Charting the Tractability Frontier of Certain Conjunctive Query Answering

0.45 MB·English

by  Jef Wijsen

Charting the Tractability Frontier of Certain Conjunctive Query
Answering
JefWijsen
Universite´ deMons,Belgium
Abstract C conf year city R conf rank
An uncertain database is defined as a relational database in PODS 2016 Rome PODS A
3 whichprimarykeysneednotbesatisfied. Arepair(orpossi- PODS 2016 Paris KDD A
1 ble world) of such database is obtained by selecting a maxi- KDD 2017 Rome KDD B
0 mal number of tuples without ever selecting two distinct tu-
2 pleswiththesameprimarykeyvalue. ForaBooleanqueryq, Figure1: Uncertaindatabase.

n the decision problem CERTAINTY(q) takes as input an un-
a certain database db and asks whether q is satisfied by every
J
repairofdb. OurmainfocusisonacyclicBooleanconjunc- sothecomplexityoftheproblemisdatacomplexity. There-

6 tive queries without self-join. Previous work [23] has intro- strictiontoBooleanqueriessimplifiesthetechnicaltreatment,

duced the notion of (directed) attack graph of such queries, butisnotfundamental.
]
B andhasprovedthatCERTAINTY(q)isfirst-orderexpressible Primary keys are underlined in the conference planning
D if and only if the attack graph of q is acyclic. The current database of Fig. 1. Maximal sets of tuples that agree on
. paper investigates the boundary between tractability and in- theirprimarykey,calledblocks,areseparatedbydashedlines.
s
tractability of CERTAINTY(q). We first classify cycles in ThereisuncertaintyaboutthecityofPODS2016,andabout
c
[ attackgraphsaseitherweakorstrong,andthenproveamong the rank of KDD. The database has four repairs. The query

othersthefollowing. Iftheattackgraphofaqueryqcontains x y(C(x,y,‘Rome’) R(x,‘A’)) (Will Rome host some A
1 ∃ ∃ ∧
a strong cycle, then CERTAINTY(q) is coNP-complete. If conference?) istrueinonlythreerepairs.
v
3 theattackgraphofqcontainsnostrongcycleandeveryweak The problem CERTAINTY(q) is in coNP for first-order
0 cycleofitisterminal(i.e.,noedgeleadsfromavertexinthe queriesq(a“no”certificateisarepairfalsifyingq). Itscom-
0 cycle to a vertex outside the cycle), then CERTAINTY(q) is plexity for conjunctive queries has attracted the attention of
1
inP. Wethenpartiallyaddresstheonlyremainingopencase, several authors, also outside the database community [5]. A
.
1 i.e., when the attack graph contains some nonterminal cycle major research objective is to find an effective method that
0 and no strong cycle. Finally, we establish a relationship be- takes as input a conjunctive query q and decides to which
3
tween the complexities of CERTAINTY(q) and evaluating q complexityclassesCERTAINTY(q)belongs,ordoesnotbe-
1
: onprobabilisticdatabases. long.Complexityclassesofinterestaretheclassoffirst-order
v
expressibleproblems(orAC0),P,andcoNP-complete.
i 1 Introduction
X Unlessspecifiedotherwise,wheneverwesay“query”inthe
r Primary key violations are a natural way for modeling un- remainderofthispaper,wemeanaBooleanconjunctivequery
a
certainty in the relational model. If two distinct tuples have withoutself-join(i.e.,withoutrepeatedrelationnames). Such
the same primary key value, then at least one of them must queriesarecalledacycliciftheyhaveajointree[3].
be mistaken, but we do not knowwhich one. Thisrepresen- Our previous work [21, 23] has revealed the frontier
tation of uncertainty is also used in probabilistic databases, between first-order expressibility and inexpressibility of
whereeachtupleisassociatedwithaprobabilityanddistinct CERTAINTY(q) for acyclic queries q. In the current work,
tupleswiththesameprimarykeyvaluearedisjointprobabilis- we study the frontier between tractability and intractability
ticevents[18,page35]. of CERTAINTY(q) for the same class of queries. That is,
In this paper, the term uncertain database is used for we aim at an effective method that takes as input a query
databases with primary key constraints that need not be sat- q and decides whether CERTAINTY(q) is in P or coNP-
isfied. A repair (or possible world) of an uncertain database complete (or neither of the two, which is theoretically pos-
db is a maximal subset of db that satisfies all primary key sible if P=coNP [14]). For queries with exactly two atoms,
(cid:54)
constraints. Semantics of querying follows the conventional suchamethodwasrecentlyfoundbyKolaitisandPema[13],
paradigm of consistent query answering [2, 4]: Given a butmovingfromtwotomorethantwoatomsisamajorchal-
Booleanqueryq,thedecisionproblemCERTAINTY(q)takes lenge.
asinputanuncertaindatabasedbandaskswhetherqissatis- Uncertaindatabasesbecomeprobabilisticbyassumingthat
fiedbyeveryrepairofdb.Noticethatqisnotpartoftheinput, the probabilities of all repairs are equal and sum up to 1. In
1
probabilistic terms, distinct tuples of the same block repre- 2 MoreRelatedWork
sent disjoint (i.e., exclusive) events, while tuples of distinct
TheinvestigationofCERTAINTY(q)waspioneeredbyFux-
blocks are independent. Such probabilistic databases have
man and Miller [9, 10], who defined a class of queries q for
been called block-independent-disjoint (BID). The tracta-
which CERTAINTY(q) is first-order expressible. This class
bility/intractabilityfrontierofqueryevaluationonBIDprob-
has later on been extended by Wijsen [21, 23], who devel-
abilisticdatabaseshasbeenrevealedbyDalvietal.[8]. Here,
opedaneffectivemethodtodecidewhetherCERTAINTY(q)
evaluating a Boolean query is a function problem that takes
is first-order expressible for acyclic queries q. In their
as input a BID probabilistic database and asks the probabil-
conclusion, Fuxman and Miller [9, 10] raised the question
ity (a real number between 0 and 1) that q is true. The de-
whether there exist queries q, without self-join, such that
cision problem CERTAINTY(q), on the other hand, simply
CERTAINTY(q) is in P but not first-order expressible. The
askswhetherthisprobabilityisequalto1.
first example of such a query was identified by Wijsen [22].
Inpreviouswork[23], weintroducedthe(directed)attack
Thecurrentpaperidentifiesalargeclassofsuchqueries(all
graphofanacyclicquery,andshowedthatCERTAINTY(q)is
acyclicquerieswithacyclicattackgraphinwhichallcycles
first-orderexpressibleifandonlyifq’sattackgraphisacyclic.
areweakandterminal).
Inthecurrentpaper,westudyattackgraphsinmoredepth.We
Kolaitis and Pema [13] recently showed that for every
willclassifycyclesinattackgraphsaseitherweakorstrong.
queryqwithexactlytwoatoms,CERTAINTY(q)iseitherin
Themaincontributionscanthenbesummarizedasfollows.
