in
reality, the metric can be expressed in the form (I; I
0
) =
P
A
i
2I4I
0
1
2
i
and we have
the following result.
Theorem 3 If the domain, D, of J is countable and fA
i
g is a listing of the elements
of D
L
, where i = 1; 2; 3; : : :, then the topology Q is generated by
the metric dened by
(I; I
0
) =
X
A
i
2I4I
0
1
2
i
:
An alternative approach to and proof of this theorem can be
found in [33] where it
is shown that, with a slightly modied form of , the process
above determines, up to
equivalence, all metrics which generate Q.
3 Further topologies on I
J
L
In addition to the query topology, Batarekh and Subrahmanian [3,
5] have discussed a
topology which they call the positive query topology and have
established equality be-
tween it and the well known Scott topology dened on the lattice
I
J
L
. Once again, their
treatment is subject to the \matching condition" and once again
we regard this as an
unnecessary restriction to impose. In this section, we therefore
propose to investigate
this topology from our viewpoint and to generalise the results
obtained by Batarekh
and Subrahmanian. Later, we establish continuity of the
immediate consequence map
of any denite logic program and also some interesting
convergence results in our gen-
eralisation, Q
+
, of the positive query topology. As already noted in the
Introduction,
these facts lead to the Kowalski - van Emden theorem and
therefore show that Q
+
can
be taken as a foundation for the declarative semantics of denite
logic programs.
3.1 The Positive Query Topology
With the notation established earlier and following [3, 5] we
dene a positive query
to be a query E = 9(L
1
^ ^ L
n
) in which each literal L
i
is in fact positive or, in
other words, is an atom. Again, we let G(E) denote the set fI 2
X; I j= Eg and
G
+
= fG(E);E is a positive query g. By denition, the positive query
topology on X is
that topology obtained by taking G
+
as a subbasis. Thus, a typical basic open set in
this topology is a set of the type G(E
1
^ ^ E
n
) which is, by denition,
T
n
i=1
G(E
i
),
where E
1
; E
2
; : : : ; E
n
are positive queries. Of course, the collection B
+
of all sets of
12
-
this latter type forms a countable basis for the positive query
topology. Thus, as in
the case of the query topology, the positive query topology is
always second countable,
and again the root cause of the problem is the existential
quantier in a positive query
which corresponds to taking a union. To overcome this diculty,
we next introduce a
generalisation of the positive query topology which we call the
positive atomic topology.
3.2 The topology Q
+
With reference to x2, let G
+
denote the set fG(p(d)); p(d) 2 D
L
g. By denition, the
positive atomic topology is the topology Q
+
dened on X by taking G
+
as a subbasis.
Thus, a basic open set in this topology is of the form G(p
1
(d
1
)^ ^ p
n
(d
n
)) by which
we mean the set
T
n
i=1
G(p(d
i
)).
The positive query topology bears the same sort of relationship
to the positive
atomic topology as exists between the query topology and the
atomic topology. In
both cases, removal of the existential quantier frees up the
creation of open sets which
become more plentiful as a result. For example, the positive
atomic topology is not
second countable unless the domain of J is countable. However,
it is not Hausdor
either and we discuss the exact separation properties enjoyed by
these topologies in
a little detail later on. We shall see that these topologies are
yet more examples of
topologies of interest in theoretical computer science which
possess only the barest
separation properties viz. they are T
0
but not even T
1
.
Remark 4 The same comment as was made in Remark 1 also holds
here in forming
the basis of Q
+
from the class G
+
. So, noting again that G(
) = I
J
L
, we have
a) I j=
for every I 2 I
J
L
.
b) In particular, j=
. Indeed, if j= p
1
(d
1
) ^ ^ p
n
(d
n
), then n = 0 and the
conjunction involved is the empty one.
The analogue of Proposition 2 for the positive query and
positive atomic topologies
is as follows.
Proposition 5 The positive atomic topology is ner than the
positive query topology.
Proof The proof of Proposition 2 carries over mutatis mutandis
to the case of the
positive query and positive atomic topologies.2
In fact, we have the following result.
Proposition 6 The positive query and positive atomic topologies
coincide i J satises
the matching condition. In particular, they coincide for the
Herbrand preinterpretation
J.
Proof For suciency, if J satises the matching condition, then
the proof of the su-
ciency part of Proposition 3 easily modies to the present
situation, and so the positive
query topology coincides with Q
+
.
Conversely, suppose J does not satisfy the matching condition.
The proof of neces-
sity given in Proposition 3 does not carry over directly, as it
made use of the Hausdor
13
-
separation property which is not satised by the topologies
currently in question. We
therefore reason as follows.
