Expressing Default Abduction Problems as Quantified Boolean

18 Pages·2012·0.23 MB·English

1
Expressing Default Abduction Problems as
Quanti(cid:12)ed Boolean Formulas
Hans Tompits esis" as explanation for observed facts, according
Institut fu(cid:127)r Informationssysteme 184/3, to known laws modelling a speci(cid:12)c domain under
Technische Universita(cid:127)t Wien, consideration [29].
Favoritenstra(cid:25)e 9{11, A{1040 Vienna, Austria Various formalisations of abductive reasoning
E-mail: [email protected] have been proposed in the literature, di(cid:11)ering ba-
sically in the way the domain knowledge is repre-
sented. Besides abduction by set-covering [30] or
Abduction is the process of (cid:12)nding explanations for
probabilisticabduction [28;30],animportantcat-
observed phenomena in accord to known laws about
egory of abductive reasoning techniques is logic-
a given application domain. This form of reasoning is
animportantprincipleofcommon-sensereasoningand based abduction, in which the application domain
is particularly relevant in conjunction with nonmono- ismodelledbymeansofalogicaltheory.Although
tonicknowledgerepresentationformalisms.Inthispa- classical logic is mainly used for this purpose [32],
per, we deal with a model for abduction in which the recent years witnessed an increasing interest in
domainknowledgeisrepresentedintermsofadefault employing nonmonotonic reasoning formalisms as
theory. We show how the main reasoning tasks asso- underlying language for logic-based abduction. As
ciated with this particular form of abduction can be a case in point, abductive logic programming [18;
axiomatisedwithinthelanguageofquanti(cid:12)edBoolean 36; 11; 19] is a well-known realisation of this
logic. More speci(cid:12)cally, we provide polynomial-time
paradigm, and successful implementations of this
constructible reductions mapping a given abduction
particular approach exist [20; 5; 8].
problemintoaquanti(cid:12)edBooleanformula(QBF)such
Anotherimportantnonmonotonicreasoningfor-
that the satisfying truth assignments to the free vari-
malism relevant for abduction is default logic [33].
ablesofthelatterdeterminethesolutionsoftheorigi-
nalproblem.Sincetherearenowe(cid:14)cientQBF-solvers A formal model for abduction from default the-
available,thisreductiontechniqueyieldsastraightfor- ories has been introduced by Eiter, Gottlob, and
ward method to implement the discussed abduction Leone [12]. In this approach, a default abduction
tasks. We describe a realisation of this approach by problem, P, consists of a default theory T =
appeal to the reasoning system QUIP. hW;(cid:1)i (where W is a set of formulas and (cid:1) is a
Keywords:Defaultlogic,abduction,compilationmeth- set of defaults), a set M of observed phenomena,
ods, problem solving, quanti(cid:12)ed Boolean logic andasetH ofhypotheses.Anexplanation forP is
any set E (cid:18) H such that M can be inferred from
theextendeddefaulttheoryT =hW[E;(cid:1)i,pro-
E
1. Introduction viding T has a consistent extension. Inference in
E
the context of default logic is understood either
The philosopher C. S. Peirce identi(cid:12)ed three as brave inference (checking whether M is con-
fundamental modes of reasoning: (i) deduction, tained in some extension of T ) or skeptical in-
E
an analytic process determining necessary conse- ference (checking whether M is contained in all
quences from general rules; (ii) induction, a syn- extensions of T ).
E
theticformofreasoningderivingprobablegeneral- Although a restriced fragment of this approach
isations from factual data; and (iii) abduction, an- has been realised by the diagnosis front-end of
otherformofsyntheticinference,yieldingrelevant the DLV system [8], no general solver for default
premisses from rules and observed consequences. abduction has been put forth so far. In this pa-
Thus, following Peirce, abduction can be charac- per, we take up this challenge and describe a gen-
terised as the \probational adoption of a hypoth- eralmethodtobuildaprototypereasoningsystem
AICommunications
ISSN0921-7126,IOSPress.Allrightsreserved
2 H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas
for the default abduction model from [12], based Reduction methods to QBFs naturally gener-
on a reduction approach. The central idea is to alise similar approaches for problems in NP; these
translate a given abduction task into a quanti(cid:12)ed latter problems can in turn be solved by translat-
Boolean formula (QBF) and then applying some ing them (in polynomial time) to sat, the satis(cid:12)-
sophisticated QBF-solver to evaluate the trans- abilityproblemofclassicalpropositionallogic(see
latedQBF.Theexistenceofe(cid:14)cientQBF-solvers, e.g., [21] for such an application in Arti(cid:12)cial In-
like,e.g.,thesystemsdevelopedbyCadolietal.[3], telligence). Besides the implementation of di(cid:11)er-
Giunchiglia et al. [15], Rintanen [35], Letz [23], ent nonmonotonic reasoning tasks as realised by
or Feldmann et al. [14], makes this reduction ap- the system QUIP, successful applications based on
proach practicably applicable. reductionstoQBFshavealsobeenappliedtocon-
Concerningtheparticularencodings,weprovide ditional planning [34].
e(cid:14)cient (polynomial-time constructible) transla- The paper is organised as follows. In the next
tions of the following reasoning tasks into QBFs: section, we give some basic notation and recapitu-
latetherelevantaspectsofdefaultlogic.Section3
(i) computingallexplanationsforagivenabduc-
introduces the default abduction framework, and
tion problem;
Section 4 lays down the elementary facts about
(ii) computing all hypotheses which are relevant
QBFs. Section 5 contains our main results, and
for a given abduction problem, i.e., which
Section 6 deals with implementational issues. Sec-
contribute to some explanation; and
tion 7 supplies some concluding remarks.
(iii) computingallhypotheseswhicharenecessary
for a given abduction problem, i.e., which
contribute to all explanations.
2. Preliminaries
Eachoftheabovetasksisdealtwithforbraveand
skeptical default inference, as well as for arbitrary 2.1. Basic Notation
explanationsandforsubset-minimalexplanations.