P or coNP-complete, and it is decidable which of the two is
the case. If CERTAINTY(q) is in P and not first-order ex-
1. Iftheattackgraphofanacyclicqueryqcontainsastrong
pressible, then it can be reduced in polynomial time to the
cycle, then CERTAINTY(q) is coNP-complete. This
problemoffindingmaximal(withrespecttocardinality)inde-
willbeTheorem2.
pendentsetsofverticesinclaw-freegraphs. Thelatterprob-
lem can be solved in polynomial time by an ingenious algo-
2. If the attack graph of an acyclic query q contains no
rithmofMinty[17]. Unfortunately,theproposedreductionis
strong cycle and all weak cycles of it are terminal (i.e.,
notapplicableonquerieswithmorethantwoatoms.
noedgeleadsfromavertexinthecycletoavertexout-
The counting variant of CERTAINTY(q), which has
sidethecycle), thenCERTAINTY(q)isinP. Thiswill
been denoted (cid:92)CERTAINTY(q), takes as input an uncertain
beTheorem3.
database db and asks to determine the number of repairs of
dbthatsatisfyqueryq. MaslowskiandWijsen[16,15]have
3. The only acyclic queries q not covered by the two pre-
recentlyshowedthatforeveryqueryq,thecountingproblem
ceding results have an attack graph with some nonter-
(cid:92)CERTAINTY(q)iseitherinFPor(cid:92)P-complete,anditisde-
minal cycle and without strong cycle. We provide sup-
cidablewhichofthetwoisthecase.
portingevidenceforourconjecturethatCERTAINTY(q)
As observed in Section 1, uncertain databases are a re-
is tractable for such queries. Our results imply that
stricted case of block-independent-disjoint (BID) probabilis-
CERTAINTY(q) is tractable for “cycle” queries q of
(cid:0) tic databases [7, 8]. This observation will be elaborated in
the form (R x ,x ) R (x ,x )... R (x ,x )
∃(cid:1)∗ 1 1 2 ∧ 2 2 3 ∧ k−1 k−1 k Section7.
R (x ,x ) . These queries arise in the work of Fux-
∧ k k 1 All aforementioned results assume queries without self-
manandMiller[10]. Thecasek=2wassolvedin[22],
join. For queries q with self-joins, only fragmentary results
butthecasek>2wasopenandwillbesettledbyCorol-
aboutthecomplexityofCERTAINTY(q)areknown[6,20].
lary1.
Theextensiontounionsofconjunctivequerieshasbeenstud-
iedin[12].
4. Theorem 6 and its Corollary 2 will establish a
relationship between the tractability frontiers of 3 Preliminaries
CERTAINTY(q) and query evaluation on probabilistic
We assume disjoint sets of variables and constants. If(cid:126)x is
databases.
a sequence containing variables and constants, then vars((cid:126)x)
denotesthesetofvariablesthatoccurin(cid:126)x,and (cid:126)x denotesthe
Ourworksignificantlyextendsandgeneralizesknownresults | |
lengthof(cid:126)x.
intheliterature.
LetU be a set of variables. A valuation overU is a total
The remainder of this paper is organized as follows. The
mappingθfromU tothesetofconstants. Suchvaluationθis
next section further discusses related work. Section 3 de-
extendedtobetheidentityonconstantsandonvariablesnot
fines the basic notions of certain conjunctive query answer-
inU.
ing. Section4definesthenotionofattackgraph. Sections5
and6showourmainintractabilityandtractabilityresultsre- Atoms and key-equal facts. Every relation name R has
spectively. Section 7 establishes a relationship between the a fixed signature, which is a pair [n,k] with n k 1: the
≥ ≥
complexities of CERTAINTY(q) and evaluating query q on integernisthearityoftherelationnameand 1,2,...,k is
{ }
probabilistic databases. Section 8 concludes the paper and theprimarykey.TherelationnameRisall-keyifn=k.IfRis
raiseschallengesforfutureresearch.Severalproofshavebeen arelationnamewithsignature[n,k],thenR(s ,...,s )isanR-
1 n
movedtoanAppendix. atom(orsimplyatom),whereeachs iseitheraconstantora
i
2
variable(1 i n).SuchatomiscommonlywrittenasR((cid:126)x,(cid:126)y) edge between F and G with label L. A join tree is shown in
≤ ≤
where the primary key value(cid:126)x=s ,...,s is underlined and Fig.2(left).
1 k
(cid:126)y=s ,...,s .Afactisanatominwhichnovariableoccurs.
k+1 n Certain query answering. Given a Boolean conjunctive
TwofactsR ((cid:126)a ,(cid:126)b ),R ((cid:126)a ,(cid:126)b )arekey-equalifR =R and
1 1 1 2 2 2 1 2 queryq,CERTAINTY(q)is(thecomplexityof)thefollowing
(cid:126)a =(cid:126)a .
1 2 set.
WewilluselettersF,G,H,Iforatoms,andA,B,Cforfacts
ofanuncertaindatabase. ForatomF =R((cid:126)x,(cid:126)y),wedenoteby CERTAINTY(q)= db dbisanuncertaindatabasesuch
{ |
key(F)thesetofvariablesthatoccurin(cid:126)x,andbyvars(F)the thateveryrepairofdbsatisfiesq
}
setofvariablesthatoccurinF,thatis,key(F)=vars((cid:126)x)and
CERTAINTY(q) is said to be first-order expressible if there
vars(F)=vars((cid:126)x) vars((cid:126)y).
∪ exists a first-order sentence ϕ such that for every uncertain
Uncertain database, blocks, and repairs. A database databasedb,db CERTAINTY(q)ifandonlyifdb=ϕ.The
∈ |
schema is a finite set of relation names. All constructs that formulaϕ,ifitexists,iscalledacertainfirst-orderrewriting
followaredefinedrelativetoafixeddatabaseschema. of q.
Anuncertaindatabaseisafinitesetdboffactsusingonly
Purified uncertain databases. Let q be a Boolean con-
therelationnamesoftheschema. Ablockofdbisamaximal
junctivequery.Anuncertaindatabasedbissaidtobepurified
set of key-equal facts of db. If A db, then block(A,db)
relativetoqifforeveryfactA db, thereexistsavaluation
∈
denotestheblockofdbcontainingA. Anuncertaindatabase ∈
θovervars(q)suchthatA θ(q) db. Intuitively,everyfact
dbisconsistentifitdoesnotcontaintwodistinctfactsthatare ∈ ⊆
inapurifieduncertaindatabaseisrelevantforthequery. This
key-equal(i.e.,ifeveryblockofdbisasingleton). Arepair
notionofpurifieddatabaseisnewandillustratednext.
ofdbisamaximalconsistentsubsetofdb.1
Example1 Theuncertaindatabase R(a,b),S(b,a),S(b,c)
Booleanconjunctivequery. ABooleanconjunctivequery { }
is not purified relative to query R(x,y),S(y,x) because it
is a finite set q={R1((cid:126)x1,(cid:126)y1), ..., Rn((cid:126)xn,(cid:126)yn)} of atoms. By containsnoR-factthat“joins”wit{hS(b,c). } (cid:67)
vars(q),wedenotethesetofvariablesthatoccurinq. Theset
qrepresentsthefirst-ordersentence Thefollowinglemmaimpliesthatinthestudyoftractabil-
ity of CERTAINTY(q), we can assume without loss of gen-
(cid:0) (cid:1)
u1... uk R1((cid:126)x1,(cid:126)y1) Rn((cid:126)xn,(cid:126)yn) , erality that uncertain databases are purified; this assumption
∃ ∃ ∧···∧
willsimplifythetechnicaltreatment. Noticethatthequeryq
where u ,...,u = vars(q). The query q is satisfied by
1 k inthelemma’sstatementisnotrequiredtobeacyclic.
{ }
uncertain database db, denoted db = q, if there exists a
|
valuation θ over vars(q) such that for each i 1,...,n , Lemma1 LetqbeaBooleanconjunctivequery. Letdb0 be
∈ { }
R(θ((cid:126)x),θ((cid:126)y)) db. We say that q has a self-join if some anuncertaindatabase.Itispossibletocomputeinpolynomial
i i i
∈
relationnameoccursmorethanonceinq(i.e.,ifR =R for time an uncertain database db that is purified relative to q
i j
some1 i< j n). suchthat
≤ ≤
The restriction to Boolean queries simplifies the technical db CERTAINTY(q) db0 CERTAINTY(q).