Let p(d) be an element of D
L
in which d is not the denotation of any vector of
variable free terms in L, so that at least one component of d is
not the denotation of
any variable free term in L. Thus, p(d) 2 D
L
n
D
L
, where
D
L
has the meaning given
to it in proving Proposition 3. Let U = G(E) be a subbasic open
neighbourhood of
I = fp(d)g in the positive query topology. Then I j= E and E is
a positive query of the
form 9(p(z
1
) ^ ^ p(z
n
)) in which p is the only predicate symbol and z
1
; : : : ; z
n
are
vectors of terms (not necessarily variables), and for each
vector z
i
, i = 1; 2; : : : ; n, not all
its components can be variable free. Moreover, because fp(d)g j=
E, for some variable
assignment v we have (p(z
1
) ^ ^ p(z
n
)) j v = p(d). Now suppose, more generally,
that G(E) is a basic neighbourhood of I = fp(d)g in the positive
query topology. Then,
by what we have already noted, E = E
1
^ ^ E
l
for some nite number of positive
queries E
1
; E
2
; : : : ; E
l
, where
(i) E
i
= 9(p(z
i
1
) ^ ^ p(z
i
n
i
)), for i = 1; 2; : : : ; l.
(ii) Each z
i
j
is a vector of terms, not necessarily variables, not all of
whose compo-
nents are variable free.
(iii) For some variable assignment v
i
, we have (p(z
i
1
) ^ ^ p(z
i
n
i
)) j v
i
= p(d), for
i = 1; 2; : : : ; l.
Now let u be an assignment of all variables to elements of D
each of which is the
denotation of a variable free term in L. Let p(z
i
j
) j u be denoted by p(d
i
j
). Then, for
i = 1; 2; : : : ; l and each j = 1; 2; : : : ; n
i
, we have p(d
i
j
) 2
D
L
and therefore p(d
i
j
) 6= p(d).
Letting
I
denote the set fp(d
i
j
); i = 1; 2; : : : ; l and j = 1; 2; : : : ; n
i
g, we see that
I
2
G(E), but p(d) 62
I
, and so
I
62 G(p(d)). Thus, we can never have G(E) G(p(d)) for
any basic neighbourhood G(E) of I = fp(d)g in the positive query
topology. It follows
therefore that G(p(d)) is not open in the positive query
topology and, hence, that the
positive query and positive atomic topologies are dierent.2
Corollary 2 If J satises the matching condition, then the class
of all sets G(E) such
that E is a ground positive query forms a countable subbasis (in
fact, basis) for the
positive query topology.
3.3 Convergence results in Q
+
We consider next certain results concerning net convergence in
I
J
L
when this set is
endowed with the topology Q
+
. Since this latter topology is non-Hausdor, we lack
uniqueness of limits of convergent nets, or of lters for that
matter. Such a lack is
often regarded in mathematics as to be sucient to render a
non-Hausdor topology
unworkable. However, in this case I
J
L
is a lattice and this fact permits us to dene the
notion of the greatest limit . Such a limit is unique, of
course, and its use allows us to
overcome many of the problems posed by the failure of the
Hausdor separation axiom.
The basic net convergence result for Q
+
is as follows. It is proved in the same way
as Proposition 4 was proved and we omit the details.
Proposition 7 In the topology Q
+
:
14
-
a) Every net I
converges to .
b) If I 6= , a net I
converges to I i every element of I is eventually an element
of I
.
Let us again use the identication of each subset I with its
characteristic function
I
, as described in x2:1, this time to transport the topologyQ
+
to 2
D
L
. As a corollary of
Proposition 7, we easily obtain in complete generality the
following result rst obtained
in [5] in the presence of the matching condition and using a
considerable amount of
lattice theory.
Theorem 4 For an arbitrary preinterpretation J of L the topology
Q
+
on I
J
L
coincides
with the Scott topology.
Proof Suppose that
I
!
I
in Q
+
and A is any element of D
L
. Suppose rst that
A 2 I (thus we are supposing in particular that I 6= ), so
that
I
(A) = 1. Since
I
! I, we see that A 2 I
eventually by Proposition 7 and so
I
(A) = 1 eventually.
Thus, if U is given as either of the two possible Scott
neighbourhoods f1g, f0; 1g of
I
(A) in 2, eventually we have
I
(A) 2 U . Now suppose that A 62 I, including the
case that I = . Then
I
(A) = 0. The only Scott neighbourhood of 0 in 2 is f0; 1g
and obviously
I
(A) 2 f0; 1g eventually. Thus, we see that for each A 2 D
L
we have
I
(A)!
I
(A) in the Scott topology on 2.
Conversely, suppose for each A 2 D
L
that
I
(A) !
I
(A) in the Scott topology
on 2; we will show then that
I
!