From a theoretical point of view, the rationale We deal with propositional languages and use
of the current approach relies on the observation the primitive sentential connectives : and ^, to-
that the evaluation problem of quanti(cid:12)ed Boolean gether with the logical constant >, to construct
formulas, qsat, is PSPACE-complete, so any de- formulas in the standard way. The operators _,
cision problem in PSPACE can be polynomially (cid:27), (cid:17), as well as the symbol ?, are de(cid:12)ned from
reduced to qsat. In fact, the evaluation problem :, ^, and > as usual. We write L to denote a
A
for QBFs having prenex normal form with i(cid:0)1 languageoveranalphabetAofpropositional vari-
quanti(cid:12)eralternationsiscompleteforthei-thlevel ables or atoms. Formulas are denoted by Greek
of the polynomial hierarchy. Since the reasoning lower-case letters (possibly with subscripts). Con-
tasksconsideredinthispaperarelocatedbetween junctionsofform (cid:30) areassumedtostandfor
i2I i
the second and fourth level of the polynomial hi- the logical constanVt > whenever I =;. A literal is
erarchy, e(cid:14)cient translations to QBFs with one, eitheranatomporanegatedatom:p.Thesetof
two, or three quanti(cid:12)er alternations must exist. all atoms occurring in a formula (cid:30) is denoted by
A similar approach for solving various reason- Var((cid:30)). Similarly, for a set S of formulas, Var(S)
ing tasks belonging to the area of nonmonotonic is the set of all atoms occurring in elements of S,
reasoninghasbeenrealisedinthesystemQUIP [6; i.e., Var(S)= Var((cid:30)). Furthermore, :S de-
(cid:30)2S
13; 4; 2; 27]. This prototype implementation cur- notes the set fS:(cid:30)j(cid:30)2Sg.
rently handles the computation of the main rea- The (propositional) derivability operator, ‘, is
soning tasks for default logic, several modal non- de(cid:12)ned in the usual way, and likewise its seman-
monotoniclogics,consistency-basedapproachesto tic counterpart j=. The deductive closure of a set
belief revision and paraconsistent reasoning, and S (cid:18) L of formulas is given by Cn (S) = f(cid:30) 2
A A
equilibrium logic, a generalisation of the stable L j S ‘ (cid:30)g. We say that S is deductively closed
A
model semantics for logic programs. We obtain a i(cid:11) S =Cn (S). Furthermore, S is consistent pro-
A
straightforwardimplementationofthetranslations viding ? 2= Cn (S). If the language is clear from
A
for default abduction problems by appeal to the the context, we usually drop the index \A" from
system QUIP. Cn ((cid:1)) and simply write Cn((cid:1)).
A
H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas 3
Given an alphabet A, we de(cid:12)ne a disjoint al- one of :(cid:12) ;:::;:(cid:12) holds). We call (cid:11) the prereq-
1 n
phabet A0 as A0 = fp0 j p 2 Ag: Accordingly, for uisite, the formulas (cid:12) ;:::;(cid:12) the justi(cid:12)cations,
1 n
(cid:11) 2 L , we de(cid:12)ne (cid:11)0 as the result of replacing in and (cid:13) the consequent of (cid:14). For notational conve-
A
(cid:11) each atom p from A by the corresponding atom nience, we also write p((cid:14)) for the prerequisite (cid:11)
p0 in A0 (so implicitly there is an isomorphism be- of (cid:14), j((cid:14)) for the set f(cid:12) ;:::;(cid:12) g, and c((cid:14)) for
1 n
tween A and A0). This is de(cid:12)ned analogously for the consequent (cid:13) of (cid:14). Accordingly, p((cid:1)) is the
sets of formulas. set of prerequisites of all default rules in (cid:1); j((cid:1))
We assume the reader familiar with the basic andc((cid:1))arede(cid:12)nedanalogously.Furthermore,we
concepts of complexity theory (see, e.g., [26] for de(cid:12)ne Var((cid:1)) = Var(p((cid:1)) [ j((cid:1)) [ c((cid:1))) and
a comprehensive introduction). Relevant for our Var(T) = Var(W)[Var((cid:1)), for T = hW;(cid:1)i. As
purposes are the elements of the polynomial hier- well, we extend our priming notation introduced
archy, given by the following sequence of classes: for (sets of) formulas to defaults and default the-
ories in the obvious way, i.e., (cid:14)0 denotes the result
(cid:6)P =(cid:5)P =P;
0 0 of replacing each atom p in default (cid:14) by p0, and
and, for i>0, likewise for sets of defaults and default theories.
WesaythatadefaulttheoryT =hW;(cid:1)iis(cid:12)nite
(cid:6)P =NP(cid:6)Pi(cid:0)1 and (cid:5)P =co-NP(cid:6)Pi(cid:0)1:
i i i(cid:11) both W and (cid:1) are (cid:12)nite.
Here, P is the class of all problems solvable on a For simplicity, defaults will also be written in
deterministic Turing machine in polynomial time; the form ((cid:11):(cid:12)1;:::;(cid:12)n=(cid:13)).
NP is similarly de(cid:12)ned but using a nondeter- The semantics of default theories is de(cid:12)ned in
ministic Turing machine as underlying computing terms of extensions. Formally, a set S (cid:18) LA is
model; and, for complexity classes C and A, the an extension of a default theory T = hW;(cid:1)i i(cid:11)
notation CA stands for the relativized version of S = i(cid:21)0Si, where S0 =W and, for i(cid:21)0,
C,consistingofallproblemswhichcanbedecided S
S =Cn(S )[fc((cid:14))j(cid:14) 2GD g;
i+1 i i
by Turing machines of the same sort and time
boundasinC,onlythatthemachineshaveaccess with
to an oracle for problems in A. As well, co-C is
GD =f(cid:14) 2(cid:1)jp((cid:14))2S ;:j((cid:14))\S =;g:
i i
theclassofallproblemswhicharecomplementary
to the problems in C. We note that (cid:6)P = NP, In general, a default theory may possess none,
1
(cid:6)P =NPNP, and (cid:5)P =co-NPNP. one,orseveralextensions.Intuitively,anextension
2 2
characterises a possible totality of beliefs to which
2.2. Default Logic an agent may refer on the basis of a given default
theory.