∈ ⇐⇒ ∈
treatment,butisnotfundamental. Sinceeveryrelationname
4 AttackGraph
hasafixedsignature,relevantprimarykeyconstraintsareim-
plicitlypresentinallqueries;moreover,primarykeyswillbe The primary key of an atom F gives rise to a functional de-
underlined. pendencyamongthevariablesthatoccurinF. Forexample,
R(x,y,z,u)givesriseto x,y x,y,z,u ,whichwillbeab-
Join tree and acyclic conjunctive query. The notions of { }→{ }
breviatedasxy xyzu(andwhichisequivalenttoxy zu).
join tree and acyclicity [3] are recalled next. A join tree for → →
The set K(q) defined next collects all functional dependen-
aconjunctivequeryqisanundirectedtreewhoseverticesare
ciesthatariseinatomsofq.
theatomsofqsuchthatthefollowingconditionissatisfied:
Definition1 LetqbeaBooleanconjunctivequery.Wedefine
ConnectednessCondition.Wheneverthesamevari-
K(q)asthefollowingsetoffunctionaldependencies.
ablexoccursintwoatomsF andG, thenxoccurs
ineachatomontheuniquepathlinkingF andG. K(q)= key(F) vars(F) F q (cid:67)
{ → | ∈ }
Commonly,anedgebetweenatomsF andGislabeledbythe Concerningthefollowingdefinition,recallfromrelational
(possiblyempty)setvars(F) vars(G). ThetermConnected- databasetheory[19,page387]thatifΣisasetoffunctional
ness Condition appears in [1∩1] and refers to the fact that the dependencies over a setU of attributes and X U, then the
⊆
setofverticesinwhichxoccursinducesaconnectedsubtree. attribute closure of X (with respect to Σ) is the set A U
{ ∈ |
Aconjunctivequeryqisacyclicifithasajointree. Thesym- Σ =X A .
| → }
L
bolτwillbeusedforjointrees. WewriteF(cid:97)Gtodenotean Definition2 LetqbeaBooleanconjunctivequery.Forevery
F q,wedefineF+,qasthefollowingsetofvariables.
1Itmakesnodifferencewhethertheword“maximal”referstocardinality ∈
ofsetsorset-containment. F+,q= x vars(q) K(q F ) =key(F) x (cid:67)
{ ∈ | \{ } | → }
3
R(u,a,x)=F R(u,a,x)=F
x
{ }
S(y,x,z)=G S(y,x,z)=G
x,y x,z
{ } { }
T(x,y)=H P(x,z)=I T(x,y)=H P(x,z)=I
Figure2: Jointree(left)andattackgraph(right)ofqueryq . TheattackfromGtoF isstrong. Allotherattacksareweak.
1
Inwords,F+,qistheattributeclosureofthesetkey(F)with LetusnextcomputetheattacksoutgoingfromH. Thepath
respecttothesetoffunctionaldependenciesthatariseinthe x,y
fromH toGinthejointreeisH{(cid:97)}G. Sincethelabel x,y
atomsofq\{F}.Notethatvariablesplaytheroleofattributes isnotcontainedinG+,q1,theattackgraphcontainsadir{ecte}d
inourframework.
edgefromH toG,.i.e.,H(cid:32)τ1 G. ThepathfromH toF inthe
Example2 Let q1 = R(u,a,x), S(y,x,z), T(x,y), P(x,z) . x,y x
Ajointreeforthisque{ryisshowninFig.2(left). Toshorte}n jointreeisH{(cid:97)}G{(cid:97)}F. Sincethelabel x iscontainedin
notation, let F =R(u,a,x), G=S(y,x,z), H =T(x,y), and H+,q1, the attack graph contains no directe{d}edge from H to
I=P(x,z),asindicatedinthefigure. Wehavethefollowing. F. Andsoon. ThecompleteattackgraphisshowninFig.2
(right). (cid:67)
K(q F )= y xyz,x xy,x xz
1
\{ } { → → → }
key(F)= u andF+,q1 = u Remarkably,itwasshownin[23]thatifτ1 andτ2 aredis-
{ } { }
tinct join trees for the same acyclic query q, then the attack
K(q G )= u ux,x xy,x xz
1\{ } { → → → } graph of τ1 is identical to the attack graph of τ2. This moti-
key(G)= y andG+,q1 = y vatesthefollowingdefinition.
{ } { }
K(q H )= u ux,y xyz,x xz
1\{ } { → → → } Definition4 LetqbeanacyclicBooleanconjunctivequery.
key(H)= x andH+,q1 = x,z Theattackgraphofqistheattackgraphofτforanyjointree
{ } { }
K(q I )= u ux,y xyz,x xy τforq. WewriteF (cid:32)q G(orsimplyF (cid:32)Gifqisclearfrom
1
\{ } { → → → }
key(I)= x andI+,q1 = x,y,z (cid:67) the context) to indicate that the attack graph of q contains a
{ } { } q
directed edge from F to G. We write F (cid:32)G if it is not the
Definition3 LetqbeanacyclicBooleanconjunctivequery. casethatF (cid:32)q G. (cid:54) (cid:67)
Letτbeajointreeforq. Theattackgraphofτisadirected
graph whose vertices are the atoms of q. There is a directed
Theattackgraphofanacyclicqueryqcanbecomputedin
edgefromF toGifF,Garedistinctatomssuchthatforevery
quadratictimeinthelengthofq[23]. Figures4and5show
label L on the unique path that links F and G in τ, we have
attack graphs, but omit join trees. The main result in [23] is
L(cid:42)F+,q.
thefollowing.
WewriteF(cid:32)τ Giftheattackgraphofτcontainsadirected
edgefromF toG. ThedirectededgeF (cid:32)τ Gisalsocalledan Theorem1([23]) The following are equivalent for all
attackfromF toG.IfF(cid:32)τ G,wesaythatF attacksG(orthat acyclicBooleanconjunctivequeriesqwithoutself-join:
GisattackedbyF). (cid:67)
1. Theattackgraphofqisacyclic.
Example3 ThisisacontinuationofExample2. Fig.2(left)
2. CERTAINTY(q)isfirst-orderexpressible.
shows a join tree τ for query q . The attack graph of τ is
1 1 1
showninFig.2(right)andiscomputedasfollows.
Finally,weprovidetwolemmasthatwillbeusefullateron.
LetusfirstcomputetheattacksoutgoingfromF. Thepath
x
from F to G in the join tree is F {(cid:97)}G. Since the label x Lemma2 LetqbeanacyclicBooleanconjunctivequery.Let
isnotcontainedinF+,q1,theattackgraphcontainsadirec{ted} F,Gbedistinctatomsofq.IfF(cid:32)G,thenkey(G)(cid:42)F+,qand
edge from F to G, i.e., F (cid:32)τ1 G. The path from F to H in vars(F)(cid:42)F+,q.
x x,y
thejointreeisF {(cid:97)}G{(cid:97)}H. Sincenolabelonthatpathis Lemma3([23]) Let q be an acyclic Boolean conjunctive
containedinF+,q1,theattackgraphcontainsadirectededge query. Let F,G,H be distinct atoms of q. If F (cid:32) G and
fromF toH. Inthesameway,onefindsthatF attacksI. G(cid:32)H,thenF (cid:32)H orG(cid:32)F.
4
5 Intractability
vars(q)
The following definition classifies cycles in attack graphs as
either strong or weak. The main result of this section is
thatCERTAINTY(q)iscoNP-completeforacyclicqueriesq F⊞,q hx,y,zi
whoseattackgraphcontainsastrongcycle.
