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Oblivious Bounds on the Probability of Boolean Functions

August 2014
ACM Transactions on Database Systems 39(1)

DOI:10.1145/2532641

Authors:

Wolfgang Gatterbauer

Northeastern University

Dan Suciu

University of Washington

This article develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an exact characterization of optimal oblivious bounds, that is, when the new probabilities are chosen independently of the probabilities of all other variables. Our motivation comes from the weighted model counting problem (or, equivalently, the problem of computing the probability of a Boolean function), which is &num;P-hard in general. By performing several dissociations, one can transform a Boolean formula whose probability is difficult to compute into one whose probability is easy to compute, and which is guaranteed to provide an upper or lower bound on the probability of the original formula by choosing appropriate probabilities for the dissociated variables. Our new bounds shed light on the connection between previous relaxation-based and model-based approximations and unify them as concrete choices in a larger design space. We also show how our theory allows a standard relational database management system (DBMS) to both upper and lower bound hard probabilistic queries in guaranteed polynomial time.

Example 7.6. Probabilities for the top 10 query answers for varying query parameters $1, $2, and tuple probabilities p. The ranking is determined by the upper dissociation bounds (upper end of the red interval) and is identical to the one determined by the actual probabilities (crosses), except in (c) where tuples 6 and 21 are flipped. MaxLin shows the maximal lineage among the query answers, which is too big in (f) for exact probabilistic inference. (b): x = 0.999 999 999 999 = 1−10 −12. (f): x = 0.999 999 999 9 = 1−10 −10 .

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Oblivious Bounds on the Probability of Boolean Functions

WOLFGANG GATTERBAUER, Carnegie Mellon University

DAN SUCIU, University of Washington

This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple

occurrences of variables as independent and assigning them new individual probabilities. We call this ap-

proach dissociation and give an exact characterization of optimal oblivious bounds, i.e. when the new prob-

abilities are chosen independent of the probabilities of all other variables. Our motivation comes from the

weighted model counting problem (or, equivalently, the problem of computing the probability of a Boolean

function), which is #P-hard in general. By performing several dissociations, one can transform a Boolean

formula whose probability is difﬁcult to compute, into one whose probability is easy to compute, and which

is guaranteed to provide an upper or lower bound on the probability of the original formula by choosing

appropriate probabilities for the dissociated variables. Our new bounds shed light on the connection be-

tween previous relaxation-based and model-based approximations and unify them as concrete choices in

a larger design space. We also show how our theory allows a standard relational database management

system (DBMS) to both upper and lower bound hard probabilistic queries in guaranteed polynomial time.

Categories and Subject Descriptors: G.3 [Probability and Statistics]; H.2.m [Database Management]:

Miscellaneous; I.1.1 [Symbolic and algebraic manipulation]: Expressions and Their Representation

General Terms: Algorithms, Theory

Additional Key Words and Phrases: Probabilistic databases, Weighted model counting, Boolean expressions,

Oblivious approximations, Relaxation

1. INTRODUCTION

Query evaluation on probabilistic databases is based on weighted model counting

for positive Boolean expressions. Since model counting is #P-hard in general, today’s

probabilistic database systems evaluate queries using one of the following three ap-

proaches: (1) incomplete approaches identify tractable cases (e.g., read-once formu-

las) either at the query-level [Dalvi and Suciu 2007; Dalvi et al. 2010] or the data-

level [Olteanu and Huang 2008; Sen et al. 2010]; (2) exact approaches apply exact

probabilistic inference, such as repeated application of Shannon expansion [Olteanu

et al. 2009] or tree-width based decompositions [Jha et al. 2010]; and (3) approxi-

mate approaches either apply general purpose sampling methods [Jampani et al. 2008;

Kennedy and Koch 2010; Re et al. 2007] or approximate the number of models of the

Boolean lineage expression [Olteanu et al. 2010; Fink and Olteanu 2011].