In default logic [33], knowledge about the world Setting GD((cid:1);S) = i(cid:21)0GDi; it holds that
is represented in terms of default theories, which S =Cn(W [c(GD((cid:1);S)S)),foranyextensionS of
are ordered pairs of form T =hW;(cid:1)i, where W is T =hW;(cid:1)i.WecalltheelementsofGD((cid:1);S)the
a set propositional formulas, called the premisses generating defaults of S.
of T, and (cid:1) is a set of default rules (or defaults, Following Marek and Truszczyn(cid:19)ski [24], exten-
forshort).ThesetW representscertain(thoughin sions can alternatively be characterised as follows.
general incomplete) information about the given For any set (cid:1) of defaults and any S (cid:18) LP, de-
applicationdomain,whilst(cid:1)representsdefeasible (cid:12)ne the reduct of (cid:1) with respect to S as the set of
knowledge,whichmaybeinvalidatedbynew,more classical inference rules
accurate information. Formally, a default, (cid:14), is an p((cid:14))
(cid:1) = (cid:14) 2(cid:1);:j((cid:14))\S =; :
inference rule of form S (cid:26)c((cid:14)) (cid:12) (cid:27)
(cid:12)
(cid:11) : (cid:12) ;:::;(cid:12) (cid:12)
1 n; Furthermore, for (cid:12)any set F of propositional for-
(cid:13)
mulas and any set K of classical inference rules,
where (cid:11);(cid:12) ;:::;(cid:12) ; and (cid:13) are formulas. Intu- let CnK(F) be the set of all formulas derivable
1 n
itively, (cid:14) expresses that (cid:13) is asserted whenever fromF usingclassicallogictogetherwiththerules
(cid:11) is believed and (cid:12) ;:::;(cid:12) are consistent with in K. Then, S is an extension of T = hW;(cid:1)i i(cid:11)
1 n
what is believed (i.e., there is no evidence that S =Cn(cid:1)S(W).
4 H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas
There are two basic inference operations in the WecallE a braveexplanationforP i(cid:11)(i)hW[
context of default logic, viz. brave inference and E;(cid:1)i (cid:24) M and (ii) hW[E;(cid:1)i has a consistent
b
skeptical inference. To wit, we say that a formula extension. Similarly, E is a skeptical explanation
(cid:30) is a brave consequence of a given default the- forP i(cid:11)(i)hW[E;(cid:1)i (cid:24) M and(ii)hW[E;(cid:1)i
s
ory T, symbolically T (cid:24) (cid:30), i(cid:11) (cid:30) belongs to has a consistent extension.
b
some extension of T, and (cid:30) is a skeptical conse-
quence of T, symbolically T (cid:24) (cid:30), i(cid:11) (cid:30) belongs The stipulation that hW [ E;(cid:1)i possesses a
s
to all extensions of T. As shown in [16], both in- consistent extension assures that explanations are
ference tasks are intractable in the general case. consistent with the knowledge represented in the
More speci(cid:12)cally, given a default theory T and a given default theory hW;(cid:1)i. This requirement is
formula (cid:30), checking whether T (cid:24) (cid:30) holds is (cid:6)P- similar to the usual consistency condition in ab-
b 2
complete, whilst checking whether T (cid:24) (cid:30) holds duction from theories in classical logic.
s
is (cid:5)P-complete. Analogously, checking whether a We note that brave explanations for a given ab-
2
given default theory possesses an extension is (cid:6)P- duction problem P = hH;M;W;(cid:1)i can be equiv-
2
alently characterised as sets E (cid:18) H such that
complete as well.
Let S be a set of formulas. We write T (cid:24) S hW [E;(cid:1)i has a consistent extension containing
b
to indicate that T (cid:24) (cid:30), for any (cid:30)2S; a similar M.Thisisasimpleconsequenceofthewell-known
b
notation applies for (cid:24) . fact that if a default theory T has some consis-
s
tent extension, then all extensions of T must be
consistent.
The assumption that hypotheses and mani-
3. The Abduction Framework
festations in a default abduction problem P =
hH;M;W;(cid:1)i are given by literals instead of ar-
FollowingEiter,Gottlob,andLeone[12],wede-
bitrary formulas is no restriction since for each
(cid:12)ne a formal model for abduction from proposi-
non-literal hypothesis or manifestation (cid:30) a new
tional default theories as follows.
propositional atom p can be introduced, and af-
(cid:30)
ter adding the formula p (cid:17) (cid:30) to W, (cid:30) can be
De(cid:12)nition 1 A defaultabductionproblem, P, is a (cid:30)
equivalently replaced by p .
quadruple hH;M;W;(cid:1)i, where H and M are sets (cid:30)
We extend our notation Var((cid:1)) and our priming
of literals, and hW;(cid:1)i is a default theory. We call
convention to default abduction problems in the
the elements of H the hypotheses or abducibles,
obvious way, i.e., for P =hH;M;W;(cid:1)i, we de(cid:12)ne
and the elements of M the observations or mani-
Var(P) = Var(H [M [W)[Var((cid:1)) and P0 =
festations of P.
hH0;M0;W0;(cid:1)0i.
The default abduction problem P is (cid:12)nite i(cid:11)
Following Occam’s principle of parsimony, one
each constituting member H, M, W, and (cid:1) of P
usually prefers simpler explanations over more
is (cid:12)nite.
complicatedones.Inthepresentcontext,thisidea
Informally, in a default abduction problem P = is realised by accepting only those explanations
hH;M;W;(cid:1)i, H speci(cid:12)es the universe of admis- which are minimal among the class of explana-
sible explanations, the elements of M are the ob- tions. More formally, we call E (cid:18) H a minimal
servedphenomena,andhW;(cid:1)irepresentsthecur- brave (resp., minimal skeptical) explanation for P
rent world knowledge. A set E (cid:18) H serves as i(cid:11)E isabrave(resp.,skeptical)explanationforP
an explanation for the observed manifestations in and there is no F (cid:26)E such that F is also a brave
M if the adjunction of E to the world knowl- (resp., skeptical) explanation for P.
edge hW;(cid:1)i entails all propositions in M. Given Another interesting property of hypotheses is
that there are two di(cid:11)erent inference operations whether they contribute to some explanation or
associatedwithdefaulttheories,thereare,accord- toall explanationsforagivenabductionproblem.
ingly, two fundamental kinds of explanations for More speci(cid:12)cally, we call a set H0 (cid:18) H relevant
observed phenomena, as de(cid:12)ned next. for P = hH;M;W;(cid:1)i under brave (resp., skepti-
cal)explanations i(cid:11)H (cid:18)E forsomebrave(resp.,
0
De(cid:12)nition 2 LetP =hH;M;W;(cid:1)ibeadefaultab- skeptical) explanation E for P. Dually, H (cid:18)H is
0
duction problem, and let E (cid:18)H. necessary for P under brave (resp., skeptical) ex-
H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas 5
Table1
Complexityresultsofabductionfromdefaulttheories.
