Unfolding Partiality and Disjunctions in Stable Model Semantics

0.64 MB·English

by  T. Janhunen

Unfolding Partiality and Disjun
tions
∗
in Stable Model Semanti
s
4
0 Tomi Janhunen and Ilkka Niemelä
0
2 Department of Computer S
ien
e and Engineering

n Helsinki University of Te
hnology
a
J P.O.Box 5400, FIN-02015 HUT, Finland

2
{Tomi.Janhunen,Ilkka.Niemela}hut.(cid:28)

]
I
A
Dietmar Seipel
.
s
c University of Würzburg
[
Am Hubland, D-97074 Würzburg, Germany
2
v seipelinformatik.uni-wuerzburg.de
9
0
0
Patrik Simons
3
0
Neotide Oy
3
0 Wol(cid:30)ntie 36
/
s
FIN-65200 Vaasa, Finland
c
:
v Patrik.Simonsneotide.(cid:28)
i
X
r
a Jia-Huai You
Department of Computing S
ien
e
University of Alberta
Edmonton, Alberta, Canada T6G 2H1
you
s.ualberta.
a
∗
A preliminary version of this paper [21℄ appears in the Pro
eedings of the 7th In-
ternational Conferen
e on the Prin
iples of Knowledge Representation and Reasoning,
KR'2000.
1
Abstra
t
The paper studies an implementation methodology for partial and
disjun
tive stable models where partiality and disjun
tions are un-
foldedfromalogi
programsothatanimplementationofstablemodels
for normal (disjun
tion-free) programs
an be used as the
ore infer-
en
e engine. The unfolding is done in two separate steps. Firstly, it is
shown that partial stable models
an be
aptured by total stable mod-
elsusing asimple linear and modular program transformation. Hen
e,
reasoning tasks
on
erning partial stable models
an be solved using
an implementation of total stable models. Disjun
tive partial stable
models have been la
king implementations whi
h now be
ome avail-
able as the translation handles also the disjun
tive
ase. Se
ondly, it
is shown how total stable models of disjun
tive programs
an be de-
termined by
omputing stable models for normal programs. Hen
e, an
implementation of stable models of normal programs
an be used as a

ore engine for implementing disjun
tive programs. The feasibility of
the approa
h is demonstrated by
onstru
ting a system for
omput-
ing stable models of disjun
tive programs using the smodels system
as the
ore engine. The performan
e of the resulting system is
om-
pared to that of dlv whi
h is a state-of-the-art system for disjun
tive
programs.
1 INTRODUCTION
Implementation te
hniques for de
larative semanti
s of logi
programs have
advan
ed
onsiderably during the last years. For example, the XSB sys-
tem [40℄ is a WAM-based fulllogi
programmingsystem supporting the well-
founded semanti
s. In addition to this kind of a skepti
al approa
h that is
based on query evaluation also a
redulous approa
h fo
using on
omputing
models of logi
programs is gaining popularity. This work has been
entered
around the stable model semanti
s [17, 18℄. There are reasonably e(cid:30)
ient
implementations available for
omputing stable models for disjun
tive and
normal (disjun
tion-free) programs, e.g., dlv [24℄, smodels [46, 45℄,
mod-
els [1℄, and assat [29℄. The implementations have provided a basis for a
new paradigm for logi
programming
alled answer set programming (a term