F+,q G+,q
Definition5 LetqbeanacyclicBooleanconjunctivequery.
y
ForeveryF q,wedefineF(cid:1),q asthefollowingsetofvari-
∈
ables.
x,y x ‘d’ y,z
F(cid:1),q= x vars(q) K(q) =key(F) x h i h i
{ ∈ | | → }
An attack F (cid:32)G in the attack graph of q is called weak if
key(G) F(cid:1),q. Anattackthatisnotweak,iscalledstrong.
⊆
A (directed) cycle of size n in the attack graph of q is a
sequence of edges F (cid:32)F (cid:32)F ...(cid:32)F (cid:32)F such that
0 1 2 n 1 0
i= jimpliesF =F. Thus,cyclemeansel−ementarycycle.
i j
(cid:54) (cid:54)
A cycle in the attack graph of q is called strong if at least
oneattackinthecycleisstrong. Acyclethatisnotstrong,is
calledweak. (cid:67)
Figure3: HelpfortheproofofTheorem2.
ItisstraightforwardthatF+,q F(cid:1),q.
⊆
Example4 Forthequeryq1inFig.2,wehavethefollowing. Theorem2 Let q be an acyclic Boolean conjunctive query
without self-join. If the attack graph of q contains a strong
K(q ) = u ux,y xyz,x xy,x xz
1 { → → → → } cycle,thenCERTAINTY(q)iscoNP-complete.
F(cid:1),q1 = u,x,y,z
{ }
G(cid:1),q1 = x,y,z Proof Since CERTAINTY(q) is obviously in coNP, it suf-
{ } fices to show that it is coNP-hard. Assume that the attack
H(cid:1),q1 = x,y,z graphofqcontainsastrongcycle. ByLemma4,wecanas-
{ }
I(cid:1),q1 = x,y,z sumeF,G qsuchthatF (cid:32)G(cid:32)F andtheattackF (cid:32)Gis
{ } ∈
strong. Foreveryvaluationθover x,y,z ,wedefine(cid:98)θasthe
The attack F (cid:32)q1 G is weak, because key(G)= x F(cid:1),q1. followingvaluationovervars(q). { }
TheattackG(cid:32)q1 F isstrong,becausekey(F)={u}⊆G(cid:1),q1.
OnecanverifythattheattackfromGtoF isth{eo}n(cid:54)⊆lystrong 1. Ifu F+,q G+,q, then(cid:98)θ(u)=‘d’forsomefixedcon-
∈ ∩
stantd;
attackintheattackgraphofq .
1
The attack cycle G(cid:32)q1 H (cid:32)q1 G is weak. The attack cycle 2. ifu F+,q G+,q,then(cid:98)θ(u)=θ(x);
F (cid:32)q1 G(cid:32)q1 F is strong, because it contains the strong attack ∈ \
G(cid:32)q1 F. Forthesamereason,theattackcycleF(cid:32)q1 H(cid:32)q1 G(cid:32)q1 3. ifu G+,q F(cid:1),q,then(cid:98)θ(u)= θ(y),θ(z) ;
∈ \ (cid:104) (cid:105)
F isstrong. (cid:67) 4. ifu (cid:0)G+,q F(cid:1),q(cid:1) F+,q,then(cid:98)θ(u)=θ(y);
∈ ∩ \
Example4showedthattheattackgraphofq hasastrong
1 5. ifu F(cid:1),q (cid:0)F+,q G+,q(cid:1),then(cid:98)θ(u)= θ(x),θ(y) ;and
cycle of length 3, and a strong cycle of length 2. This is no
∈ \ ∪ (cid:104) (cid:105)
coincidence,asstatedbythefollowinglemma.
6. ifu F(cid:1),q G+,q,then(cid:98)θ(u)= θ(x),θ(y),θ(z) .
(cid:54)∈ ∪ (cid:104) (cid:105)
Lemma4 LetqbeanacyclicBooleanconjunctivequery. If
Notice that (cid:98)θ(u) can be a sequence of length two or three;
theattackgraphofqcontainsastrongcycle,thenitcontains
twosequencesofthesamelengthareequaliftheycontainthe
astrongcycleoflength2.
sameelementsinthesameorder. TheVenndiagramofFig.3
Thefollowingproofestablishesthatforeveryacyclicquery willcomeinhandy: everyregioncontainsaboxedlabelthat
q whose attack graph contains a strong cycle, there exists a indicateshow(cid:98)θ(u)iscomputedforvariablesuinthatregion.
polynomial-timemany-onereductionfromCERTAINTY(q ) Forexample,assumeubelongstotheregionwithlabel x,y
0
to CERTAINTY(q), where q0 = R0(x,y),S0(y,z,x) . Since (i.e.,u F(cid:1),q (cid:0)F+,q G+,q(cid:1)),then(cid:98)θ(u)= θ(x),θ(y)(cid:104). (cid:105)
{ } ∈ \ ∪ (cid:104) (cid:105)
CERTAINTY(q0) was proved coNP-hard by Kolaitis and Weshowthreesublemmasthatwillbeusedlateroninthe
Pema[13],weobtainthedesiredcoNP-hardlowerboundfor proof.
CERTAINTY(q). Astheproofisratherinvolved,weprovide
inFig.3amnemonicfortheconstructioninthebeginningof Sublemma1 Let θ1,θ2 be two valuations over x,y,z . If
{ }
the proof. To further improve readability, some parts of the H qsuchthatF =H =G, then θ(cid:98)1(H),θ(cid:98)2(H) isconsis-
∈ (cid:54) (cid:54) { }
proofwillbestatedassublemmas. tent.
5
ProofSublemma1LetH qsuchthatF=H=G. Assume respectively. LetF =R (x,y)andG =S (y,z,x). Inthere-
0 0 0 0
∈ (cid:54) (cid:54)
thefollowing. mainder of the proof, we establish a polynomial-time many-
one reduction from CERTAINTY(q ) to CERTAINTY(q).
0
Foreveryu key(H),θ(cid:98)1(u)=θ(cid:98)2(u). (1) coNP-hardnessofCERTAINTY(q)thenfollowsfromcoNP-
∈
hardnessofCERTAINTY(q ),whichwasestablishedin[13].
0
Itsufficestoshowthefollowing. Let db be an uncertain database. By Lemma 1, we can
0
assumethatdb ispurifiedrelativetoq . LetV bethesetof
0 0
Foreveryu∈vars(H),θ(cid:98)1(u)=θ(cid:98)2(u). (2) valuationsθover x,y,z suchthatθ(q0) db0. Sincedb0is
{ } ⊆
purified,thefollowingholds.
Weconsiderfourcases.
Case θ (x)=θ (x) and θ (y)=θ (y). If θ (z)=θ (z), db0 = θ(F0) θ V θ(G0) θ V
1 2 1 2 1 2 { | ∈ }∪{ | ∈ }
thenθ =θ ,and(2)holdsvacuously. Assumenextθ (z)=
1 2 1
(cid:54) Letdb= (cid:98)θ(H) H q,θ V . SinceV canbecomputed
θ2(z). Thenitfollowsfrom(1)thatnovariableofkey(H)be- { | ∈ ∈ }
inpolynomialtimeinthesizeofdb ,thereductionfromdb
longstoaregionoftheVenndiagram(seeFig.3)thatcontains 0 0
z. Since z occurs in all regions outside F(cid:1),q, we conclude todbisinpolynomialtime. Sinceqcontainsnoself-join,the
key(H) F(cid:1),q. Since K(q) contains key(H) vars(H), it setdbispartitionedbythethreedisjointsubsetsdefinednext.
follows⊆vars(H) F(cid:1),q. Sincezdoesnotoccu→rinsideF(cid:1),q
⊆ dbF = (cid:98)θ(F) θ V
intheVenndiagram,weconclude(2). { | ∈ }
dbG = (cid:98)θ(G) θ V
Case θ (x)=θ (x) and θ (y)=θ (y). By (1), no vari- { | ∈ }
able of ke1y(H) be2longs to a1 reg(cid:54)ion2of the Venn diagram dbrest = {(cid:98)θ(H)|H∈q,F (cid:54)=H(cid:54)=G,θ∈V}
that contains y. It follows key(H) F+,q. Consequently,
Sincedb isconsistentbySublemma1,everyrepairofdb
vars(H) F+,q. Since neither y nor⊆z occurs inside F+,q in rest
isthedisjointunionofdb ,arepairofdb ,andarepairof
⊆ rest F
theVenndiagram,weconclude(2).
db . Inthenextstepoftheproof,weestablishaone-to-one
G
Case θ1(x) = θ2(x) and θ1(y) = θ2(y). First assume relationshipbetweenrepairsofdb0andrepairsofdb.