AbductionProblemP
arbitraryexplanations minimalexplanations
DecisionProblem brave skeptical brave skeptical
consistency (cid:6)P2 (cid:6)P3 (cid:6)P2 (cid:6)P3
relevance (cid:6)P2 (cid:6)P3 (cid:6)P3 (cid:6)P4
necessity (cid:5)P2 (cid:5)P3 (cid:5)P2 (cid:5)P3
planations i(cid:11) H (cid:18) E for each brave (resp., skep- Abstracting from the particular kind of expla-
0
tical)explanationE forP.Bothconceptsaresim- nation, the main decision problems in the context
ilarly de(cid:12)ned for abduction problems under mini- of abductive reasoning are the following:
mal explanations.
consistency: Does a given (cid:12)nite default abduc-
For illustration, let us consider the following
tion problem possess an explanation?
simple example.
relevance: Given some (cid:12)nite default abduction
problemP =hH;M;W;(cid:1)iandasetH (cid:18)H,
Example 1 Let hW;(cid:1)i be the default theory con- 0
is H relevant for P?
sisting of the following items: 0
necessity: Given some (cid:12)nite default abduction
problemP =hH;M;W;(cid:1)iandasetH (cid:18)H,
W = l ^ (s _ (c ^ r)) (cid:27) m; 0
is H necessary for P?
(cid:8)(cid:0) (cid:1) 0
(:s _ :c) ;
Concerning the computational complexity of
(cid:9)
D= (w ::j=r); (w ::r=j) : these tasks, as shown in [12], these problems are
(cid:8) (cid:9) locatedbetweenthesecondandfourthlevelofthe
Furthermore, assume we are interested in (cid:12)nding polynomial hierarchy. Table 1 gives the speci(cid:12)c
explanations for the manifestation M = fmg us- results; each entry C represents completeness of
ingH =fl;s;c;wgasthesetofhypotheses.Then, the corresponding problem for the class C.1 As
by straightforward calculations, it can be shown canbeseenfromtheseresults,skepticalabduction
that the brave explanations for P =hH;M;W;(cid:1)i is always one level harder than brave abduction.
are given by fl;sg, fl;s;wg, and fl;c;wg, whilst This contrasts with complexity results for usual
only fl;sg and fl;s;wg represent the skeptical ex- nonmonotonicreasoningformalisms,whereskepti-
planations for P. The reason for this di(cid:11)erence cal reasoning has complementary complexity than
is that m is not contained in all extensions of brave reasoning. Also, minimality of explanations
hW [fl;c;wg;(cid:1)i, which are given by is a source of complexity for relevance but nei-
therforconsistencynorfornecessity.Therea-
S1=Cn(fl;c;w;r;mg) and sonforthisphenomenonisthefactthatanabduc-
tion problem P has a minimal brave (resp., mini-
S =Cn(fl;c;w;jg):
2
malskeptical)explanationi(cid:11)ithasabrave(resp.,
Observe that flg is the only set necessary for skeptical) explanation; likewise, a set is necessary
P under brave explanations, whilst fl;sg is neces- for P under minimal brave (resp., minimal skep-
saryforP underskepticalexplanations.Moreover, tical) explanations i(cid:11) it is necessary for P under
fcgisrelevantforP underbraveexplanations but brave (resp., skeptical) explanations. Finally, the
not under skeptical ones. Also, fl;sg and fl;c;wg
1Strictlyspeaking,[12]de(cid:12)nesrelevantandnecessaryhy-
aretheminimalbraveexplanationsforP,whereas
potheses as single formulas (actually, literals) whereas we
fl;sg is the single minimal skeptical explanation
allow here sets for this purpose; but this di(cid:11)erence is im-
for P. 2 material.
6 H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas
results for relevance contrast with correspond- 1. if (cid:8)=>, then (cid:23) ((cid:8))=1;
I
ing results for abduction from theories in classical 2. if (cid:8) = p, for some atom p, then (cid:23) ((cid:8)) = 1 if
I
logic where minimality is not a source of complex- p2I, otherwise (cid:23) ((cid:8))=0;
I
ity [10]. 3. if (cid:8) = :(cid:9), then (cid:23) ((cid:8)) = 1 if (cid:23) ((cid:9)) = 0,
I I
The main contribution of this paper is to show otherwise (cid:23) ((cid:8))=0;
I
howeachoftheabovedecisionproblems,aswellas 4. if (cid:8)=((cid:8) ^(cid:8) ), then (cid:23) ((cid:8))=1 if (cid:23) ((cid:8) )=
1 2 I I 1
eachcorrespondingsearchproblem,canbemapped (cid:23) ((cid:8) )=1, otherwise (cid:23) ((cid:8))=0;
I 2 I
in polynomial time into a quanti(cid:12)ed Boolean for- 5. if (cid:8) =8p(cid:9), then (cid:23) ((cid:8)) =1 if (cid:23) ((cid:9)[p=>]) =
I I
mula such that the models of the latter determine (cid:23) ((cid:9)[p=?])=1, otherwise (cid:23) ((cid:8))=0.
I I
theanswersoftheformer.Infact,thetranslations
Thetruthconditionsfor?, _, (cid:27),(cid:17),and9follow
arerealisedinsuchawaythatthestructureofthe
from the above in the usual way. Obviously, we
constructed formulas precisely matches the com-
have that
plexityoftheencodedreasoningtasks.Inthenext
section,webrie(cid:13)yrecapitulatetherelevantaspects (cid:23) (8p(cid:9))=(cid:23) ((cid:9)[p=>] ^ (cid:9)[p=?]) and
I I
of quanti(cid:12)ed Boolean logic.