oined by Vladimir Lifs
hitz). The basi
idea is that a problem is solved by
devising a logi
program su
h that the stable models of the program provide
the answers to the problem, i.e., solving the problem is redu
ed to a stable
2
model
omputationtask [27, 32, 34, 13, 5℄. This approa
h has led to interest-
ing appli
ations in areas su
h as planning [8, 11, 2℄, model
he
king [30, 20℄,
and software
on(cid:28)guration [49℄.
This paper addresses two issues in the stable model semanti
s: partial-
ity and disjun
tions. The idea is to develop methodology su
h that e(cid:30)
ient
pro
edures for
omputing (total) stable models that are emerging
an be ex-
ploited when dealing with partial stable models and disjun
tive programs.
Sometimes it is natural to use partial stable models to represent a domain.
Even when working with total stable models, partial stable models
ould
be useful, e.g., for debugging purposes to show what is wrong in a program
without any total stable models. However, little has been done on imple-
menting the
omputation of partial stable models and most of the work has
fo
used on query evaluation w.r.t. the well-founded semanti
s. In the paper
we show that total stable models
an
apture partial stable models using
a simple linear program transformation. This transformation works also in
the disjun
tive
ase showing that implementations of total stable models,
e.g. dlv,
an be used for
omputing partial stable models. Using a suitable
transformation of queries, a me
hanism for query answering
an be realized
as well.
Our translation is interesting in many respe
ts. First, it should be noted
that the translation does not follow dire
tly from the
omplexity results al-
readyavailable. Ithasbeenshown, e.g., thattheproblemofdΣe
pidingwhether
a query is
ontained in some model (possibility inferen
e) is 2-
omplete for
both partial and total stable models [12, 15℄. This implies that there exists a
polynomial time redu
tion from possibility inferen
e w.r.t. partial models to
possibility inferen
e w.r.t. total models. However, this kind of a translation
is guaranteed to preserve only the yes/no answer to the possibility inferen
e
problem. Se
ond, not all translations are satisfa
tory from a
omputational
point of view. In pra
ti
e, when a program is
ompiled into another form
to be exe
uted,
ertain
omputational properties of the translation play an
important role:
• e(cid:30)
ien
y of the
ompilation (in whi
h order of polynomial),
• modularity (are independent, separate
ompilations of parts of a pro-
gram possible), and
• stru
tural preservation (are the
omposition and intuition of the origi-
nalprogrampreserved so that debugging and understandingof runtime
3
behavior are made possible).
All this points to the importan
e of (cid:28)nding good translation methods to
enable the use of an existing inferen
e engine to solve other interesting prob-
lems.
The e(cid:30)
ien
y of pro
edures for
omputing stable models of normal pro-
grams has in
reased substantially in re
ent years. An interesting possibility
to exploit the
omputational power of su
h a pro
edure is to use it as a
ore
engine for implementing other reasoning systems. In this paper, we follow
this approa
h and develop a method for redu
ing stable model
omputation
of disjun
tive programs to the problem of determining stable models for nor-
mal programs. This isΣnpon-trivial as de
iding whether a disjun
NtiPve program
has a stable model is 2-
omplete [12℄ whereas the problem is -
omplete
in the non-disjun
tive
ase [31℄. The method has been implemented using
the smodels system [46, 45℄ as the
ore engine. The performan
e of the im-
plementation is
ompared to that of dlv, whi
h is a state-of-the-art system
for
omputing stable models for disjun
tive programs.
There are a number of novelties in the work. Maximal partial stable
models for normal programs are known as regular models, M-stable models,
and preferred extensions [10, 39, 50℄. Although this semanti
s has a sound
and
omplete top-down query answering pro
edure [10, 16, 28℄, so far very
little e(cid:27)ort has been given to a serious implementation. For disjun
tive pro-
grams, to our knowledge, no implementation has ever been attempted. As
a result, we obtain (perhaps) the (cid:28)rst s
alable implementation of the regu-
lar model/preferred extension semanti
s, and the (cid:28)rst implementation ever
for partial stable model semanti
s for disjun
tive programs. Our te
hni-