This paper provides a new algebraic framework for approximating the probability

of positive Boolean expressions. While our method was motivated by query evaluation

on probabilistic databases, it is more general and applies to all problems that rely

on weighted model counting, e.g., general probabilistic inference in graphical mod-

Authors’ addresses: W. Gatterbauer (corresponding author), Tepper School of Business, Carnegie Mellon

University; email: gatt@cmu.edu; D. Suciu, Computer Science and Engineering Department, University of

Washington, Seattle.

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DOI 10.1145/0000000.0000000 http://doi.acm.org/10.1145/0000000.0000000

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

arXiv:1409.6052v1 [cs.AI] 21 Sep 2014

1:2 W. Gatterbauer and D. Suciu

Conjunctive dissociation Disjunctive dissociation

ϕ=ϕ1∧ϕ2ϕ=ϕ1∨ϕ2

ϕ0=ϕ1[x0

1/x]∧ϕ2[x0

2/x]ϕ0=ϕ1[x0

1/x]∨ϕ2[x0

2/x]

0p1

Assignment for p

oblivious

lower

bounds

oblivious

upper

bounds

0p1

Assignment for p

oblivious

lower

bounds

oblivious

upper

bounds

Optimal obli- Upper p0

1·p0

2=p p0

1=p p0

2=p

vious bounds Lower p0

1=p p0

2=p(1−p0

1)·(1−p0

2)=1−p

Model-based Upper p0

1=p p0

2=1 (optimal) p0

1=p p0

2=1 (non-optimal)

bounds Lower p0

1=p p0

2=0 (non-optimal) p0

1=p p0

2=0 (optimal)

Relaxation & Comp. p0

1=p p0

2=P[x|ϕ1]p0

1=p p0

2=P[x|¬ϕ1]

Fig. 1. Dissociation as framework that allows to determine optimal oblivious upper and lower bounds for

the probabilities p0=hp0

1, p0

2iof dissociated variables. Oblivious here means that we assign new values

after looking at only a limited scope of the expression. Model-based upper conjunctive and lower disjunctive

bounds are obliviously optimal (they fall on the red line of optimal assignments), whereas lower conjunctive

and upper disjunctive are not. Relaxation & Compensation is a form of dissociation which is not oblivious

(p2is calculated with knowledge of ϕ1) and does not, in general, guarantee to give an upper or lower bound.

els [Chavira and Darwiche 2008].1An important aspect of our method is that it is not

model-based in the traditional sense. Instead, it enlarges the original variable space

by treating multiple occurrences of variables as independent and assigning them new

individual probabilities. We call this approach dissociation2and explain where exist-

ing relaxation-based and model-based approximations ﬁt into this larger space of ap-

proximations. We characterize probability assignments that lead to guaranteed upper

or lower bounds on the original expression and identify the best possible oblivious

bounds, i.e. after looking at only a limited scope of the expression. We prove that for

every model-based bound there is always a dissociation-based bound that is as good or

better. And we illustrate how a standard relational DBMS can both upper and lower

bound hard probabilistic conjunctive queries without self-joins with appropriate SQL

queries that use dissociation in a query-centric way.

We brieﬂy discuss our results: We want to compute the probability P[ϕ]of a Boolean

expression ϕwhen each of its Boolean variables xiis set independently to true with

some given probability pi=P[xi]. Computing P[ϕ]is known to be #P-hard in gen-

eral [Valiant 1979] and remains hard to even approximate [Roth 1996]. Our approach

is to approximate P[ϕ]with P[ϕ0]that is easier to compute. The new formula ϕ0is

derived from ϕthrough a sequence of dissociation steps, where each step replaces d

distinct occurrences of some variable xin ϕwith dfresh variables x0

1, x0

2,...x0

d. Thus,

1Note that weighted model counting is essentially the same problem as computing the probability P[ϕ]of

a Boolean expression ϕ. Each truth assignment of the Boolean variables corresponds to one model whose

weight is the probability of this truth assignment. Weighted model counting then asks for the sum of the

weights of all satisfying assignments.

2Dissociation is the breaking of an existing association between related, but not necessarily identical items.

