Semantical and Computational Aspects of H o r n Approximations"
M a r c o Cadoli
Dipartimento di Informatica e Sistemistica
Universita di Roma "La Sapienza"
via Salaria 113, 1-00198 Roma, Italia
cadoliCassi.ing.uniromal.it
Abstract
In a recent s t u d y Selman a n d K a u t z proposed a
m e t h o d , called Horn approximation, for speedi n g u p inference i n p r o p o s i t i o n a l Knowledge
Bases. T h e i r technique is based on the compilation of a p r o p o s i t i o n a l f o r m u l a i n t o a pair of
H o r n f o r m u l a e : a H o r n Greatest Lower B o u n d
( G L B ) a n d a H o r n Least U p p e r B o u n d ( L U B ) .
In this paper we address t w o questions t h a t
have been o n l y m a r g i n a l l y addressed so far: 1)
w h a t is t h e semantics of the H o r n a p p r o x i m a tions? 2) w h a t is the exact c o m p l e x i t y of findi n g H o r n a p p r o x i m a t i o n s ? W e o b t a i n semantical as well as c o m p u t a t i o n a l results. T h e m a j o r
results o f the f o r m e r k i n d are: H o r n G L B s are
closely related to models of the c i r c u m s c r i p t i o n ;
reasoning w r t the H o r n L U B can b e m a p p e d
i n t o classical reasoning. T h e m a j o r results of
the l a t t e r k i n d are: f i n d i n g a H o r n G L B is
" m i l d l y " harder t h a n solving the o r i g i n a l i n ference p r o b l e m ; finding the H o r n L U B is a
search p r o b l e m t h a t cannot be parallelized. We
believe t h a t our results provide useful criteria
t h a t m a y help finding a knowledge c o m p i l a t i o n
policy.
1
Introduction
In a recent s t u d y [Selman a n d K a u t z , 1991; K a u t z and
Selman, 1992] Selman a n d K a u t z proposed a m e t h o d ,
called Horn approximation, for speeding up inference
in p r o p o s i t i o n a l K n o w l e d g e Bases. P r o p o s i t i o n a l inference is the p r o b l e m of checking whether
holds,
where
and
are p r o p o s i t i o n a l f o r m u l a e . T h e s t a r t i n g p o i n t o f t h e i r technique stems f r o m the fact t h a t
inference for general p r o p o s i t i o n a l f o r m u l a e is c o - N P complete — h e n c e p o l y n o m i a l l y unfeasible— while it is
doable in p o l y n o m i a l t i m e w h e n
is a H o r n f o r m u l a .
T h e f a s c i n a t i n g question t h e y address is the f o l l o w i n g :
is it possible to compile a p r o p o s i t i o n a l f o r m u l a
into
•Work supported by the E S P R I T Basic Research Action
N.3012-COMPULOG and by the Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo of the C N R (Italian Research Council).
Cadoli
39
Selman and Kautz's proposal is to approximate inference
wrt a propositional formula
by using its Horn GLBs
and LUBs. In this way inference could be unsound or
incomplete, but it is anyway possible to spend more time
and use a general inference procedure to determine the
answer directly from the original formula. The general
inference procedure could still use the approximations
to prune its search space (see [Selman and Kautz, 1991,
page 905]). It is also important to notice that Horn GLBs
and LUBs can be computed off-line, hence this form of
approximate reasoning is actually a compilation.
Table 1 summarizes the major properties of Horn
GLBs and LUBs stated in [Selman and Kautz, 1991;
Kautz and Selman, 1992]. The four columns refer respectively to:
• logical relation wrt
(i. e. what kind of inference
can be performed using this approximation?);
• size of the formula wrt the size
• number of possible approximations of this kind;
• computational complexity of the search problem of
finding the approximation.
Table 1 : Some properties o f H o r n G L B s a n d L U B s .
H o r n a p p r o x i m a t i o n s have t w o c o m p u t a t i o n a l problems:
1) c o m p u t i n g t h e m is an N P - h a r d task a n d 2) due to its
e x p o n e n t i a l size, it m a y be impossible to store the H o r n
L U B . A b o u t the f i r s t aspect Selman a n d K a u t z notice
t h a t since a p p r o x i m a t i o n s could be c o m p u t e d off-line,
the c o m p u t a t i o n a l cost of finding t h e m w i l l be a m o r t i z e d
over the t o t a l set of subsequent queries to the Knowledge
Base. W i t h respect to the second aspect, they propose in
[ K a u t z a n d S e l m a n , 1992] a technique for "compressing"
the H o r n L U B i n t o a (quasi-)equivalent f o r m u l a . Due
to reasons related to circuit c o m p l e x i t y theory, it is not
possible to a p p l y the technique in general (see [ K a u t z
a n d S e l m a n , 1992] for f u r t h e r details).