al work on the relationship between stable and partial stable models via a
translational approa
h provides a
ompelling argument for the naturalness
of partial stable models: stable models and partial stable models share the
same notion of unfoundedness,
arefully studied earlier in [14, 26℄. Finally,
we demonstrate how key tasks in
omputing disjun
tive stable models
an
be redu
ed to stable model
omputation for normal programs by suitable
program transformations. In parti
ular, we develop te
hniques for mapping
a disjun
tive program into a normal one su
h that the set of stable models
of the normal program
overs the set of stable models of the disjun
tive one
and in many
ase even
oin
ides with it. Moreover, we devise a method
where the stability of a model
andidate for a disjun
tive program
an be
determined by transforming the disjun
tive program into a normal one and
4
he
king the existen
e of a stable model for it. Finally, in the experimental
part of this paper, we present a new way of en
oding quanti(cid:28)ed Boolean
formulas as disjun
tive logi
programs. This transformation is more e
onom-
i
al in the number of propositional atoms and disjun
tive rules than earlier
transformations presented in the literature [12, 25℄.
The rest of the paper is stru
tured as follows. We (cid:28)rst review the basi
de(cid:28)nitions and
on
epts in Se
tion 2. It is then shown in Se
tion 3 that par-
tial stable models
an be
aptured with total stable models using a simple
program transformation. In Se
tion 4, we des
ribe the method for
om-
puting disjun
tive stable models using an implementation of non-disjun
tive
programs as a
ore engine. After this, we present some experimental results
in Se
tion 5 and (cid:28)nish with
on
luding remarks in Se
tion 6.
As a
omment on the histori
al development of the translation given in
Se
tion 3, the
hara
terization of partial stable models as stable models of
the transformed program was (cid:28)rst sket
hed for normal programs in a proof
by S
hlipf [42, Theorem 3.2℄. Fordisjun
tive programs, itwas dis
overed and
proven in [43℄, and independently in [21℄. In the
urrent paper we present a
proof based on unfounded sets, whi
h was given in [21℄, as this proof reveals
some of the properties of unfounded sets whi
h are of interest in their own
right. Yet another approa
h to
omputing the partial stable models of a
disjun
tive program based on a program transformation has been developed
P
byRuizandMinker[37℄: adisjun
tiveprogram istranslatedintoapositive
P3S P
disjun
tive program with
onstraints, the 3S(cid:21)transformation of , su
h
P3S
that the total minimal models of that additionally ful(cid:28)ll the
onstraints
P

oin
ide with the partial stable models of .
2 DEFINITIONS AND NOTATIONS
P P
A disjun
tive logi
program (or, just disjun
tive program ) isa set of rules
of the form
a a b ,...,b , c ,..., c
1 k 1 m 1 n
∨···∨ ← ∼ ∼ (1)
k 1 m,n 0 a b c
i i i
where ≥ , ≥ and 's, 's and 's are atoms from the Herbrand
Hb(P) P
1
base of . Let us also distinguish sub
lasses of disjun
tive programs.
P
1
Forthesakeof
onvenien
e,weassumethatagivenprogram isalreadyinstantiated
by the underlying Herbrand universe, and is thus ground.
5
k = 1 P P
If for ea
h rule of , then is a disjun
tion-free or normal program.
n = 0 P P
If for ea
h rule of , then is
alled positive.
Hb(P) a
Literals are either atoms from or expressions of the form ∼ where
a Hb(P) A Hb(P) A a a A
∈ . For a set of atoms ⊆ , we de(cid:28)ne ∼ as {∼ | ∈ }.
A B, C A = B C
Let us introdu
e a shorthand ← ∼ for rules where 6 ∅, and
Hb(P) A
are subsets of . In harmony with (1), the set of atoms in the head
B C
of the rule is interpreted disjun
tively while the set of literals ∪ ∼ in
the body of the rule is interpreted
onjun
tively. We wish to further simplify
A B, C A B C
the notation ← ∼ in some parti
ular
ases. When , or is a
a a a B = C = B
singleton { }, we write instead of { }. If ∅ or ∅ we omit and
C
∼ (respe
tively) as well as the separating
omma in the body of the rule.
2.1 PARTIAL AND TOTAL MODELS
We review the basi
model-theoreti