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Oblivious Bounds on the Probability of Boolean Functions 1:3

after applying dissociation repeatedly, we transform ϕinto another expression ϕ0and

approximate P[ϕ]with P[ϕ0]. The question that we address in this paper is: how should

we set the probabilities of the dissociated variables x0

iin order to ensure that P[ϕ0]

is a good approximation of P[ϕ]? In particular, we seek conditions under which ϕ0is

guaranteed to be either an upper bound P[ϕ0]≥P[ϕ]or a lower bound P[ϕ0]≤P[ϕ].

Our main result can be summarized as follows: Suppose that xoccurs positively

in ϕ. Dissociate xinto two variables x0

1and x0

2such that the dissociated formula is

ϕ0=ϕ0

1∧ϕ0

2, where x0

1occurs only in ϕ0

1and x0

2occurs only in ϕ0

2; in other words,

ϕ≡ϕ0

1[x0

1/x]∧ϕ0

2[x0

2/x](we will later deﬁne this as “conjunctive dissociation”). Let

p=P[x],p0

1=P[x0

1],p0

2=P[x0

2]be their probabilities. Then (1) P[ϕ0]≥P[ϕ]iff p0

1·p0

2≥p,

and (2) P[ϕ0]≤P[ϕ]iff p0

1≤pand p0

2≤p. In particular, the best upper bounds are

obtained by choosing any p0

1, p0

2that satisfy p0

1·p0

2=p, and the best lower bound is

obtained by setting p0

1=p0

2=p. The “only if” direction holds assuming ϕ0satisﬁes

certain mild conditions (e.g., it should not be redundant), and under the assumption

that p0

1, p0

2are chosen obliviously, i.e. they are functions only of p=P[x]and indepen-

dent of the probabilities of all other variables. This restriction to oblivious probabilities

guarantees the computation of the probabilities p0

1, p0

2to be very simple.3Our result ex-

tends immediately to the case when the variable xis dissociated into several variables

1, x0

2, . . . , x0

d, and also extends (with appropriate changes) to the case when the expres-

sions containing the dissociated variables are separated by ∨rather than ∧(Fig.1).

Example 1.1 (2CNF Dissociation). For a simple illustration of our main result, con-

sider a Positive-Partite-2CNF expression with |E|clauses

ϕ=^

(i,j)∈E

(xi∨yj)(1)

for which calculating its probability is already #P-hard [Provan and Ball 1983]. If we

dissociate all occurrences of all mvariables xi, then the expression becomes:

ϕ0=^

(i,j)∈E

(x0

i,j ∨yj)(2)

which is equivalent to Vjyj∨Vi,j x0

i,j . This is a read-once expression whose prob-

ability can always be computed in PTIME [Gurvich 1977]. Our main result im-

plies the following: Let pi=P[xi],i∈[m]be the probabilities of the original vari-

ables and denote p0

i,j =P[x0

i,j ]the probabilities of the fresh variables. Then (1) if

∀i∈[m] : p0

i,j1·p0

i,j2·· ·p0

i,jdi=pi, then ϕ0is an upper bound (P[ϕ0]≥P[ϕ]); (2) if

∀i∈[m] : p0

i,j1=p0

i,j2=. . . =p0

i,jdi=pi, then ϕ0is a lower bound (P[ϕ0]≤P[ϕ]). Fur-

thermore, these are the best possible oblivious bounds, i.e. where p0

i,j depends only on

pi=P[xi]and is chosen independently of other variables in ϕ.

We now explain how dissociation generalizes two other approximation methods in

the literature (Fig.1 gives a high-level summary and Sect.5 the formal details).

Relaxation & Compensation. This is a framework by Choi and Darwiche [2009;

2010] for approximate probabilistic inference in graphical models. The approach per-

forms exact inference in an approximate model that is obtained by relaxing equiv-