(cid:23) (9p(cid:9))=(cid:23) ((cid:9)[p=>] _ (cid:9)[p=?]):
I I
We say that (cid:8) is true under I i(cid:11) (cid:23) ((cid:8)) = 1,
I
4. Quanti(cid:12)ed Boolean Logic otherwise(cid:8)isfalseunderI.If(cid:23) ((cid:8))=1,thenI is
I
amodel of(cid:8).Thesetofallmodelsof(cid:8)isdenoted
Quanti(cid:12)edBoolean logicisanextensionofclas- by Mod((cid:8)). If Mod((cid:8)) 6= ;, then (cid:8) is satis(cid:12)able.
sicalpropositionallogicinwhichformulasareper- If (cid:8) is true under every interpretation, then (cid:8) is
mitted to contain quanti(cid:12)cations over proposi- valid. As usual, we write j=(cid:8) to express that (cid:8) is
tional variables. More formally, the language of valid.
quanti(cid:12)ed Boolean logic comprises the symbols of It is easily seen that the truth value of a QBF
propositional logic, together with unary operators (cid:8) under interpretation I depends only on the free
of form 8p (where p is some atom), called univer- variables in (cid:8). Hence, without loss of generality,
sal quanti(cid:12)ers. Similar to (cid:12)rst-order logic, 9p is for determining the truth value of QBFs, we may
de(cid:12)ned as the operator :8p: and is called an ex- restrictourattentiontointerpretationswhichcon-
istential quanti(cid:12)er. Formulas of this language are tain only atoms occurring free in the given QBF.
referredtoasquanti(cid:12)ed Boolean formulas (QBFs) In particular, closed QBFs are either true under
and are denoted by Greek upper-case letters. every interpretation or false under every interpre-
Informally, a QBF of form 8p9q(cid:8) means that tation, i.e., they are either valid or unsatis(cid:12)able.
for all truth assignments of p there is a truth as- So, for closed QBFs there is no need to refer to
signment of q such that (cid:8) is true. For instance, particularinterpretations.Aswell,ifaclosedQBF
under this reading, it is easily seen that the QBF (cid:8) is valid, we say that (cid:8) evaluates to true, and,
9p9q((p (cid:27) q) ^ 8r(r (cid:27) q)) evaluates to true. correspondingly, if (cid:8) is unsatis(cid:12)able, we say that
The precise semantical meaning of QBFs is de- (cid:8) evaluates to false. Two sets of formulas (i.e., or-
(cid:12)nedasfollows.First,someancillarynotation.An dinary propositional formulas or QBFs) are logi-
occurrence of a propositional variable p in a QBF cally equivalent i(cid:11) they possess the same models.
(cid:8) is free i(cid:11) it does not appear in the scope of a Thus,formulas(cid:8)and(cid:9)arelogicallyequivalenti(cid:11)
quanti(cid:12)er Qp (Q 2 f8;9g), otherwise the occur- (cid:8)(cid:17)(cid:9) is valid.
rence of p is bound. If (cid:8) contains no free variable Inthesequel,weusethefollowingabbreviations
occurrences, then (cid:8) is closed, otherwise (cid:8) is open. in the context of QBFs: Let R = f(cid:30) ;:::;(cid:30) g
1 n
Furthermore, (cid:8)[p =(cid:30) ;:::;p =(cid:30) ] denotes the re- and S = f ;:::; g be indexed sets of formu-
1 1 n n 1 n
n
sultofuniformlysubstitutingin(cid:8)eachfreeoccur- las. Then, R(cid:20)S abbreviates ((cid:30) (cid:27) ), and
i=1 i i
renceofavariablepibyaformula(cid:30)i,for1(cid:20)i(cid:20)n. R < S is a shorthand for (RV(cid:20) S) ^ :(S (cid:20) R).
By an interpretation, I, we understand a set of Furthermore, for an indexed set P = fp ;:::;p g
1 n
atoms. Informally, an atom p is true under I i(cid:11) of variables and Q2f8;9g, we let QP (cid:8) stand for
p2I.Ingeneral,thetruthvalue,(cid:23) ((cid:8)),ofaQBF the formula Qp Qp (cid:1)(cid:1)(cid:1)Qp (cid:8). To ease notation,
I 1 2 n
(cid:8) under an interpretation I is recursively de(cid:12)ned we usually identify a (cid:12)nite set of formulas with
as follows: the conjunction of its elements. Finally, the logi-
H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas 7
cal complexity of a formula (cid:8), or the degree of (cid:8), 5.1. Preparatory Characterisations
symbolicallyd((cid:8)),isthenumberofoccurrencesof
the primitive operators :, ^, and 8p (where p is
We start with some basic results concerning the
some variable) occurring in (cid:8).
operator (cid:20). To wit, we (cid:12)rst describe a particular
The operators (cid:20) and < are fundamental tools
substitution theorem involving (cid:20), and afterwards
for expressing certain tests on sets of atoms. In
we discuss two basic QBF modules required sub-
particular, the following properties hold: Let P =
sequently.
fp ;:::;p g be a set of indexed atoms, and let
1 n
I ;I (cid:18)P be two interpretations. Then,
1 2 Proposition 2 Let H = f(cid:30) ;:::;(cid:30) g and G =
1 n
(i) I (cid:18)I i(cid:11) I0 [I is a model of P0 (cid:20)P; and fg ;:::;g gbeindexedsetsofformulasandatoms,
1 2 1 2 1 n
(ii) I (cid:26)I i(cid:11) I0 [I is a model of P0 <P. respectively, such that G \ Var(H) = ;, and let
1 2 1 2
(cid:8) be a QBF containing one or more desig-
We say that QBF (cid:8) is in prenex form if (cid:8) = G(cid:20)H
Q p :::Q p (cid:30), where Q 2 f8;9g, for 1 (cid:20) i (cid:20) natedoccurrencesofG(cid:20)H suchthattheelements
1 1 n n i
of G occur free in (cid:8) and, moreover, only in
n, and (cid:30) is some propositional formula. Using G(cid:20)H
quanti(cid:12)er-shiftingrulessimilartothoseofclassical the designated subformulas G(cid:20)H of (cid:8)G(cid:20)H. Fur-
(cid:12)rst-orderlogic,anyQBFcanbee(cid:11)ectivelytrans- thermore, let I (cid:18)Var((cid:8)G(cid:20)H)nG and J (cid:18)G, and
formed into an equivalent QBF in prenex form. In let HJ = gi2J(cid:30)i.
fact, this transformation can be done in polyno- Then, (cid:8)VG(cid:20)H is true under I[J i(cid:11) (cid:8)HJ is true
mial time. under I, where (cid:8)HJ comes from (cid:8)G(cid:20)H by replac-
QBFs are closely related to the constituting ing all designated occurrences of G(cid:20)H by H .