I
in 2
D
L
i.e. that I
! I in I
J
L
. If I = , this
last conclusion is immediate from Proposition 7; otherwise
suppose A is an arbitrary
element of I so that
I
(A) = 1. Let U = f1g be chosen as a Scott neigbourhood of
I
(A). Since
I
(A)!
I
(A), eventually
I
(A) 2 U i.e. eventually
I
(A) = 1 or, in
other words, eventually A 2 I
as required.
We now see that convergence
I
!
I
in Q
+
is equivalent to pointwise convergence
in the Scott topology on 2. It follows, therefore that Q
+
is coincident with the product
of the Scott topologies on 2 and in turn that it is coincident
with the Scott topology
on 2
D
L
by [15, Page 122].2
From this theorem it follows that all the general facts about
the Scott topology
apply to Q
+
. For example, it must be the case that Q
+
is T
0
but not T
1
, although it is
easy to see this directly.
Continuing with our main development, we have the following
further consequences
of Proposition 7. First, if I
! I in Q
+
and J I, then we have I
! J . Next,
for any net I
of interpretations in X = I
J
L
consider the set fJ 2 X; I
! Jg. This
set contains and is hence never empty. We are led naturally,
therefore, to dene the
notion of the greatest limit, gl(I
), of the net I
by gl(I
) =
S
fJ 2 X; I
! Jg. It
follows immediately then that we have I
! gl(I
), so that gl(I
) is well named.
Proposition 8 Let I
be a net of interpretations in I
J
L
.
a) If I
is monotonic increasing in the sense that I
I
0
whenever
0
, then
gl(I
) =
S
I
.
b) If I
is monotonic decreasing in the sense that I
I
0
whenever
0
, then
gl(I
) =
T
I
.
15
-
Proof a) Case 1. Suppose gl(I
) = and
S
I
6= .
In this case, some I
is non-empty, say I
0
is non-empty. Let A 2 I
0
. Then A 2 I
whenever
0
by the monotonicity hypothesis. Hence, I
! I
0
and so gl(I
) 6= .
This contradiction means that
S
I
= and so we get gl(I
) =
S
I
in this case.
Case 2. Suppose that gl(I
) = I 6= .
Let A 2 I. Then A 2 I
eventually, since I
! gl(I
), and so A 2
S
I
and
therefore I
S
I
. Conversely, suppose A 2
S
I
. Then A 2 I
0
for some
0
and by
monotonicity A 2 I
whenever
0
. Therefore, I
!
S
I
and we have
S
I
I.
Thus, I =
S
I
as required.
b) Case 1. Suppose gl(I
) = and
T
I
6= .
Let A 2
T
I
. Then A 2 I
for all and so A is eventually in I
. It follows that
I
!
T
I
and hence gl(I
) 6= . This contradiction gives
T
I
= and so we get
gl(I
) =
T
I
in this case.
Case 2. Suppose gl(I
) = I 6= .
Let A 2 I. Then A 2 I
eventually i.e. there is
0
such that A 2 I
whenever
0
. Now let
1
be any element of the index set of the given net. Since is
directed by , there is
2
2 such that
0
2
and
1
2
. But then I
0
I
2
and
I
1
I
2
. By the fact that A 2 I
whenever
0
we conclude that A 2 I
2
, and in
turn we deduce that A 2 I
1
. In other words, A 2 I
for every . Therefore, A 2
T
I
and so gl(I
)
T
I
. In addition, we see that
T
I
6= in this case.
Conversely, if A 2
T
I
, then A 2 I
for all and so A is eventually in I
. Therefore,
I
!
T
I
and we have
T
I
I as required.2
In fact, we also have the following result.
Proposition 9 a) If I
is monotonic increasing, then I
!
S
I
in Q.
b) If I
is monotonic decreasing, then I
!
T
I
in Q.
Proof a) If
S
I
= , then I
= for all and the result is clear. So suppose that
S
I
6= . If A 2
S
I
, then A 2 I
0
for some =
0
. Thus, by monotonicity, A 2 I
whenever
0
and so A is eventually in I
. If A 62
S
I
, if such A exists, then A 62 I
for any and so A is eventually not in I
. Therefore, I
!
S
I
in Q as required.
b) Suppose A 2
T
I
, if such A exists. Then A 2 I
for all and so A is eventually
in I
. Suppose A 62
T
I
, if such A exists. Then A 62 I
0
for some =
0
. But then
by monotonicity, A 62 I
whenever
0
, and so A is eventually not in I
as required.
Therefore, I
!