O t h e r c o m p u t a t i o n a l properties o f H o r n a p p r o x i m a tions are s t u d i e d in [Greiner a n d Schuurmans, 1992;
R o t h , 1993].
I n this paper w e address t w o i m p o r t a n t questions t h a t
have n o t been addressed so far:
1. is it possible to describe H o r n a p p r o x i m a t i o n s w i t h a
semantics t h a t does n o t rely on the syntactic n o t i o n
o f H o r n clause?
2. w h a t is t h e exact c o m p l e x i t y of finding H o r n approximations?
40
Automated Reasoning
An answer to the first question shows the exact meaning
of the approximate answers. An answer to the second
question tells in which cases it is reasonable —from the
computational point of view— to use Horn approximations.
We obtain two different kinds of results:
semantical
• Horn GLBs of
are closely related to models
of the circumscription of
• reasoning wrt Horn GLBs is the same as reasoning by counterexamples using only minimal
models;
• while skeptical reasoning wrt the Horn GLBs of
a formula
is the same as ordinary reasoning
wrt
brave reasoning wrt the Horn GLBs of
is the same as reasoning wrt CIRC
• compiling more knowledge does not always give
better Horn GLBs;
• reasoning wrt the Horn L U B can be mapped
into classical reasoning;
• the Horn L U B of
is related to
CWA
computational
• finding a Horn GLB is "mildly" harder than
solving the original inference problem;
• reasoning wrt the Horn L U B is exactly as hard
as solving the original inference problem;
• finding a Horn UB is a search problem that
cannot be parallelized.
We believe that our results provide useful criteria that
may help finding a knowledge compilation policy. In
particular, we show that an interesting tradeoff seems to
emerge between the computation done during the compilation time and the computation done during the query
answering time.
The structure of the paper is as follows: in Section 2
and 3 we study Horn GLBs and LUBs, respectively; we
discuss our results in Section 4.
T h e o r e m 1 Let
be a p r o p o s i t i o n a l f o r m u l a a n d
a
Horn G L B of
The m i n i m u m model o f
is minimal
for
T h e o r e m 1 implies t h a t if we have a H o r n G L B of
t h e n we can o b t a i n in linear t i m e (see [ D o w l i n g a n d
Gallier, 1984]) a m i n i m a l m o d e l of
M o r e technically, the t h e o r e m shows a p o l y n o m i a l r e d u c t i o n from
the search p r o b l e m of f i n d i n g a m i n i m a l m o d e l of
to
the search p r o b l e m o f f i n d i n g a H o r n G L B o f
The
present a u t h o r analyzed in [ C a d o l i , 1992] the c o m p u t a t i o n a l c o m p l e x i t y of the search p r o b l e m of f i n d i n g a
m i n i m a l m o d e l of a p r o p o s i t i o n a l f o r m u l a . One of the
results of t h a t paper is t h a t f i n d i n g a m i n i m a l m o d e l of
a formula
is h a r d (using m a n y - o n e reductions) w i t h
respect to the class
It is i m p o r t a n t to
remark that
h a r d problems are in a precise
sense c o m p u t a t i o n a l l y harder t h a n N P - c o m p l e t e or coN P - c o m p l e t e problems 2 . W e recall t h a t the p r o b l e m o f
deciding whether
holds is c o - N P - c o m p l e t e .
As shown in [Cadoli, 1992],
hardness of
f i n d i n g a m i n i m a l m o d e l holds even if a m o d e l of
is
k n o w n . T h i s fact can be c o m p a r e d w i t h a consideration
i n [Selman a n d K a u t z , 1 9 9 1 , T h e o r e m l ] :
i s satisfiable iff
is satisfiable, hence f i n d i n g a H o r n G L B is
N P - h a r d . We can now say t h a t even if we k n o w t h a t
is
satisfiable a n d have one of its models in h a n d , f i n d i n g a
Horn G L B is still
h a r d . W e recall t h a t f i n d i n g a m o d e l ( n o t necessarily m i n i m a l ) of a p r o p o s i t i o n a l
f o r m u l a is per se an N P - h a r d task.
C o r o l l a r y 2 F i n d i n g a H o r n G L B of a p r o p o s i t i o n a l
formula
is
hard.
This holds even if a
model of E is already k n o w n .