J
membersofthepolynomialhierarchy,asdescribed
by the following well-known result: Proof.Foraformula(cid:9)(cid:11) containingdesignatedoc-
currences of a formula (cid:11) as consecutive parts, de-
Proposition 1 ([40]) Givenapropositionalformula
(cid:12)ne d0((cid:9) ) like d((cid:9) ), but count the speci(cid:12)c oc-
(cid:30) whose atoms are partitioned into i (cid:21) 1 sets (cid:11) (cid:11)
V ;:::;V , deciding whether 9V 8V 9V :::QV (cid:30) currencesof(cid:11)asatomicformulas.Theproofofthe
ev1aluatesito true is (cid:6)P-complete,1wh2ere3Q = 9iif proposition proceeds by induction on d0((cid:8)G(cid:20)H).
i
i is odd and Q = 8 if i is even. Dually, decid-
ing whether 8V 9V 8V :::Q(cid:22)V (cid:30) evaluates to true Induction Base. Assume d0((cid:8) ) = 0. Then,
1 2 3 i G(cid:20)H
is (cid:5)P-complete, where Q(cid:22) =8 if i is odd and Q(cid:22) =9 (cid:8) = G (cid:20) H and (cid:8) = H . Since, for each
i G(cid:20)H HJ J
if i is even. 1(cid:20)i(cid:20)n, the truth value of g (cid:27) (cid:30) under inter-
i i
From this proposition it follows that any deci- pretation I [J coincides with the truth value of
sion problem in (cid:6)P can be e(cid:14)ciently reduced to (cid:30)i under I if gi 2 J, and gi (cid:27) (cid:30)i is trivially true
i
the evaluation problem of prenex QBFs of form under I[J if gi 2= J, we have that
9V18V29V3:::QVi(cid:30), with Q as above, and sim- n
ilarly for problems in (cid:5)P. In the next section, (cid:23) ( (g (cid:27) (cid:30) ))=(cid:23) ( (cid:30) );
i I[J i i I i
we describe polynomial-time constructible trans- i^=1 g^i2J
lations from the main abductive reasoning tasks
whichprovesthat(cid:8) istrueunderI[J i(cid:11)H
intoQBFssuchthatthequanti(cid:12)erstructureofthe G(cid:20)H J
is true under I.
resultant formulas precisely matches the inherent
complexity of the encoded problem.
Induction Step. Assume d0((cid:8) )>0, and let
G(cid:20)H
the statement hold for all formulas (cid:9) , con-
G(cid:20)H
5. Translations
taining speci(cid:12)ed occurrences of G(cid:20)H, such that
d0((cid:9) )<d0((cid:8) ).
Inthissection,wepresente(cid:14)cientreductionsof G(cid:20)H G(cid:20)H
Since d0((cid:8) )>0, (cid:8) is a compound for-
the main reasoning tasks in the context of abduc- G(cid:20)H G(cid:20)H
tionfromdefaulttheoriesintoQBFs.Thesereduc- mula having as its main operator either :, ^, or
tions are constructed in such a way that the solu- a universal quanti(cid:12)er (recall that _, (cid:27), (cid:17), and
tions of the given abduction problems are deter- existential quanti(cid:12)ers are de(cid:12)ned operators). We
mined by the models of the corresponding QBFs. only deal with the case where (cid:8)G(cid:20)H is a univer-
In what follows, we tacitly assume that all con- sallyquanti(cid:12)edformula;theremainingcasesfollow
sidered default abduction problems are (cid:12)nite. by similar arguments.
8 H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas
Assume therefore that (cid:8)G(cid:20)H = 8p(cid:9)G(cid:20)H. Re- 9U R ^ (cid:30)i (1)
call that (cid:23)I[J(8p(cid:9)G(cid:20)H) is given by (cid:16) g^i2I (cid:1)
(cid:23)I[J((cid:9)G(cid:20)H[p=>] ^ (cid:9)G(cid:20)H[p=?]): isvalid.ButQ= gi2I(cid:30)i,hence,bythesemantics
ofexistentialquanVti(cid:12)cation,wegetthat(1)isvalid
Sincebothd0((cid:9)G(cid:20)H[p=>])andd0((cid:9)G(cid:20)H[p=?])are i(cid:11) R[Q is satis(cid:12)able. It follows that I is a model
strictly less than d0((cid:8)G(cid:20)H), by induction hypoth- of C[R;S;G] i(cid:11) R[Q is satis(cid:12)able.
esis it follows that Part 2 is an immediate consequence of Part 1,
by observing that R[Q j= i(cid:11) (R[f: g)[Q
(cid:23) ((cid:9) [p=>])=(cid:23) ((cid:9) [p=>]) and
I[J G(cid:20)H I HJ is unsatis(cid:12)able, and since
(cid:23) ((cid:9) [p=?])=(cid:23) ((cid:9) [p=?]);
I[J G(cid:20)H I HJ :9V (R ^ : ) ^ (G(cid:20)S)
from which we immediately get, by the semantics is equival(cid:0)ent to (cid:1)
of conjunction and universal quanti(cid:12)cation, that
8V (R ^ (G(cid:20)S)) (cid:27) .
(cid:23)I[J(8p(cid:9)G(cid:20)H)=(cid:23)I(8p(cid:9)HJ); (cid:0) (cid:1)
Note that C[R;S;G] is an open QBF having G
i.e., (cid:8) is true under I [J i(cid:11) (cid:8) is true un-
G(cid:20)H HJ asitssetoffreevariables.Thesevariablesfacilitate
der I.
the selection of those elements of S which deter-
minethesetsQ(cid:18)S suchthatR[Qissatis(cid:12)able.
Next, we describe two fundamental QBF mod- Infact,C[R;S;G]isdesignedtoexpressall poten-
ules, capturing the following tasks, respectively: tial subsets Q (cid:18) S such that R[Q is satis(cid:12)able.
Similar considerations apply to D[R;S; ;G].
1. Given(cid:12)nitesetsRandS ofpropositionalfor-
mulas, compute all Q(cid:18)S such that R[Q is
5.2. Encodings for Default Logic
satis(cid:12)able.