T
I
in Q.2
We now turn our attention to the immediate consequence map,
T
J
P
or simply T ,
associated with any normal logic program, P , and we establish
next the important fact
that T
J
P
is always continuous in the topology Q
+
for any denite logic program P . As
usual, our general reference for matters concerned with logic
programming is [26], and
for convenience we recall that T
J
P
: I
J
P
! I
J
P
is dened as follows, where I
J
P
denotes I
J
L
and L is the underlying language of P :- T
J
P
(I) is dened to be the set of all p(d) 2 D
L
such that there is a program clause p(t) L
1
(t
1
); : : : ; L
n
(t
n
) in P and a variable
assignment v satisfying J
v
(t) = d i.e. p(t j v) = p(d) and I j=
v
L
1
(t
1
) ^ ^ L
n
(t
n
).
It is clear that T
J
P
(I) 2 I
J
P
for each I 2 I
J
P
.
16
-
Theorem 5 The immediate consequence map T
J
P
is continuous in the topology Q
+
for
any denite logic program P .
Proof Let I 2 I
J
P
and let I
! I be a net converging to I in Q
+
; we must show
that T (I
)! T (I) in Q
+
, where T denotes T
J
P
. If T (I) = , then it is immediate from
Proposition 7 a) that T (I
)! T (I). So suppose that T (I) 6= and let A = p(d) belong
to T (I). Thus, by the denition of T , there is a program clause
p(t) p
1
(t
1
); : : : ; p
n
(t
n
)
and a variable assignment v such that p(t j v) = A and I j=
v
p
1
(t
1
) ^ ^ p
n
(t
n
).
Case 1. The program clause is a unit clause i.e. n = 0.
In this case the body of the program clause is the empty
conjunction
of atoms.
But I
j=
for every by Remark 4, and so A 2 T (I
) for every . Hence, A 2 T (I
)
eventually.
Case 2. The program clause is not a unit clause.
Since I j=
v
p
1
(t
1
)^ ^p
n
(t
n
) we have that p
1
(t
1
j v); : : : ; p
n
(t
n
j v) 2 I. But I
! I
and so eventually p
i
(t
i
j v) 2 I
for i = 1; 2; : : : ; n. Thus, I
j=
v
p
1
(t
1
) ^ ^ p
n
(t
n
)
eventually and hence A 2 T (I
) eventually.
Hence, in any case A is eventually in T (I
) and so T (I
)! T (I) as required.2
Remark. Since T
J
P
is actually lattice continuous, this last result can also be
proved
by noting that Q
+
coincides with the Scott topology, and then using a well
known
theorem relating topological continuity and lattice
continuity.
We now consider sequences of interpretations, iterates of the
immediate consequence
map, T , and the connections that exist between xed points of T
and the iterates of T .
The following result is elementary.
Proposition 10 Let I 2 I
J
P
and let T denote T
J
P
. Then in either Q or Q
+
we have
T
n
(I) convergent to an interpretation K i T
n+1
(I) converges to K.
Corollary 3 For any I 2 I
J
P
, we have that gl(T
n
(I)) = gl(T
n+1
(I)) = gl(T (T
n
(I))).
The form of Kleene's rst recursion theorem appropriate for the
development of our
theory is contained in the next theorem and its corollary,
Corollary 4, below.
Theorem 6 Let P be a denite program and let T denote T
J
P
.
a) Suppose I 2 I
J
P
is such that fT
n
(I)g is monotone increasing. Then gl(T
n
(I)) is
a xed point of T . Conversely, every xed point of T arises as
gl(T
n
(I)) for some such
I.
b) If T is continuous in Q and I 2 I
J
P
is such that fT
n
(I)g is monotone decreasing,
then gl(T
n
(I)) is a xed point of T .
Proof a) Here gl(T
n
(I)) =
S
1
n=1
T
n
(I). Hence, since every net converges to its great-
est limit, we have that T
n
(I) !
S
1
n=1
T
n
(I). Therefore, by Theorem 5, T (T
n
(I)) !
T (
S
1
n=1
T
n
(I)) and so T
n+1
(I)! T (
S
1
n=1
T
n
(I)). By Proposition 10 it now follows that
T
n
(I)! T (
S
1
n=1
T
n
(I)) and we obtain the inclusion
T (
[
1
n=1
T
n
(I)) gl(T
n
(I)) =
[
1
n=1
T
n
(I):
17
-
In the other direction, if A 2
S
1
n=1
T
n
(I), then A 2 T
m
(I) for some m and we
can assume m > 1 since fT
n
(I)g is monotonic increasing. Hence, A 2 T (T
m1
(I))
T (
S
1
n=1
T
n
(I)) since T is monotonic, P being a denite program. Therefore,
we obtain
the reverse inclusion
S
1
n=1
T
n
(I) T (
S
1
n=1
T
n
(I)) as required.
For the converse statement, suppose that I is a xed point of T .
Then T
n
(I) = I
for all n and so the sequence fT
n
(I)g is monotonic increasing. Therefore gl(T
n
(I)) =
S
1
n=1
T
n
(I) = I and this establishes the claim.
b) Here gl(T
n
(I)) =
T
1
n=1
T
n
(I) and so by Proposition 9 b) we have T
n
(I) !