We notice t h a t the above corollary gives us j u s t a lower
b o u n d . It is reasonable to ask how easy is to f i n d a H o r n
G L B , i . e . t o give a n upper b o u n d t o the p r o b l e m . I n
[Selman a n d K a u t z , 1 9 9 l ] a n a l g o r i t h m for c o m p u t i n g
a H o r n G L B of a f o r m u l a
is s h o w n . T h e a l g o r i t h m
performs a n e x p o n e n t i a l n u m b e r o f p o l y n o m i a l steps. I t
is possible to show t h a t a H o r n G L B can be f o u n d in
p o l y n o m i a l t i m e b y a d e t e r m i n i s t i c T u r i n g machine w i t h
access to an NP oracle, i. e. to prove t h a t the p r o b l e m
is in the class
. T h i s means t h a t we o n l y need a
p o l y n o m i a l n u m b e r o f queries t o the G L B i n order t o
" p a y off" the overhead of the knowledge c o m p i l a t i o n .
2.2
From minimal models to G L B s
We now show t h a t if we have a m i n i m a l m o d e l M of a
formula
t h e n we can easily b u i l d a very g o o d a p p r o x imation of a Horn G L B of
I n p a r t i c u l a r w e show t h a t
w e can b u i l d i n linear t i m e a H o r n L B o f
whose m i n i m u m m o d e l i s M . T h i s result allows u s t o p e r f o r m , i n
is the class of decision problems that can be
computed by a polynomial-time deterministic machine which
can use for free an oracle (or subroutine) that answers a set
of NP-complete queries (e. g. satisfiability checks) whose cardinality is bound by a logarithmic function. We refer the
reader to [Johnson, 1990] for a thorough description of all
the complexity classes that are cited in this paper.
2
B o t h NP-complete and co-NP-complete problems can be
solved w i t h a single call to an oracle in N P .
Cadoli
41
42
Automated Reasoning
m a n , 1992] in general it is n o t possible to store efficiently
the H o r n L U B o f
I n p a r t i c u l a r the size o f
can b e
e x p o n e n t i a l in t h e size of
a n d this seems to be independent on the representation used for
(see [ K a u t z
a n d Selman, 1992] for f u r t h e r details). As a consequence
any m e t h o d for efficiently representing the H o r n L U B
is i n c o m p l e t e . In [Selman a n d K a u t z , 1 9 9 1 , page 908]
t h e a u t h o r s propose t o approximate the H o r n L U B w i t h
H o r n upper b o u n d s o f l i m i t e d l e n g t h . T h i s idea i s used i n
[Greiner a n d Schuurmans, 1992], where H o r n U B s w i t h
a l i m i t e d n u m b e r of H o r n clauses are s t u d i e d . In Subsection 3.1 we investigate a b o u t this idea a n d analyze its
c o m p u t a t i o n a l properties. O t h e r c o m p u t a t i o n a l properties of H o r n L U B s are addressed in Subsection 3.2. In
Subsection 3.3 we make a brief semantical r e m a r k .
Cadoli
43
3.3
A semantical r e m a r k
Equation (1) gives a sound and complete characterization of inference wrt
in terms of classical propositional inference. In this subsection we make a brief
remark about the relation existing between Horn LUBs
and closed-world assumption [Reiter, 1978].
O b s e r v a t i o n 6 Let M be the minimum model of
M is the intersection of all the minimal models of
Therefore M is a model of
iff the closed-world assumption
is consistent.
We notice that
may be consistent even i f i s
non-Horn: The CWA of
is consistent. Relations between Horn LUBs and closed-world reasoning
are implicit in the works [Borgida and Etherington, 1989;
Selman and Kautz, 199l].
4
Discussion
The computational results that we have seen in Sections 2 and 3 show that when we deal with knowledge
compilation there exists an interesting tradeoff between
computation during compile time (off-line) and computation during query-answering time (on-line).
In Section 2 we have seen that the computational effort
of finding a Horn G L B is justified only if a significant
number of queries to it will be done. In particular we
have seen that the compilation is more expensive than
a set of query answering tasks. The size of such a set
has a lower bound which is a function logarithmic in the
size of the input and an upper bound which is a function
polynomial in the size of the input.
In Section 3 we have obtained similar results, showing
that high-quality Horn UBs need a significant computational effort.
Since compilation causes anyway loss of information
(either soundness or completeness), the computational
effort spent in the compilation must be compared to the
quality of the inference obtained. It is an open issue to
find an adequate formal framework for comparing the
two aspects.
Acknowledgments
I am grateful to H. Kautz and B. Selman for fruitful discussions on their method. I had interesting discussions
also with L. C. Aiello, M. Lenzerini, E. Omodeo and M.
Schaerf, who also read previous versions of the paper.
Thanks to D. Schuurmans for sending me a preliminary
version of his paper.
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