2. Given (cid:12)nite sets R and S of propositional In order to express default abduction problems
formulas, and some propositional formula , in terms of QBFs, we need suitable QBF encod-
compute all Q(cid:18)S such that R[Qj= . ings capturing the underlying default inference
mechanism. To this end, we use results from [6],
These tasks can be characterised as follows.
where QBF reductions for default reasoning are
given. More speci(cid:12)cally, that paper contains two
Proposition 3 Let R and S = f(cid:30) ;:::;(cid:30) g be (cid:12)-
1 n
di(cid:11)erent kinds of encodings for default logic: one
nite sets of propositional formulas, let be a
translation is based on the characterisation of
propositional formula, and let G=fg ;:::;g g be
1 n extensions following the method of Marek and
a set of new variables. Furthermore, consider any
Truszczyn(cid:19)ski [24], and the other translation ex-
Q(cid:18)S and any I (cid:18)G such that (cid:30) 2Q i(cid:11) g 2I,
i i ploits the so-called full-set characterisation of ex-
for 1(cid:20)i(cid:20)n.
tensions, due to Niemela(cid:127) [25]. Here, we adopt the
Then,
former method because it is more suitable for our
1. R[Qissatis(cid:12)ablei(cid:11)I isamodeloftheQBF purposes.
Thefollowingresultcontainstherequiredreduc-
C[R;S;G]=9U R ^ (G(cid:20)S) ;
tions.
(cid:0) (cid:1)
whereU =Var(R[S)andsuchthatG\U = Proposition 4 ([6; 39]) Let T = hW;(cid:1)i be a de-
;; and fault theory with (cid:1) = f(cid:14) ;:::;(cid:14) g, let (cid:30) be a
1 n
2. R[Qj= i(cid:11) I is a model of the QBF propositional formula, and let C = fc ;:::;c g,
1 n
D = fd ;:::;d g, and D0 = fd0;:::;d0 g be sets
D[R;S; ;G]=8V (R ^ (G(cid:20)S)) (cid:27) ; 1 n 1 n
of pairwise distinct new variables.
(cid:0) (cid:1)
where V = Var(R[S [f g) and such that Furthermore, consider the QBFs from Fig. 1
G\V =;. with V = Var(W [ c((cid:1))), P = Var((cid:30)) n V,
U = Var(p((cid:14) ))nV, and Z = Var((cid:12) )nV, for
i i i;j j
Proof. Consider Q (cid:18) S and I (cid:18) G such that (cid:30)i 2 (cid:12)j 2j((cid:14)i) and 1(cid:20)i(cid:20)n.
Q i(cid:11) gi 2 I, for 1 (cid:20) i (cid:20) n. For proving Part 1, Then, for any F (cid:18)(cid:1) and any I (cid:18)D such that
according to Proposition 2, we have that I is a (cid:14) 2 F i(cid:11) d 2 I, for 1 (cid:20) i (cid:20) n, the following
i i
model of 9U R ^ (G(cid:20)S) i(cid:11) conditions hold:
(cid:0) (cid:1)
H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas 9
E[W;(cid:1);D]=9D0((D (cid:20)D0) ^ (cid:8) ^ (cid:8) ^ (cid:8) );
1 2 3
M [W;(cid:1);(cid:30);D]=E[W;(cid:1);D] ^ 8V 8P W ^ (D (cid:20)c((cid:1))) (cid:27) (cid:30) ;
b
(cid:16)(cid:0) (cid:1) (cid:17)
M [W;(cid:1);(cid:30);D]=E[W;(cid:1);D]^9V 9P W ^ (D (cid:20)c((cid:1))) ^ :(cid:30) ;
s
(cid:16) (cid:17)
n
(cid:8) = d0 (cid:17) 9V 9Z W ^ D (cid:20)c((cid:1)) ^ (cid:12) ;
1 i i;j j
i^=1h (cid:12)j2^j((cid:14)i) (cid:16) (cid:0) (cid:1) (cid:17)i
n
(cid:8) = (d0 ^ :d ) (cid:27) 9V 9U W ^ D (cid:20)c((cid:1)) ^ :p((cid:14) )) ;
2 i i i i
i^=1h (cid:16) (cid:0) (cid:1) (cid:17)i
(cid:8) =8C (C (cid:20)D0) (cid:27) 8V W ^ (C (cid:20)c((cid:1))) (cid:27) (D (cid:20)c((cid:1))) _
3
n h (cid:16)(cid:0) (cid:1) (cid:17)
n
d0 ^:c ^8V 8U W ^ (C (cid:20)c((cid:1))) (cid:27) p((cid:14) ) :
i i i i
i_=1(cid:16) (cid:16)(cid:0) (cid:1) (cid:17)(cid:17)io
Fig.1.QBFmodulesforexpressingdefaultreasoning.
1. Cn(W [c(F)) is an extension of T i(cid:11) I is a and M [W;(cid:1);(cid:30);D]. More speci(cid:12)cally, the vari-
s
modelofE[W;(cid:1);D],providing(C[D[D0)\ ables in D0 take care of selecting the members
Var(T)=;; in the reduct (cid:1) , and the variables in D check
S
2. Cn(W [c(F)) is an extension of T contain- whether the consequent of a selected rule is con-
ing (cid:30) i(cid:11) I is a model of M [W;(cid:1);(cid:30);D], pro- tained in S =Cn(cid:1)S(W). To that end, the follow-
b
viding D\(Var(T)[Var((cid:30))) = (C [D0)\ ingtestsareperformed,expressedbythesubmod-
Var(T)=;; and ules (cid:8)1, (cid:8)2, and (cid:8)3:
3. Cn(W [c(F)) is an extension of T not con-
{ (cid:8) tests whether each default in the guessed
1
taining (cid:30) i(cid:11) I is a model of M [W;(cid:1);(cid:30);D],
s set (cid:1) is consistent with the guess for the
S
assuming the same proviso as in 2.
extension S;
{ (cid:8) tests whether no applicable default in (cid:1)
Here, the module E[W;(cid:1);D] characterises the 2 S
is missing with respect to the guessed assign-
extensionsofthegivendefaulttheoryT =hW;(cid:1)i,
ment for D; and
and
{ (cid:8) utilises the test whether, for each default,
3
D[W;c((cid:1));(cid:30);D]= it holds that its consequent is actually con-
tained in Cn(cid:1)S(W).