on
epts by following the presentation
P I P
in [15℄. Let be any disjun
tive program. A partial interpretation for is
T,F Hb(P) T F =
atpair h f i of subsetus of su
h that ∩ ∅. The atoms in the sets
I = T I = F I = Hb(P) (T F)
, and − ∪ are
onsidterfed to bue true, false,
and unde(cid:28)ned, respe
tively. We introdu
e
onstants , , and , to denote
I P
the respe
tive three truth valuesu. A partial interpretation for is a total
P I = Hb(P)
interpretation for whenever ∅, i.e., if everytatom of is either
I
true or false. When no
onfufsion arises, wetuse ualone to spe
ify a total
I P I = Hb(P) I I =
interpretation for (then − and ∅ hold).
P
Given a pIatrtiIafl interIpuretation for , the truth vatluefs of atuoms are de-
termined by , and as explained above while , and have their
E I(E)
(cid:28)xed truth values. For morEe
omIplex logi
al exIp(resas)ions , we use t tfo
denuote the trutIh(av)aluefoft inu . The value ∼ is de(cid:28)ned to be , ,
or whenever is , , or , respe
tively. To handle
onjun
tions and
dfi<sjuun
<tiotns, we introdu
e an orderingLon=thle,t.h.r.e,el truth values by setting
1 n
. By default, a set of literals { } denotes the
onjun
-
l1 ln WL l1 ln
tion ∧···∧ while denotes the
orresponding disjun
tion ∨···∨ .
I(L) I(WL)
The truth values and are de(cid:28)ned as the respe
tive minimum
I(l ),...,I(l ) A B, C
1 n
and maximum among the truth values . A rule ← ∼
I I(WA) I(B C)
is satis(cid:28)ed in if and only if ≥ ∪∼ . A partial interpretation
M P P P M
for is a partial muodel of if all rules of are satis(cid:28)ed in , and for
M =
a total model, also ∅ holds. Let us then introdu
e an orderingt amongt
M M M M
1 2 1 2
partial mf odels off a disjun
tive program: ≤ if and only if ⊆
M M M P
1 2
and ⊇ . A partial model of is a minimal one if there is no
6
M′ P M′ < M M′ M M′ = M
partial model of su
h that (i.e., ≤ and 6 ). In
N N N N
1 2 1 2

ase of total models, we have ≤ if and only if ⊆ . Moreover,
N P
a totalNm′odePl of is
Non′sideNred to be a minimal one if there is no total
model of su
h that ⊂ .
2.2 STABLE MODELS
I P
Given a partial interpretation for a disjun
tive program , we de(cid:28)ne a
P
redu
tion of as follows:
PI = A B A B, C P C If .
{ ← | ← ∼ ∈ and ⊆ }
Note that this transformation
oin
ides with the Gelfond-Lifs
hitz redu
tion
P P I
of (the GL-redu
tion of ) when is a total interpretation.
N
De(cid:28)nition 2.1 (Total stable model) A total interpretation for a dis-
P N
jun
tive program is a stable model if and only if is a minimal total model
PN
of .
The original de(cid:28)nition of partial stable models [35, 36℄ is based on a
P
weaker redu
tion. Given a disjun
tive program and an interpretation
I P P
I
, the redu
tion is the set of rules obtained from by repla
ing any
c I( c)
∼ in the body of a rule by ∼ . As noted in [35℄, the only pra
ti
al
PI P P
I I
di(cid:27)Aeren
eBb,etCweenP and isI(thaCt) = huas rules that
oIrr(espCo)nd=tto rules
oAf ←B ∼PI ∈ satIi(sfyCin)g= ∼f . Note that if ∼ P , then
I
← ∈ , and if ∼ , then the partial models of are not
P
I