T
1
n=1
T
n
(I) in Q. Therefore, by the hypothesis that T is continuous in
this topology,
T (T
n
(I)) ! T (
T
1
n=1
T
n
(I)) i.e. T
n+1
(I) ! T (
T
1
n=1
T
n
(I)). Hence, by Proposition 10
we have T
n
(I)! T (
T
1
n=1
T
n
(I)). Finally, using the uniqueness of limits in a Hausdor
space we obtain the required equality
T
1
n=1
T
n
(I) = T (
T
1
n=1
T
n
(I)).2
Corollary 4 Suppose P is a denite logic program and T denotes
T
J
P
. Then
a) lfp(T ), the least xed point of T , equals T " !.
b) If T is continuous in Q, then P is canonical (see [7,
18]).
Proof a) If I is any xed point, then I. Hence T
n
() I for all n, and so
S
1
n=1
T
n
() I. But T
n
() is monotonic increasing and so gl(T
n
()) =
S
1
n=1
T
n
() I.
By Theorem 6, gl(T
n
()) is a xed point and, by the last inclusion, is now seen to
be
the least such. By denition, T " ! =
S
1
n=1
T
n
() and so T " ! = lfp(T ) as stated.
b) Let I be any xed point of T . Then I D
L
and so I
T
1
n=1
T
n
(D
L
). By
Proposition 8 b) this intersection is equal to gl(T
n
(D
L
)) which is a xed point according
to the present hypothesis and Theorem 6 b). It is now clear that
gl(T
n
(D
L
)) is the
greatest xed point of T and, by denition, T # ! =
T
1
n=1
T
n
(D
L
) = gl(T
n
(D
L
)).
Therefore, the greatest xed point of T is equal to T # ! and so
P is canonical as
required.2
To nish this section we establish the Kowalski - van Emden
theorem as a corollary
of the results above.
Corollary 5 Let J be the Herbrand preinterpretation for a denite
logic program P , let
T denote T
P
and let M
P
denote the least Herbrand model of P . Then M
P
= lfp(T ) =
T " !.
Proof Now M
P
is a model of P and so we have T (M
P
) M
P
, and hence T (T (M
P
))
T (M
P
) using monotonicity of T . Therefore, T (M
P
) is also a model of P . Hence,
M
P
T (M
P
) by denition of M
P
and it follows that T (M
P
) = M
P
and so M
P
is a
xed point of T . Clearly M
P
is equal to the least xed point of T since every xed
point is a model.2
4 Continuity in the topology Q
4.1 Sucient conditions for continuity in Q
In this section, we study in some considerable detail the map
T
J
P
: I
J
P
! I
J
P
and its
continuity with respect to the topology Q; we also consider some
of the consequences of
18
-
continuity. In general, T
J
P
is not continuous, as we shall see, relative to Q even when
P
is a denite program. Thus, one of the main tasks to be
undertaken is the determination
of conditions which will ensure continuity and which, if
possible, are simple to check in
terms of syntax, and this task we address rst.
Following [26] we recall that a normal program clause C is a
clause of the form
p(t) L
1
(t
1
); : : : ; L
n
(t
n
), where n 0 and the L
i
are literals; this is sometimes
abbreviated to C
+
C
. A normal program is a nite set of normal program clauses.
As shown in [26, Chapter 4], once one allows literals (i.e.
negated atoms) to be present
in the bodies of program clauses, one can allow arbitrary rst
order formulae to be
present. We shall not do that, however, and from now on the
terms program clause
and program will mean normal program clause and normal program
respectively.
Since we are not concerned here with resolution, we can always
arrange the body of
C so that the positive literals are grouped at the left hand end
and the negative ones
at the right hand end by relabelling the literals in the body if
necessary, thus C takes
the form
p(t) p
1
(t
1
); : : : ; p
l
(t
l
);:p
l+1
(t
l+1
); : : : ;:p
n
(t
n
);
where 0 l n. We will often suppose this has been done for each
program clause
in any normal logic program. Actually, on occasions it will be
preferable to write C in
the form
p(t) p
1
(t
1
); : : : ; p
l
(t
l
);:q
1
(s
1
); : : : ;:q
m
(s
m
);
where 0 l and 0 m, for purposes of notation and indexing.
For the rest of this section, P will denote an arbitrary normal
logic program and
J will denote a preinterpretation of the underlying language L
of P . The following
denition is a natural companion to the notions D
L
and p(d) in that it is a combination
of both syntactic and semantical entities.
Denition 1. Let C be a clause p(t) L
1
(t
1
); : : : ; L
n
(t
n
) in P , where n 0. We
will call p(t j v) L
1
(t
1
j v); : : : ; L
n
(t
n
j v), where v is a given variable assignment, the
J - instance of C determined by v and denote it by J
v
(C) or C j J
v
. If J is understood,
we write simply C j v and refer to it as the v - instance of C.