θ1(z)=θ2(z).(cid:54) By (1), no variable of key(H) belongs to a Thefunctionmapwillmaprepairsofdb0 torepairsofdb.
region of the Venn diagram that contains x. Consequently, For every repair r0 of db0, map(r0) is the disjoint union of
key(H) G+,q. Itfollowsvars(H) G+,q. Sincexdoesnot threesets,asfollows.
occurin⊆sideG+,qintheVenndiagra⊆m,weconclude(2).
Nextassumeθ1(z)=θ2(z). By(1),novariableofkey(H) map(r0) = {(cid:98)θ(F)|θ(F0)∈r0,θ∈V}
belongstoaregionof(cid:54) theVenndiagramthatcontainsxorz. (cid:98)θ(G) θ(G0) r0,θ V
∪ { | ∈ ∈ }
Consequently, key(H) F(cid:1),q G+,q. It follows vars(H) ∪ dbrest
F(cid:1),q G+,q. Sinceneit⊆herxno∩rzoccursinsideF(cid:1),q G+⊆,q
∩ ∩ Clearly, the first of these three sets is contained in dbF, and
intheVenndiagram,weconclude(2).
thesecondindb . BySublemmas2and3,foreveryθ V,
G
∈
Caseθ (x)=θ (x)andθ (y)=θ (y). By(1),novariable
1 2 1 2
of key(H) be(cid:54)longs to a region(cid:54) of the Venn diagram that θ(F0)∈r0 ⇐⇒ (cid:98)θ(F)∈map(r0) (3)
contains x or y. Consequently, key(H) F+,q G+,q. It θ(G0) r0 (cid:98)θ(G) map(r0) (4)
followsvars(H) F+,q G+,q.Sincenone⊆ofx,y,∩orzoccurs ∈ ⇐⇒ ∈
inside F+,q G+⊆,q in t∩he Venn diagram, we conclude (2). Toprovethe =-directionof(3)(theotherimplicationsare
⇐
Thisconclud∩estheproofofSublemma1. straightforward),assumeA map(r0)withA=(cid:98)θ(F). Bythe
∈
(cid:97) definitionofmap,wecanassumeθ V suchthatθ(F ) r
(cid:48) (cid:48) 0 0
∈ ∈
andθ(cid:98)(F)=A.From(cid:98)θ(F)=θ(cid:98)(F),itfollowsbySublemma2
(cid:48) (cid:48)
Sublemma2 Letθ1,θ2betwovaluationsover{x,y,z}. thatθ(F0)=θ(cid:48)(F0),henceθ(F0)∈r0.
Thefollowingsublemmastatesthatmapisabijectionfrom
1. θ(cid:98)1(F)andθ(cid:98)2(F)arekey-equal θ1(x)=θ2(x). thesetofrepairsofdb0tothesetofrepairsofdb.
⇐⇒
2. θ(cid:98)1(F)=θ(cid:98)2(F) θ1(x)=θ2(x)andθ1(y)=θ2(y). Sublemma4 1. Ifr0 isarepairofdb0,thenmap(r0)isa
⇐⇒ repairofdb.
Sublemma3 Letθ ,θ betwovaluationsover x,y,z .
1 2
{ } 2. Foreveryrepairrofdb, thereexistsarepairr0 ofdb0
1. θ(cid:98)1(G)andθ(cid:98)2(G)arekey-equal θ1(y)=θ2(y)and suchthatr=map(r0).
⇐⇒
θ (z)=θ (z).
1 2 3. If r ,r are distinct repairs of db , then map(r ) =
0 (cid:48)0 0 0 (cid:54)
map(r ).
2. θ(cid:98)1(G)=θ(cid:98)2(G) θ1=θ2. (cid:48)0
⇐⇒
ToconcludetheproofofTheorem2,weshow:
We continue the proof of Theorem 2. Let q = R (x,y),
0 0
{
S (y,z,x) . The signatures of R and S are [2,1] and [3,2] db CERTAINTY(q ) db CERTAINTY(q).
0 0 0 0 0
} ∈ ⇐⇒ ∈
6
BySublemma4,itissufficienttoprovethatforeveryrepair
r ofdb ,
0 0 R (x,y,u ,u ) R (x,y,u ,u )
3 3 4 4 4 3
r =q map(r ) =q.
0 0 0
| ⇐⇒ |
= Assume r =q . We can assume θ V such that
0 0
⇒ | ∈
θ(q0) r0. Obviously,(cid:98)θ(q) map(r0). R1(x,u1,u2,z) R5(y,u5,u6)
⊆ ⊆
= Assumemap(r ) =q. Wecanassumeavaluation
0
⇐ | R (u,z)
µovervars(q)suchthatµ(q) map(r ). 0
0
⊆
Letτbeajointreeforq.LetH0(cid:97)L1H1...(cid:97)L(cid:96)H(cid:96)betheunique R2(x,u2,u1,z) R6(y,u6,u5)
pathinτbetweenFandG,whereH =FandH =G.Fori
0 (cid:96)
∈
0,...,(cid:96) ,wecanassumeθi V suchthatµ(Hi)=θ(cid:98)i(Hi)
{ } ∈ ∈
map(r0). Let i 0,...,(cid:96) 1 . We show θi(x)=θi+1(x) Figure4: Attackgraph. Allcyclesareweakandterminal.
andθ(y)=θ (∈y){. SinceF−(cid:32)}G(cid:32)F,thelabelL contains
i i+1 i
a variable u such that u F+,q and a variable w such that
i i i
wi(cid:54)∈G+,q(possiblyui=w(cid:54)∈i). Rar4e(xte,rym,ui4n,aul3a)n,dRw5(eya,ku.5,u6), R6(y,u6,u5)}. All attack cycle(cid:67)s
Since u vars(H) vars(H ), it must be the case that
i i i+1
∈ ∩
θ(cid:98)i(ui)=µ(ui)=θ(cid:100)i+1(ui). Sinceyoccursineveryregionout- Example6 IntheattackgraphofFig.5,allcyclesareweak,
side F+,q in the Venn diagram (Fig. 3) and ui F+,q, it is butnocycleisterminal. (cid:67)
(cid:54)∈
correcttoconcludeθ(y)=θ (y).
i i+1
Likewise, since w vars(H) vars(H ), it must be the Theorem3 Let q be an acyclic Boolean conjunctive query
i i i+1
∈ ∩ without self-join. If all cycles in the attack graph of q are
casethatθ(cid:98)i(wi)=µ(wi)=θ(cid:100)i+1(wi). Sincexoccursinevery
regionoutsideG+,q intheVenndiagramandw G+,q, itis weakandterminal,thenCERTAINTY(q)isinP.
i
(cid:54)∈
correcttoconcludeθi(x)=θi+1(x). Notice that if a query q has exactly two atoms, then q is
Consequently, θ0(x) = θ(cid:96)(x) and θ0(y) = θ(cid:96)(y). From acyclicandeverycycleinq’sattackgraphmustbeterminal.