onstrained by the rule in
luded in .
M
De(cid:28)nition 2.2 (Partial stable model) A partial interpretation for a
P P M
disjun
tive program is a partial stable model of if and only if is a
P
M
minimal partial model of .
M P
M
In the above de(cid:28)nition, the relation between and is similar to
the one for the total stable model, both for the purpose of preserving the
stability
ondition. While maximizing falsity and minimizing true atoms, a
partial stable model does not insist that every atom must be either true or
false. (Partial)stablemodelsareintimatelyrelatedtounfoundedsets[14,26℄.
I
De(cid:28)nition 2.3 (Unfounded sets) Let be a partial interpretation for a
P U Hb(P)
disjun
tive program . A set ⊆ of ground atoms is an unfounded
7
P I
set for w.r.t. , if at least one of the following
onditions holds for ea
h
A B, C P A U =
rule ← ∼ ∈ su
h that ∩ 6 ∅:
f t
B I = C I =
UF1: ∩ 6 ∅ or ∩ 6 ∅,
B U =
UF2: ∩ 6 ∅, or
t u
(A U) (I I ) =
UF3: − ∩ ∪ 6 ∅.
t
U P I I U I =
An unfounded set for w.r.t. is -
onsistent if and only if ∩ ∅.
The
onditions UF1 and UF3 above
oin
ide with the
onditions
I(B C) = f I(W(A U)) = f
∪∼ and − 6 ,
U
respe
tively. The intuition is that the atoms of an unfounded set
an be
A B, C
assumedtobefalsewithoutviolatingthesatis(cid:28)abilityofanyrule ← ∼
U
of the program whose head
ontains some atoms of . For any su
h rule,
I
either the rule body is false in (UF1), or the rule body
an be falsi(cid:28)ed
U I
by falsifying the atoms in (UF2), or the head of the rule is not false in
(UF3). In parti
ular, unfounded sets w.r.t. partial/total models
an be used
for
onstru
ting smallerpartial/totalmodels (re
allthe de(cid:28)nitionof minimal
partial and total models) in a way that is made pre
ise by what follows.
M = T,F
Lemma 2.4 Let h i be a partial model of a positive disjun
tive
P U P M M
UprograMm and aMn u′ n=fouTndedU,seFt foUr w.r.t. . Then, if P is total or
is -
onsistent, h − ∪ i is a partial model of .
M P U M′
PROOF. Let , , and be de(cid:28)ned asabove. Additionally,we assume
M U M
thatA(a) Bis toPtal, or (b) is -
Mon′sistent. Let us thenMa′s(sWumAe)t<haMt s′(oBm)e
rule ←M′(oWf A)is<nott satiMs(cid:28)e′(dBi)n= t whi
h mMea′(nWs tAh)at= f M′(B) = u.
Thus (i) and , or (ii) and .
Our proof splits in two separate threads.
A U = M′(WA) < t
I. Assume that ∩ ∅ holds. Consider the
ase (i). Now
A (T U) = A U =
iAmpliTes= ∩ − M(W∅.A)Si<n
et ∩ ∅ holds, tooM, w′(eBo)b=taint
∩ B ∅ soTthatU B . TOn theMo(tBhe)r=hatnd,
implies ⊆ − . Thus ⊆ and holds as well.
M(WA) < M(B)
But then M′(WA) = f , a
onAtradiF
tionU. The
ase (iAi) is aFnalyzed
AnextU. N=ow M(WA)im=pflies ⊆ ∪ asMwe′(lBl a)s= u⊆ , sin
e
B∩(F U∅).=Thus . MBorFeo=ver, from M(B) > fwe obtain
∩ ∪ ∅. Thuswe obtain ∩ 6 ∅sothat holds. To
M(WA) < M(B)

on
lude, we have established that , a
ontradi
tion.
8
A U =
II. Otherwise ∩ 6 ∅ holdsA. TheBn at least one oMf (tBhe)u=nfofundedness

onditions isBappliF
able tBo ←F U. If UF1Mis,′(B) = f holds. It
follows that ⊆ and ⊆ ∪ . Thus
ontradi
ting
B U =
bothB(i) a(nFd (iUi).) =If UF2 is apMpli′
(aBb)le=, wfe have ∩ 6 ∅. It follows
that ∩ ∪ 6 ∅ so that , a
ontradi
tion.
M(W(A U)) > f
Thus UF3 must apply, i.e., − holds. Let us then