In general, we talk of J -
instances of C or just instances of C. If we have written C as
C
+
C
, then we may
write C
+
j v C
j v for C j v.
Given a v-instance p(t j v) L
1
(t
1
j v); : : : ; L
n
(t
n
j v) of a clause C, we shall refer
to p(t j v) as the head of that instance and to L
1
(t
1
j v); : : : ; L
n
(t
n
j v) as the body .
This terminology will not conict with that usually applied to
clauses, since the context
will always determine which is meant. Finally, two J - instances
C j v and C j u of the
same clause C will be said to be dierent if either (i) C
+
j v and C
+
j u are dierent
elements of D
L
, or (ii) a J - instance of a literal present in C
j v resp. present in
C
j u is dierent from each J - instance of each literal present in
C
j u resp. present
in C
j v. Two instances of a clause C which are not dierent will be
said to be equal
or the same.
Denition 2. A program clause C : p(t) L
1
(t
1
); : : : ; L
n
(t
n
) in P , where n 0,
will be said to be of nite type relative to an element p(d) of
D
L
and J if C has only
nitely many dierent J- instances with head p(d). The program P
will be said to be
of nite type relative to p(d) and J if each clause in P is of
nite type relative to p(d)
19
-
and J . Finally, P will be said to be of nite type relative to J
if P is of nite type
relative to p(d) and J for each p(d) 2 D
L
.
We will allow this terminology to extend to the situation where
there are no instances
of a clause C resp. of any clause in P with p(d) in the head. In
this case, p(d) does
not occur as an instance of the head of C resp. does not occur
as an instance of the
head of any clause in P .
Thus, if the clause C : p(t) L
1
(t
1
); : : : ; L
n
(t
n
) is of nite type relative to some
p(d) and J , then in general many dierent variable assignments
give the same value
with respect to J to each literal L
i
(t
i
) in the body of C. In other words, we get
L
i
(J
v
(t
i
)) = L
i
(J
v
0
(t
i
)) for possibly many dierent variable assignments v; v
0
, where
i = 1; 2; : : : ; n.
Example 2.
Take C to be the clause p(x) :p(f(y)) and take J to be the
preinterpretation in
which the domain of J is the set of integers and in which J
assigns f to the function
f
0
, where f
0
(d) = d + 1. Then it is readily seen that C is not of nite type
relative to
any p(d) 2 D
L
. On the other hand, if in J we assign f to a constant function
f
0
, then
C is of nite type relative to every p(d) 2 D
L
.
We turn now to the proof of one of our main results.
Theorem 7 Let P be a normal program and J be a preinterpretation
of the language
L of P . Let I 2 I
J
P
and suppose that P has the property that whenever p(d) 2 D
L
and
T
J
P
(I) j= :p(d), then P is of nite type relative to p(d) and J.
Then T
J
P
is continuous
at I in the topology Q. In particular, if P is of nite type
relative to J, then T
J
P
is
continuous in Q.
Proof Let I 2 I
J
P
be any interpretation satisfying the hypothesis of the theorem,
and let
G
2
= G(L(d)) be an arbitrary subbasic neighbourhood of T (I) in Q,
where T denotes
T
J
P
. We need to nd a neighbourhood G
1
of I in Q such that T (G
1
) G
2
.
SinceG
2
= G(L(d)) is a neighbourhood of T (I), we have that T (I) j=
L(d). Suppose
rst that L is positive so that G
2
is of the form G(p(d). Thus, T (I) j= p(d) and so
p(d) 2 T (I). Therefore, there is a program clause C in P of the
form
p(t) p
1
(t
1
); : : : ; p
l
(t
l
);:q
1
(s
1
); : : : ;:q
m
(s
m
)
(in which l +m 0) and a variable assignment v with the following
properties.
a) If l +m = 0, then C j v is the unit clause p(d)
and, trivially, I j=
.
b) If l +m > 0, then C j v is, say,
p(d) p
1
(d
1
); : : : ; p
l
(d
l
);:q
1
(e
1
); : : : ;:q
m
(e
m
)
and I j= p
1
(d
1
)^ ^p
l
(d
l
)^:q
1
(e
1
)^ ^:q
m
(e
m
), where d
i
= J
v
(t
i
) for i = 1; 2; : : : ; l
and e
j
= J
v
(s
j
) for j = 1; 2; : : : ; m. If a) is the case, put G
1
= G(
) = I
J
P
. Then
G
1
is a neighbourhood of I and if I
0
2 G
1
, then I
0
j=
and so p(d) 2 T (I
0
) i.e.