θ(cid:98)0(H0),θ(cid:98)(cid:96)(H(cid:96)) map(r0), H0 =F, and H(cid:96) =G, it follows Therefore Theorems 2 and 3 together imply the dichotomy
∈
θ0(F0),θ(cid:96)(G0) r0 by (3) and (4). Since θ0 and θ(cid:96) agree theoremofKolaitisandPema[13].
∈
on each variable in vars(F0) vars(G0)= x,y , it follows ToproveTheorem3,weneedfourhelpinglemmas.Insim-
r0 =q0. Thisconcludesthep∩roofofTheore{m2.} (cid:3) plewords,thefirstlemmastatesthatifwereplaceavariable
|
with a constant in an acyclic query, then no new attacks are
generated,andweakattackscannotbecomestrong.
6 Tractability
We conjecture that if the attack graph of an acyclic query q Definition7 Let q be a Boolean conjunctive query. If
containsnostrongcycle,thenCERTAINTY(q)isinP. (cid:126)x = x1,...,x(cid:96) is a sequence of distinct variables and(cid:126)a =
(cid:104) (cid:105)
a ,...,a a sequence of constants, then q denotes the
1 (cid:96) [(cid:126)x (cid:126)a]
Conjecture1 LetqbeanacyclicBooleanconjunctivequery (cid:104)queryobta(cid:105)inedfromqbyreplacingeachoccur(cid:55)→renceofx with
i
without self-join. If all cycles in the attack graph of q are a, for all i 1,...,(cid:96) . If θ is a valuation, then θ is
i [(cid:126)x (cid:126)a]
weak,thenCERTAINTY(q)isinP. the valuation∈s{uch that}θ ((cid:126)x)=(cid:126)a and θ (y)=θ(cid:55)→(y) if
[(cid:126)x (cid:126)a] [(cid:126)x (cid:126)a]
y vars((cid:126)x). (cid:55)→ (cid:55)→ (cid:67)
(cid:54)∈
NoticethatbyTheorem1,weknowthatConjecture1holds
inthespecialcasewhereq’sattackgraphcontainsnocycleat Lemma5 Let q be an acyclic Boolean conjunctive query
all. Theorem 2 and Conjecture 1 together imply that for ev- withoutself-join. LetF,G q. Letz vars(q)andletcbea
∈ ∈
eryacyclicqueryq,CERTAINTY(q)iseitherinPorcoNP- constant. Letq(cid:48)=q[z c],F(cid:48)=F[z c],andG(cid:48)=G[z c]. Then,
thefollowinghold. (cid:55)→ (cid:55)→ (cid:55)→
complete. In the following section, a somewhat weaker ver-
sionofConjecture1isproved.
1. q isacyclic.
(cid:48)
6.1 AllCyclesareWeakandTerminal
2. IfF (cid:32)q(cid:48) G,thenF (cid:32)q G.
We show a weaker version of Conjecture 1. In this weaker (cid:48) (cid:48)
version,thepremise“allcyclesareweak”isstrengthenedinto 3. IfF (cid:32)q(cid:48) G andF (cid:32)q Gisaweakattack,thenF (cid:32)q(cid:48) G is
“allcyclesareweakandterminal.” (cid:48) (cid:48) (cid:48) (cid:48)
aweakattack.
Definition6 Acycleinadirectedgraphiscalledterminalif Lemma6 LetqbeanacyclicBooleanconjunctivequery. If
thegraphcontainsnodirectededgefromavertexinthecycle eachcycleintheattackgraphofqisterminal,theneachcycle
to a vertex outside the cycle. A cycle is nonterminal if it is intheattackgraphhaslength2.
notterminal. (cid:67)
Lemma7 Let q be an acyclic Boolean conjunctive query
Example5 Figure 4 shows the attack graph of the suchthateachcycleoftheattackgraphofqisterminaland
acyclic query R (x,u ,u ,z), R (x,u ,u ,z), R (x,y,u ,u ), eachatomofqbelongstoacycleoftheattackgraph.
1 1 2 2 2 1 3 3 4
{
7
1. If the same variable x occurs in two distinct cycles of Lemma 7 and [13, Theorem 2], CERTAINTY(q) is in P
i
the attack graph, then for each atom F in these cycles, for 1 i (cid:96). From the following sublemma, it follows that
≤ ≤
x key(F). CERTAINTY(q)isinP.
∈
2. IfF (cid:32)q Gisaweakattack,thenkey(G) vars(F). Sublemma5 Thefollowingareequivalent:
⊆
1. db CERTAINTY(q).
The following lemma applies to queries with an atom ∈
whoseprimarykeycontainsnovariables. 2. (cid:83) db =q.
1≤i≤(cid:96)(cid:84) i(cid:85)|
Lemma8 LetqbeaBooleanconjunctivequerywithoutself- Forthestepoftheinduction,assumethatFisanunattacked
join.LetF qsuchthatkey(F)=0/.Letq =q F .Let(cid:126)ybe atominq’sattackgraph. Let(cid:126)xbeasequenceofdistinctvari-
(cid:48)
∈ \{ }
asequenceofdistinctvariablessuchthatvars((cid:126)y)=vars(F). ablessuchthatvars((cid:126)x)=key(F). ByCorollary8.11in[23],
Letdbbeanuncertaindatabasethatispurifiedrelativetoq, thefollowingareequivalent.
andletDbetheactivedomainofdb. Thenthefollowingare
1. db CERTAINTY(q).
equivalent: ∈
2. Forsome(cid:126)a D(cid:126)x,db CERTAINTY(q ).
1. db CERTAINTY(q). ∈ | | ∈ [(cid:126)x(cid:55)→(cid:126)a]
∈
Let(cid:126)ybeasequenceofdistinctvariablessuchthatvars((cid:126)y)=
2. db(cid:54)=0/ and for all(cid:126)b∈D|(cid:126)y|, if F[(cid:126)y (cid:126)b] ∈db, then db∈ vars(F) key(F).Letq(cid:48)=q F .ByLemma1,itispossible
CERTAINTY(q(cid:48)[(cid:126)y (cid:126)b]). (cid:55)→ tocomp\uteinpolynomialtim\{ea}databasedb(cid:48) thatispurified
(cid:55)→ relativetoq suchthat
[(cid:126)x (cid:126)a]
TheproofofTheorem3cannowbegiven. (cid:55)→
db CERTAINTY(q ) db CERTAINTY(q ).