onsider
ases (a) and (b) separately.
M M(W(A U)) = t
(a) If istotal, we have ne
essarily − . Thisimplies
a A U T a T U
thatMso′m(We Aat)o=mt ∈ − belongs to . Thus also ∈ −
and , a
ontradi
tion with both (i) and (ii).
U M U T = M(W(A U)) = t
(b) If is -
onsistent, we have ∩ ∅. By −
a A U a F a F U
there is an atoMm′(W∈A) >−f su
h that 6∈ . Then 6∈ ∪
wMh′i(
WhAim)p=lieus M′(B) =. Tthus (ii)isimpossibAlean(Td (i)Uim)p=lies
and . It follows that ∩ − ∅
B T U U T = A T =
and ⊆ − . Sin
e ∩ B ∅,Tthe former implieMs (W∩A) < ∅t
whileMt(hBe)la=ttetr implMies(WthAat) < M⊆(B.)Consequently,
and , i.e., , a
ontradi
tion.
2
M U
Let us yeMt emphasize the
onteMnt′o=f LMemmUa 2.4 when is total (aPnd
need not be -
onsistent). Then − is also a total model of . A

ouple of examples on unfounded sets follow.
Example 2.5 Consider a disjun
tive program
P = a b c, a
{ ∨ ← ∼ }
I = , a P
andan interpretation h∅ { }i. The onlyrule in has its bodyunde(cid:28)ned
I a I b
in , hen
e UF1 is not appli
able. The set { } is unfounded w.r.t. sin
e
I
is unde(cid:28)ned in and not in the set, hen
e UF3 is appli
able. On the other
b I c
hand, the set { } is not unfounded w.r.t. whereas { } is unfounded w.r.t.
I c a b
. On
e belongs to an unfounded set, the atoms and
an both get in
U = a,b,c I
due to UF2. Hen
e, we hfave { f} as an unfounded set w.r.t. .
U I I
Comparing with , we (cid:28)nd that does not maximize the atoms that
P
should be false. This program has exa
tly one stable model (whi
h is als2o
a partial stable model) in whi
h all three atoms are false.
9
Unlike the
ase for normal programs, the union of unfounded sets may
not be an unfounded set.
P
Example 2.6 Consider a program
ontaining only one rule
a b
∨ ←
I = a,b ,
and an interpretation h{ } ∅i. The program has two non-empty un-
I a b a b
founded sets w.r.t. , { } and{ }. Either or depends on the otherone not
in the set foar UF3b to be appli
able. Howevae,rb, UF3 be
omes not appli
ab2le
when both and are in, thus the union { } is not an unfounded set.
I P
Aninterpretation for be
omesparti
ularlyinterestingwhentheunion
U P I P
of all unfounded sets for w.r.t. is also an unfounded set for w.r.t.
I P U P
. In this
ase, the program possesses the greatest unfounded set for
I
w.r.t. .
I
De(cid:28)nition 2.7 A total interpretation is said to be unfounded free for a
P U P I
progratm if and only if there is no unfounded set for w.r.t. su
h that
U I =
∩ 6 ∅.
The notion of unfounded freeness
aptures the stable model beautifully.
M
Theorem 2.8 [26℄ Let be a total interpretation for a disjun
tive program
P
. Then, the following are equivalent
M P
• is a stable model of .
f
M P M
• is the greatest unfounded set for w.r.t. .
M P
• is unfounded free for .
On the other hand, Eiter et al. [14℄ show that partial stable models
an
be de(cid:28)ned essentially without referen
e to three-valued logi
.
M
Theorem 2.9 [14℄ If is a partial interpretation for a disjun
tive program
P M P
, then is a partial stable model of if and only if
Mt PM
• is a minimal total model of and
f
M M P M
• is a maximal -
onsistent unfounded set for w.r.t. .
10