T (I
0
) j= p(d). Thus, T (I
0
) 2 G
2
and we have T (G
1
) G
2
as required. In case b), put
G
1
= G(p
1
(d
1
) ^ ^ p
l
(d
l
) ^ :q
1
(e
1
) ^ ^ :q
m
(e
m
)) which is a neighbourhood of I.
20
-
If I
0
2 G
1
, then I
0
j= p
1
(d
1
)^ ^ p
l
(d
l
)^:q
1
(e
1
)^ ^:q
m
(e
m
) and so p(d) 2 T (I
0
)
i.e. T (I
0
) j= p(d) and so T (I
0
) 2 G
2
. Thus, again we have T (G
1
) G
2
as required.
This covers the case when L is positive.
Now suppose that L is negative, so that G
2
= G(:p(d)). In this case we have
T (I) j= :p(d) i.e. p(d) 62 T (I).
Case 1. Suppose that p(d) is not the head of any instance of any
program clause in
P .
In this case put G
1
= I
J
P
itself so that G
1
is a neighbourhood of I. Now let I
0
2 G
1
.
Then by the hypothesis of this case, we have p(d) 62 T (I
0
) and so T (I
0
) j= :p(d). Thus,
T (I
0
) 2 G
2
and again we have T (G
1
) G
2
, as required.
Case 2. Suppose that p(d) is the head of some instance of some
clause in P .
If the clause C just referred to in the hypothesis of this case
were a unit clause
p(t) , so that the instance in question were p(d) , then T (I)
would contain p(d)
which is not the case. Thus, any such clause C in this case is
not a unit clause; let
p(t) p
1
(t
1
); : : : ; p
l
(t
l
);:q
1
(s
1
); : : : ;:q
m
(s
m
)
(with l+m 1) be a typical clause C in P with p(d) as an instance
of its head. Since
p(d) 62 T (I) we must have I 6j=
v
p
1
(t
1
) ^ ^ p
l
(t
l
) ^ :q
1
(s
1
) ^ ^ :q
m
(s
m
) for any
variable assignment v such that p(t j v) = p(d). In other words,
if v is any variable
assignment such that p(t j v) = p(d), then
I 6j= p
1
(d
1
) ^ ^ p
l
(d
l
) ^ :q
1
(e
1
) ^ ^ :q
m
(e
m
) (1)
where d
i
= J
v
(t
i
), for i = 1; 2; : : : ; l, and e
j
= J
v
(s
j
), for j = 1; 2; : : : ; m. It follows that
some of the p
i
(d
i
) do not belong to I, for 0 i l, and some of the q
j
(e
j
) do belong to
I, for 0 j m. Let p
1
(d
1
); : : : ; p
k
(d
k
), where 0 k l, be the totality of the p
i
(d
i
)
in (1) which do not belong to I, and let :q
1
(e
1
); : : : ;:q
k
0
(e
k
0
), where 0 k
0
m, be
the totality of the :q
j
(e
j
) in (1) for which q
j
(e
j
) does belong to I, in some reordering of
the body if necessary. Then I j= :p
1
(d
1
)^ ^ :p
k
(d
k
)^ q
1
(e
1
)^ ^ q
k
0
(e
k
0
). Carry
out this process for each instance of each program clause with
p(d) in the head. By
our hypothesis concerning P being of nite type relative to p(d)
and J , there are only
nitely many instances to consider. So let fp
1
(d
1
); : : : ; p
(d
)g be the entire collection
of all instances of positive literals (i.e. atoms) which occur
amongst the bodies of the
instances of the clauses referred to above and which do not
belong to I. Similarly, let
f:q
1
(e
1
); : : : ;:q
(e
)g be the entire collection of all instances of negative
literals which
occur amongst the bodies of the instances of the clauses
referred to above for which
q
1
(e
1
); : : : ; q
(e
) do belong to I. Then
I j= :p
1
(d
1
) ^ ^ :p
(d
) ^ q
1
(e
1
) ^ ^ q
(e
):
Put G
1
= G(:p
1
(d
1
)^ ^:p
(d
)^q
1
(e
1
)^ ^q
(e
)) so that G
1
is a neighbourhood
of I in Q. Now suppose that I
0
2 G
1
. Then I
0
j= :p
i
(d
i
), for i = 1; : : : ; , and
I
0
j= q
j
(e
j
), for j = 1; : : : ; . Thus, if p(d) C
j v is any instance of any program
clause with p(d) in its head, C
j v will either contain at least one p
i
(d
i
), where
i = 1; : : : ; , or at least one :q
j
(e
j
), where j = 1; : : : ; . It follows that I
0
6j= C
j v and
21
-
so p(d) 62 T (I
0
) i.e. T (I
0
) j= :p(d). Therefore, T (I
0
) 2 G
2
and we have T (G
1
) G
2
as
required.