Proof Theorem 3 Given uncertain database db, we need ∈ [(cid:126)x(cid:55)→(cid:126)a] ⇐⇒ (cid:48)∈ [(cid:126)x(cid:55)→(cid:126)a]
toshowthatitcanbedecidedinpolynomialtime(inthesize ByLemma8,thefollowingareequivalent:
of db) whether db CERTAINTY(q). Let D be the active
∈ 1. db CERTAINTY(q ).
domainofdb.ByLemma1,wecanassumethatdbispurified (cid:48)∈ [(cid:126)x(cid:55)→(cid:126)a]
relativetoq. 2. db =0/ andforall(cid:126)b D(cid:126)y,ifF db,thendb
Theproofrunsbyinductiononthelengthofq.Forthebase (cid:48)(cid:54) ∈ | | [(cid:126)x(cid:126)y (cid:126)a(cid:126)b]∈ (cid:48) (cid:48)∈
CERTAINTY(q ). (cid:55)→
oftheinduction,weconsiderthecasewheretheattackgraph (cid:48)[(cid:126)x(cid:126)y (cid:126)a(cid:126)b]
(cid:55)→
ofqcontainsnounattackedatom(i.e.,noatomhaszeroinde- By Lemma 5, all cycles in the attack graph of q
(cid:48)[(cid:126)x(cid:126)y (cid:126)a(cid:126)b]
gree). CERTAINTY(q)isobviouslyinPifq= . Assume are weak and terminal. By the induction hypothesis(cid:55)→, it
{}
nextthatqisnonempty.
follows that CERTAINTY(q ) is in P. Since the
Sinceallcyclesofq’sattackgraphareterminalandevery sizes of D(cid:126)x and D(cid:126)y are poly(cid:48)[(cid:126)xn(cid:126)yo(cid:55)→m(cid:126)a(cid:126)bia]lly bounded in the size
atomhasanincomingattack,everyatomofqbelongstosome | | | |
ofdb,itiscorrecttoconcludethatCERTAINTY(q)isinP.(cid:3)
cycleoftheattackgraph.ByLemma6,theattackgraphofqis
asetofdisjointweakcyclesF (cid:32)G (cid:32)F ,...,F (cid:32)G (cid:32)
1 1 1 (cid:96) (cid:96)
F for some (cid:96) 1. For i 1,...,(cid:96) , let q = F,G , and 6.2 NonterminalWeakCycles
(cid:96) i i i
≥ ∈{ } { }
let(cid:126)x be a sequence of distinct variables that contains every
i Theorems 2 and 3 leave open the complexity of
variable x vars(q) such that for some j=i, x vars(q ).
i j CERTAINTY(q) when the attack graph of q contains
∈ (cid:54) ∈
ByLemma7,vars(x) key(F) key(G).
i i i oneormorenonterminalweakcyclesandnostrongcycle. In
⊆ ∩
For i 1,...,(cid:96) , let db be the subset of db containing
i this section, we zoom in on acyclic queries AC(k), defined
∈{ }
epvaerrtiytifoanctofAdwbiitahmthaexsimamalesrueblasetitoPnonfamdbeiassucFhi othraGtfio.rCsaolml ae nnoexnttefromrinka∈l w{2e,a3k,.c.y.c}l,eswahnodsenoattsatcroknggracpyhclec.onBtayinsshokw(k2−in1g)
(cid:126)a D(cid:126)xi , for all A P, there exists a valuation θ such that
∈ | | ∈ tractability of CERTAINTY(AC(k)), we obtain more sup-
A=θ (F) or A=θ (G). The sequence(cid:126)a is called
[(cid:126)xi (cid:126)a] i [(cid:126)xi (cid:126)a] i porting evidence for Conjecture 1. As a side result, we will
thevecto(cid:55)→rofpartitionP. (cid:55)→
solveacomplexityissueraisedbyFuxmanandMiller[10].
In words, each partition of db groups facts that can be
i
obtained from Fi or Gi by replacing the variables of(cid:126)xi with Definition8 Fork 2,letC(k)andAC(k)denotethefollow-
≥
the same fixed constants. For example, the attack graph in ingBooleanconjunctivequerieswithoutself-join.
Fig. 4 contains an attack cycle involving R (x,y,u ,u ) and
3 3 4
C(k) = R (x ,x ),R (x ,x ),...,R (x ,x ),
R4(x,y,u4,u3). The sequence (cid:104)x,y(cid:105) contains the variables { 1 1 2 2 2 3 k−1 k−1 k
that also occur in other cycles. The facts R3(a,b,c,d) and Rk(xk,x1)},
R4C(al,eba,rley,,ft)wbootfhacbteslothnagttboetlhoengpatrotitdiiosntinwcitthpavretcittioorn(cid:104)sao,bf(cid:105)d.b AC(k) = {R1(x1,x2),R2(x2,x3),...,Rk−1(xk−1,xk),
i R (x ,x ),S (x ,x ,...,x ) ,
k k 1 k 1 2 k
cannot be key-equal. It follows that each repair of db is a }
i
disjointunionofrepairs,oneforeachpartitionofdb. where x ,...,x are distinct variables and R ,...,R ,S dis-
i 1 k 1 k k
Let db be the smallest subset of db that contains tinctrelationnames. Fori 1,...,k ,relationnameR isof
i i i
every p(cid:84)artit(cid:85)ion P satisfying P CERTAINTY(q). By signature[2,1],andS isof∈s{ignature}[k,k]. (cid:67)
i k
∈
8
R1(x1,x2) R2(x2,x3) b′ b′
S (x ,x ,x )
3 1 2 3 a c a c
c b a c b a
′ ′ ′ ′
R3(x3,x1) Figure 7: Graph representation of two repairs (of the uncer-
tain database of Fig. 6) that falsify AC(3). The left cycle is
anticlockwiseandnotencodedinS .
3
Figure 5: Attack graph of AC(3). All cycles are weak and
nonterminal
For example, assume R(a,b),R (c,d) db with i < j.
i j
∈
Since a type(x), b type(x ), and c type(x ), it fol-
i i+1 j
∈ ∈ ∈
lowsthatb=a=candthatb=cimplies j=i+1.
(cid:54) (cid:54)
R 1 2 R 1 2 The R-facts of db can be viewed as edges of a directed
1 2 i
a b b c b′ graph (1 i k). This is illustrated in Fig. 6 for k = 3.
≤ ≤
a b b c Let G=(V,E) be the directed graph such that V =D and
′ ′
a b b c E = (a,b) R(a,b) dbforsomei . Then, G is k-partite
′ ′ i
a c { | ∈ }
with vertex classes type(x ), ..., type(x ). Furthermore,
1 k
R 1 2 S 1 2 3 whenever (a,b) E and a type(x), then b type(x ) if
3 3 i i+1
∈ ∈ ∈
c a a b c i<k(andb type(x )ifi=k). Itfollowsthatthelengthof
′ 1
∈
c a′ a b′ c c′ b a′ everyelementarycycleinGmustbeamultipleofk.Sincedb
c a a b c ispurified,novertexhaszerooutdegree. WedefineC asthe
′ ′
setofcyclesoflengthksuchthatifdbcontainsS (a ,...,a ),
k 1 k
Figure 6: At the left: uncertain database that is purified thenC containsthecyclea1,a2,...,ak,a1.
relative to AC(3). At the right: graph representation of Sincedbispurified, Gisavertex-disjointunionofstrong
Rcl1o∪ckRw2is∪e.R3. Note that the three cycles encoded in S3 are ceodmgeploenaednstfsrSo1m,.a..v,eSr(cid:96)te(xfoirnsSoimtoea(cid:96)≥ver0t)exsuicnhStjh.2atfori(cid:54)= j,no
In what follows, some vertices and edges of G will be
marked. Itisstraightforwardthatdb CERTAINTY(AC(k))
Obviously,aqueryqisacyclicifitcontainsanatomFsuch (cid:54)∈
isequivalenttothefollowing.
that vars(F)=vars(q). Therefore, AC(k) is acyclic because
theS -atomcontainsallvariablesthatoccurinthequery. On Itispossibletomarkexactlyoneoutgoingedgefor
k
theotherhand,C(k)isacyclicifk=2andcyclicifk 3. eachvertexofGwithoutmarkingalledgesofsome (5)
≥
For i 1,...,k , the attack graph of AC(k) contains at- cycleinC.
∈{ }
tacks from the R-atom to every other atom. Figure 5 shows
i We provide a polynomial-time algorithm for testing con-
the attack graph of AC(3). All attack cycles are weak, but
dition (5). Marking one outgoing edge for each vertex will
Theorem 3 does not apply because the cycles are nontermi-
createacycleofmarkededgesineachstrongcomponent.
nal.