8V 8P W ^ (D (cid:20)c((cid:1))) (cid:27) (cid:30)
(cid:16)(cid:0) (cid:1) (cid:17) The latter formula encodes a result due to Got-
checks whether a given formula (cid:30) is contained in tlob [17] stating that a formula 2= Cn(cid:1)S(W) i(cid:11)
a selected extension. Note that, dually, there exists a subset B (cid:18) f(cid:13) j (cid:11) =(cid:13) 2 (cid:1) g such
i i i S
that (i) W [B 6j= , and (ii) for each (cid:11) =(cid:13) 2(cid:1)
C[W [f:(cid:30)g;c((cid:1));D]= i i S
with (cid:13) 2= B, we have that W [B 6j=(cid:11) .
i i
9V 9P W ^ (D (cid:20)c((cid:1))) ^ :(cid:30) ; Encodings for brave and skeptical default infer-
(cid:16) (cid:17) ence are obtained from Proposition 4 in the fol-
which is equivalent to :D[W;c((cid:1));(cid:30);D], checks lowing way.
whether (cid:30) is not contained in the selected ex-
Corollary 1 Let T, (cid:30), and D be as in Proposi-
tension. The selection of an extension S, in turn,
tion 4.
is done by means of the variables in D, which
Then,
constitute the set of free variables of each of the
three main modules E[W;(cid:1);D], M [W;(cid:1);(cid:30);D], 1. T (cid:24) (cid:30) i(cid:11) j=9DM [W;(cid:1);(cid:30);D]; and
b b b
10 H.Tompits/ExpressingDefaultAbductionProblemsasQuanti(cid:12)edBooleanFormulas
2. T (cid:24) (cid:30) i(cid:11) j=:9DM [W;(cid:1);(cid:30);D]. 1. E isabraveexplanationforP i(cid:11)I isamodel
s s
of the QBF Tb [P;U ], given by
cons H
We note that 9DM [W;(cid:1);(cid:30);D] can be trans-
b
formed in polynomial time into a QBF of prenex 9D M+b [W;(cid:1);M;D;UH]^
form 9X 8X , whilst :9DM [W;(cid:1);(cid:30);D] can (cid:16)
1 2 s
9V W ^ (U (cid:20)H) ^ (D (cid:20)c((cid:1))) ;
be transformed, in the same manner, into a QBF H
ofprenexform8Y 9Y ’.Hence,inviewofPropo- (cid:0) (cid:1)(cid:17)
1 2 and
sition1,theseencodingspreciselymatchtheinher-
2. E is a skeptical explanation for P i(cid:11) I is a
ent computational complexity of brave and skep-
model of the QBF Ts [P;U ], given by
tical default reasoning, respectively. cons H
9D E+[W;(cid:1);D;U ]^
H
5.3. Encodings of the Basic Abduction Tasks (cid:16)
9V W ^ (U (cid:20)H) ^ (D (cid:20)c((cid:1))) ^
H
(cid:0) (cid:1)(cid:17)
The following de(cid:12)nition gives the core modules :9DM+[W;(cid:1);M;D;U ]:
s H
for our encodings.
Proof. For Part 1, recall that E is a brave expla-
De(cid:12)nition 3 Let P = hH;M;W;(cid:1)i be a default
nation for P i(cid:11) there is a consistent extension of
abduction problem with (cid:1) = f(cid:14) ;:::;(cid:14) g, and
1 n hW [E;(cid:1)i containing M. Consider any J (cid:18) D
let D = fd ;:::;d g and U = fu j h 2
1 n H h and any F (cid:18) (cid:1) such that di 2 J i(cid:11) (cid:14)i 2 F, for
Hg be sets of pairwise distinct variables with
1 (cid:20) i (cid:20) n. We show that Cn(W [E[c(F)) is a
(D[UH)\Var(P) = ;. Furthermore, let E[(cid:1);(cid:1);(cid:1)] consistent extension of hW [E;(cid:1)i containing M
and M(cid:27)[(cid:1);(cid:1);(cid:1);(cid:1)] be the translations from Proposi- i(cid:11) I[J is a model of
tion 4, for (cid:27) 2 fb;sg, and consider the values
E[W ^ (UH (cid:20) H);(cid:1);D] and M(cid:27)[W ^ (UH (cid:20) M+b [W;(cid:1);M;D;UH]^
H);(cid:1);M;D]:
9V W ^ (U (cid:20)H) ^ (D (cid:20)c((cid:1))) :
H
Then,
(cid:0) (cid:1)
Fromthis,bythesemanticsofexistentialquanti(cid:12)-
E+[W;(cid:1);D;UH] cation, Part 1 follows.
Firstofall,accordingtoPart1ofProposition3,
istheresultofdroppinginE[W ^(U (cid:20)H);(cid:1);D]
H
we obviously have that Cn(W [E[c(F)) is con-
allquanti(cid:12)erswhichbindvariablesfromU .Like-
H
sistent i(cid:11) I[J is a model of
wise,
M+[W;(cid:1);M;D;U ] 9V W ^ (UH (cid:20)H) ^ (D (cid:20)c((cid:1))) :
(cid:27) H
(cid:0) (cid:1)
It remains to show that Cn(W [E[c(F)) is an
is the result of dropping in formula M [W ^
(cid:27)
extension of hW [E;(cid:1)i containing M i(cid:11) I [J is
(U (cid:20) H);(cid:1);M;D] all quanti(cid:12)ers which bind
H
a model of
variables from U .
H
M+[W;(cid:1);M;D;U ]: (2)
Observe that the modules E+[W;(cid:1);D;U ] and b H
H
M+[W;(cid:1);M;D;U ] have D [ U as their sets This can be seen as follows.
(cid:27) H H
of free variables. In fact, the variables in U will Tobeginwith,observethat,byconstruction,(2)
H
beusedforselectingthoseelementsfromH which is a QBF of form (cid:8)W^(UH(cid:20)H), resulting from
represent a solution to the given abduction prob- Mb[W ^ (UH (cid:20) H);(cid:1);M;D] by dropping all
lem. quanti(cid:12)ers which bind variables from UH. Since
The next results gives the speci(cid:12)c encodings for E and I are chosen in such a way that E =
brave and skeptical explanations. uh2Ih, Proposition 2 implies that I [ J is a
Vmodel of (2) i(cid:11) J is a model of (cid:8)W^E, where
Theorem 1 Under the circumstances of De(cid:12)ni- the latter formula is the result of replacing all oc-
tion 3, and supposing further that V = Var(H [ currences of W ^ (U (cid:20) H) in (cid:8) by
H W^(UH(cid:20)H)
W [c((cid:1))), the following conditions hold, for any W ^ E. Now, (cid:8) is a QBF di(cid:11)ering from
W^E
E (cid:18) H and any I (cid:18) U such that h 2 E i(cid:11) M [W ^E;(cid:1);M;D]onlybythepresenceofquan-
H b
u 2I: ti(cid:12)ersbindingvariablesfromVar(HnE).Butsuch
h