From: AAAI-88 Proceedings. Copyright ©1988, AAAI (www.aaai.org). All rights reserved.
2
The Circumscription
Existent
ial Formulae
of
and distributing
the existential
quantifiers
junction,
this formula is transformed
to
Let 4(P) be a first-order
or a second-order
formula with
equality, involving the sequence P = (Pi, . . . , Ph) of predicate symbols and possibly other predicate
symbols from
Following
McCarthy
a fixed underlying
vocabulary
u.
[Li85], we define the cir[McC80, McC86] and Lifschitz
cumscription
ofP in 4(P) to be the following second-order
formula 4*(P):
3~13~2(f@1,~2)
V
Theorem
1. Suppose that d(P) is an existential
firstorder sentence of the form 3x$, where x = (xi, . ..x~) is
a sequence of variables and $ is quantifier-free
formula.
Then the circumscription
qS+(P) of P in #(P) is equivalent
to a first-order
formula. 0
The proof of Theorem
1 constructs
a first-order
formula
equivalent
to the circumscription.
We start by bringing
$ in its complete disjunctive
normal form, that is, $ is
written
as the disjunction
of several formulae
Bi, where
each Bi is the conjunction
of literals, where a literal can
be either an atomic formula, or its negation, or an equality between two variables, or an inequality
(#) between
two variables;
moreover,
‘each disju.nct contains at least
one of the literals xi = Xj or xi #.-Xj
for any two variables xi, xj (that is, it determines
an equality type). Next,
we distribute
the existential
quantifiers
over the disjunction, and thus we have to show that each disjunct of the
form 3~6iA(‘v’P’)[(P
< P),--+ -(V~=,
3xoj)] is first order.
However, since only existential. quantifiers occur in this disjunct and Bi has a fixed equality type (in other words, the
mapping from variables to constants is fixed up to renamings), the assertion concerning. P above can be replaced
by a first-order
formula stating that P is a cert.ain finite
set and no proper subset of it satisfies Vj”=, 3x6j (the’ latter statement
can be expressed by an exponentially
long
first-order
formula, ranging over subsets of the set. of constants determined
by the equality type).
This completes
the construction.
3Xl3X2(fqXl,
using the procedure
quantifier-free
part
466
Knowledge
the circumscription
x2)
A P(Xl)
Representation
of P in the
A P(x2))
described above. After
in complete
disjunctive
x2)
bringing
the
normal form
A p(x2)
A P(n)
A (xl
A p(x2)
=
A (XI
of P in the above formula
dis-
~2))
#
x2)).
is equiva.lent
to
h(f+l,
xl) A P(Q) A ((VY)(~(Y)
[3xdx2(R(xl,
where P’ = (Pi, . . . , Pi) is a sequence of predicates and
P’ < P means that Pi c Pi, 1 2 i < k, and there is a
j 5 E such that Pi is a proper subset of Pj.
Several interesting
cases have been pointed out in the
recent past, in which the circumscription
of a first-order
formula collapses to a first-order
formula.
Lifschitz [Li85]
showed that this holds for the class of separable formula.e,
a natural and fairly wide class that includes all quantifierfree formulae.
Such results, reducing the logical complexity of circumscription
from second-order
to first-order,
are
potentially
valuable, in view of the intractability
of secondorder logic on the one hand and the completeness theorem
for first-order
logic on the other.
We show below that the same holds for all exktential
formulae.
We compute
3~13~2(R(x1,
The circumscription
4(P) A (‘dP’)[(P’
< w + ~w%
Example:
formuia
A P(w)
over the
x2) A P(xI)
((VY>(~(Y>
(-(xl,
c-)
xl))
(Y
=
A (+(x2,
++ Y = x:l)>V
A P(x2)
~1
V
Y
=
x2)))].
A (xl
# x2)A
22)))A
0
We notice that computing
a first-order
sentence equivalent to the circumscription
of P in an existential
first-order
formula I#( P) seems to increase the size of 4(P) exponentially, a phenomenon
not observed in the other known cases
of first-order
circumscription
studied in [Li85]. It would be
interesting
to determine
whether this is inherent to existential first-order
formulae, or a particular
creation of our
proof.
In the full paper we shall also prove that Theorem
1
can be extended in several directions:
It holds for formulae
containing not only relation symbols, but also function a.nd
with
constant symbols. Also, it holds for circumscription
variables (a more general variant).
Finally, it is also true
of existential
second-order
formulae,
tl1a.t is, second-order
formulae whose second-order
and first-order
qua.ntifiers are
all existential.
3
Circumscription
Boundedness
and
The positive results for the existential
formulae in the preceding section suggest that one should examine next the
class of universal first-order
formulae.
Other properties
of
the circumscription
of universal formulae have been studied before and it is known, for example,
that this cla.ss
of formulae behaves nicely with respect to the satisfiability of circumscription
(cf. [BSSS], [EMR$5],
[LiSG]). As
mentioned
in the introduction,
however, Lifschitz
[Li$5]
observed that there are universal formulae
(actually
conjunctions
of function-free
Horn clauses) whose circumscription is not first-order
expressible.
In view of this, the best
possible result one could hope for is a computationally
useful characterization
of the universal sentences that have a
first-order
circumscription.
In this section we establish a connection between the circumscription
of a conjunction
of function-free
Horn clauses
and the convergence of the corresponding
logic program.
More specifically,
we show that the circumscription
of a
conjunction
of Horn clauses is first-order
if and only if
the corresponding
progra.m is bounded.
Boundedness
is a
property
of logic programs that has been showed recently
by Gaifman et al. [GMSV87]
to be an undecidable
problem. Thus, it is not possible to give a computationally
usehi characterization
of which universa.i first-order
forIn spite of
mulae possess a first-order
circumscription.
these negative consequences,
our result suggests that it
may be possible to identify
wide subcla.sses of universal
formulae on which there are algorithms
that detect when
there a different formalization of common-sense reasoning,
which on the one hand is computationally
more tract,ahle
than circumscription, and on the other retains most salient
features of it ?
We are grateful to Vladimir Lifschitz for several useful comments and suggestions on an
earlier version of this paper. The research of the second
author was partially supported by a NSF grant
Acknowledgments.
6
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4.69