Since a basic neighbourhood of T (I) is a nite intersection of
subbasic neighbour-
hoods of T (I), it suces to consider the two subbasic
neighbourhoods G(p(d)) and
G(:p(d)) as above. Hence, T is continuous at I in Q.2
Recall that a variable symbol (or simply variable) in a program
clause is called local
if it appears only in the body of that clause, see [27]. The
following corollary generalises
[6, Theorem 12].
Corollary 6 Suppose that J is a preinterpretation of L in which
each function symbol
f is assigned to an injective function f
0
. If P is a normal program, with underlying
language L, and P has no local variables, then T
J
P
is continuous in Q. In particular, if
J is the Herbrand preinterpretation of the underlying language
of a normal program P
and P has no local variables, then T
J
P
is continuous in Q.
Proof Let C be a program clause p(t) p
1
(t
1
); : : : ; p
n
(t
n
) in P and suppose that
x
1
; : : : ; x
k
are all the variable symbols which appear in p(t). By the
hypothesis that P
contains no local variable symbols, there are no other variable
symbols in C. Let v and
v
0
be two variable assignments and suppose that p(t j v) = p(t j
v
0
). By the hypothesis
concerning injectivity, which holds in particular in any
Herbrand preinterpretation, we
see that v(x
i
) = v
0
(x
i
) for i = 1; : : : ; k. From this it follows that C
j v = C
j v
0
and
in turn it follows that P is of nite type relative to J . The
conclusion now follows from
Theorem 7.2
Example 3. Consider the program P as follows
p(o)
p(s(x)) p(x)
Take J to be the preinterpretation dened in Example 1. It is
clear that P is of nite
type relative to J and so T
J
P
is continuous in Q in this case. Similarly, the same
conclusion is drawn if we take J to be the Herbrand
preinterpretation of P .2
We next give a useful result which is a partial converse to
Theorem 7, stated in
contrapositive form.
Theorem 8 Let P be a normal program and J be a preinterpretation
of the language
L of P . Suppose that P has a clause C of the form
p(t) p
1
(t
1
); : : : ; p
l
(t
l
);:q
1
(s
1
); : : : ;:q
m
(s
m
)
which has innitely many dierent J - instances with the same head
p(d), say, but is
such that no instance of any literal occurs innitely often in
the bodies. Then T
J
P
is
discontinuous in Q at any I 2 I
J
P
such that T
J
P
(I) j= :p(d).
Proof First we note that the hypothesis that C has innitely many
dierent instances
with the same head forces l +m 1.
Suppose I 2 I
J
P
and T (I) j= :p(d), where T denotes T
J
P
as usual. Consider the
neighbourhood G
2
= G(:p(d)) of T (I). Let G
1
= G(r
1
(f
1
) ^ ^ r
(f
) ^ :r
0
1
(f
0
1
) ^
22
-
^ :r
0
(f
0
)) be an arbitrary basic neighbourhood of I in Q. Thus, I j=
r
1
(f
1
) ^ ^
r
(f
)^ :r
0
1
(f
0
1
)^ ^ :r
0
(f
0
). By the hypothesis, there must be an instance C j v of
the clause C, say p(d) p
1
(d
1
); : : : ; p
l
(d
l
);:q
1
(e
1
); : : : ;:q
m
(e
m
), such that no r
i
(f
i
)
occurs (either positively or negatively) in the body of C j v,
for i = 1; : : : ; , and no
r
0
j
(f
0
j
) occurs (either positively or negatively) in the body of C j v,
for j = 1; : : : ; .
Let I
0
= fp
1
(d
1
); : : : ; p
l
(d
l
); r
1
(f
1
); : : : ; r
(f
)g; then I
0
2 G
1
and I
0
j= C
j v. Hence
p(d) 2 T (I
0
) i.e. T (I
0
) j= p(d) and so T (I
0
) 62 G
2
. Therefore, T (G
1
) 6 G
2
and so
T is discontinuous at I, as required, since a basic
neighbourhood G
1
suces for this
argument.2
We now consider some examples which illustrate this theorem, the
previous one and
Corollary 6.
Example 4.
1) Let P be the program
p(o)
p(s(x)) p(x)
p(s
3
(o)) p(s
3
(x)):
Let J be the Herbrand preinterpretation. By Theorem 8, T
J
P
is discontinuous at ,
fp(o)g, fp(s(o))g and fp(o); p(s(o))g since all these satisfy
T
J
P
(I) j= :p(s
3
(o)) and the
remaining hypotheses of Theorem 8 are satised.
On the other hand, if I is any interpretation other than the
four already listed and
T
J
P
(I) j= :p(s
n
(o)), for some n, then n 6= 3. But then P is of nite type
relative to J
and p(s
n
(o)) and therefore T
J
P