ForeachstrongcomponentS,considerthefollowingcases
i
CERTAINTY(C(k)) was claimed coNP-hard for all k 2
≥ successivelyandexecutethefirstonethatapplies.
in [10]. Later, however, Wijsen [22] found a mistake in the
proof of that claim and showed that CERTAINTY(C(k)) is CaseSicontainsacycleoflengthkthatdoesnotbelong
tractableifk=2. ThecomplexityofCERTAINTY(C(k))for to C. Such a cycle is illustrated by Fig. 7 (left). Mark all
k 3willbesettledbyCorollary1. vertices and edges of the cycle. Notice that the number of
≥ cyclesoflengthkisatmost V k,whichispolynomialinthe
| |
Theorem4 Fork 2,CERTAINTY(AC(k))isinP. sizeofdb.
≥
CaseS containsanelementarycycleoflength(strictly)
i
Proof (Extended sketch.) Let db be an uncertain database
greater than k. Such a cycle is illustrated by Fig. 7 (right).
withschema R ,..., R , S . ByLemma1, wecanassume
1 k k Markallverticesandedgesofthecycle. Toseethatthisstep
{ }
withoutlossofgeneralitythatdbispurifiedrelativetoAC(k).
isinpolynomialtime,noticethatthefollowingareequivalent:
LetDbetheactivedomainofdb.Foreveryi 1,...,k ,de-
∈{ }
finetype(xi)asthesubsetofDthatcontainsaifforsomeval- Sicontainsanelementarycycleoflengthgreaterthank.
•
uationµ,µ (AC(k)) db. SinceAC(k)hasnoself-join,
we can ass[uxim(cid:55)→ea]without l⊆oss of generality that i= j implies 2AstrongcomponentofagraphGisamaximalstronglyconnectedsub-
graphofG. Agraphisstronglyconnectedifthereisapathfromanyvertex
(cid:54)
type(xi) type(xj)=0/. toanyother.
∩
9
S containsapatha ,a ,...,a ,a suchthata =a ThefollowingdefinitionextendsthefunctionPrtoBoolean
i 1 2 k k+1 1 k+1
• (cid:54)
and S contains a path from a to a that contains no first-orderqueriesq.
i k+1 1
edgefrom a ,a ,...,a V.
1 2 k
{ }×
Definition10 Let(db,Pr)beaprobabilisticdatabase. Letq
Thelatterconditioncanbetestedinpolynomialtime,because beaBooleanfirst-orderquery. Wedefine
thereareatmost V k+1distinctchoicesfora ,a ,...,a ,a
1 2 k k+1
| |
andpathscanbefoundinpolynomialtime. Pr(q)= ∑ Pr(w).
w worlds(db):w=q
Case neither of the above two cases applies. Conclude ∈ |
that(5)isfalse. In words, Pr(q) sums up the probabilities of the possible
worldsthatsatisfyq. (cid:67)
Ifafterthepreviousstepeverystrongcomponentcontains
acycleofmarkededges,thenitiscorrecttoconcludethat(5)
Of special interest is the application of Definition 10
istrue. NoticethateverycycleofC nowcontainsatleastone
in case q is a single fact, or a Boolean combination of
unmarkededge. Wecanachieve(5)bymarking,foreachyet
facts. Notice that if (db,Pr) is a probabilistic database and
unmarkedvertex,theverticesandedgesonashortestpathto
A ,...,A aredistinctfactsbelongingtoasameblockofdb,
somemarkedvertex. Thiscanbedonewithoutcreatingnew 1 n
cyclesofmarkededges. (cid:3) then Pr(A1∨A2∨···∨An)=∑ni=1Pr(Ai), because no possi-
bleworldcancontaintwodistinctfactsthatbelongtoasame
block.
SincequeryC(k)isacyclicifk 3, attackgraphsarenot
≥
definedforC(k)ifk 3. Nevertheless,thefollowinglemma
≥ Definition11 Probabilisticdatabase(db,Pr)iscalledblock-
immediatelyimpliesthatifCERTAINTY(AC(k))istractable,
independent-disjoint (BID)ifthefollowingholds: whenever
thensoisCERTAINTY(C(k)).
A ,...,A are facts of db taken from n distinct blocks (for
1 n
Lemma9 LetqbeaBooleanconjunctivequerywithoutself- somen≥1),thenPr(A1∧A2∧···∧An)=∏ni=1Pr(Ai). (cid:67)
join. If q q and every atom in q q is all-key, then there
exists an(cid:48)A⊆C0 many-one reduction f\rom(cid:48) CERTAINTY(q) to Theorem 2.4 in [8] implies that every BID probabilistic
(cid:48) database (db,Pr) is uniquely determined if Pr(A) is given
CERTAINTY(q).
for everyfact A db. Thisallows for anefficient encoding:
∈
ratherthanspecifyingPr(w)foreveryw worlds(db),itsuf-
Corollary1 Fork 2,CERTAINTY(C(k))isinP. ∈
≥ fices to specify Pr(A) for every A db. In the complexity
∈
Unsurprisingly,thereexistacyclicqueriesq AC(k) k resultsthatfollow,thisefficientencodingisassumed.
2 whose attack graph contains some nonterm(cid:54)∈in{al cycle|an≥d NoticethatwecanturnanuncertaindatabasedbintoaBID
n}ostrongcycle. ThecomplexityofCERTAINTY(q)forsuch probabilisticdatabasebyassumingthattheprobabilitiesofall
queriesqisopen. repairsareequalandsumupto1. Aconsistentsubsetofdb
thatisnotmaximal,wouldthenhavezeroprobability.
7 UncertaintyandProbability
In this section, we study the relationship between the com- FunctionIsSafe(q)Determinewhetherqissafe[8]
plexities of CERTAINTY(q) and evaluating q on probabilis- Input: qisaBooleanconjunctivequerywithoutself-join.
ticdatabases. Themotivationisthat,oninputofanuncertain Result: Booleanin true,false .
databasedb,theproblemCERTAINTY(q)issolvedifwecan begin { }
determine whether query q evaluates to probability 1 on the R1: if q =1andvars(q)=0/ then
| |
probabilistic database obtained from db by assuming a uni- returntrue;
formprobabilitydistributionoverthesetofrepairsofdb. We R2: ifq=q1 q2withq1=0/=q2,vars(q1) vars(q2)=0/
∪ (cid:54) (cid:54) ∩
show,however,thatthisapproachprovidesnonewinsightsin then
thetractabilityfrontierofCERTAINTY(q). returnIsSafe(q1) IsSafe(q2);
∧
7.1 BackgroundfromProbabilisticDatabases /* a is an arbitrary constant */
R3: if(cid:84)F qkey(F)=0/ then
Inthissection,wereviewanimportantresultfromprobabilis- select∈x (cid:84) k(cid:54) ey(F);
F q
ticdatabasetheory. ∈ ∈
returnIsSafe(q );
[x a]
(cid:55)→
Definition9 Apossibleworldwofuncertaindatabasedbis R4: ifthereexistsF qsuchthatkey(F)=0/=vars(F)then
∈ (cid:54)
selectF qsuchthatkey(F)=0/=vars(F);select
a consistent subset of db. The set of possible worlds of db ∈ (cid:54)
x vars(F);
is denoted worlds(db). Notice that possible worlds, unlike ∈
returnIsSafe(q );
repairs,neednotbemaximalconsistent. [x(cid:55)→a]
ifnoneoftheabovethen
A probabilistic database is a pair (db,Pr) where db is an
returnfalse;
uncertaindatabaseandPr:worlds(db) [0,1]isatotalfunc-
→
tion such that ∑ Pr(w)=1. We will assume that
w worlds(db)
thenumbersinthe∈codomainofPrarerational. (cid:67)
10