3Our usage of the term oblivious is inspired by the notion of oblivious routing algorithms [Valiant 1982]

which use only local information and which can therefore be implemented very efﬁciently. Similarly, our

oblivious framework forces p0

1, p0

2to be computed only as a function of p, without access to the rest of ϕ. One

can always ﬁnd values p0

1, p0

2for which P[ϕ] = P[ϕ0]. However, to ﬁnd those value in general, one has to ﬁrst

compute q=P[ϕ], then ﬁnd appropriate values p0

1, p0

2for which the equality P[ϕ0] = qholds. This is not

practical, since our goal is to compute qin the ﬁrst place.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:4 W. Gatterbauer and D. Suciu

alence constraints in the original model, i.e. by removing edges. The framework al-

lows one to improve the resulting approximations by compensating for the relaxed

constraints. In the particular case of a conjunctive Boolean formula ϕ=ϕ1∧ϕ2,re-

laxation refers to substituting any variable xthat occurs in both ϕ1and ϕ2with two

fresh variables x0

1in ϕ1and x0

2in ϕ2.Compensation refers to then setting their prob-

abilities p0

1=P[x0

1]and p0

2=P[x0

2]to p0

1=pand p0

2=P[x|ϕ1]. This new probability

assignment is justiﬁed by the fact that, if xis the only variable shared by ϕ1and ϕ2,

then compensation ensures that P[ϕ0] = P[ϕ](we will show this claim in Prop. 5.1). In

general, however, ϕ1, ϕ2have more than one variable in common, and in this case we

have P[ϕ0]6=P[ϕ]for the same compensation. Thus in general, compensation is applied

as a heuristics. Furthermore, it is then not known whether compensation provides an

upper or lower bound.

Indeed, let p0

1=p,p0

2=P[x|ϕ1]be the probabilities set by the compensation method.

Recall that our condition for P[ϕ0]to be an upper bound is p0

1·p0

2≥p, but we have

1·p0

2=p·P[x|ϕ1]≤p. Thus, the compensation method does not satisfy our oblivious

upper bound condition. Similarly, because of p0

1=pand p0

2≥p, these values fail to

satisfy our oblivious lower bound condition. Thus, relaxation is neither a guaranteed

upper bound, nor a guaranteed lower bound. In fact, relaxation is not oblivious at all

(since p0

2is computed from the probabilities of all variables, not just P[x]). This enables

it to be an exact approximation in the special case of a single shared variable, but fails

to guarantee any bounds, in general.

Model-based approximations. Another technique for approximation described

by Fink and Olteanu [2011] is to replace ϕwith another expression whose set of mod-

els is either a subset or superset of those of ϕ. Equivalently, the upper bound is a

formula ϕUsuch that ϕ⇒ϕU, and the lower bound is ϕLsuch that ϕL⇒ϕ. We show

in this paper, that if ϕis a positive Boolean formula, then all upper and lower model-

based bounds can be obtained by repeated dissociation: the model-based upper bound

is obtained by repeatedly setting probabilities of dissociated variables to 1, and the

model-based lower bound by setting the probabilities to 0. While the thus generated

model-based upper bounds for conjunctive expressions correspond to optimal oblivious

dissociation bounds, the model-based lower bounds for conjunctive expressions are not

optimal and can always be improved by dissociation.

Indeed, consider ﬁrst the upper bound for conjunctions: the implication ϕ⇒ϕU

holds iff there exists a formula ϕ1such that ϕ≡ϕ1∧ϕU.4Pick a variable x, denote p=

P[x]its probability, dissociate it into x0

1in ϕ1and x0

2in ϕU, and set their probabilities

as p0

1= 1 and p0

2=p. Thus, ϕUremains unchanged (except for the renaming of xto

2), while in ϕ1we have set x1= 1. By repeating this process, we eventually transform

ϕ1into true (Recall that our formula is monotone). Thus, model-based upper bounds

are obtained by repeated dissociation and setting p0

1= 1 and p0

2=pat each step. Our

results show that this is only one of many oblivious upper bounds as any choices with

1p0

2≥plead to an oblivious upper bound for conjunctive dissociations.

Consider now the lower bound: the implication ϕL⇒ϕis equivalent to ϕL≡ϕ∧

ϕ2. Then there is a sequence of ﬁnitely many conjunctive dissociation steps, which

transforms ϕinto ϕ∧ϕ2and thus into ϕL. At each step, a variable xis dissociated

into x0

1and x0

2, and their probabilities are set to p0

1=pand p0

2= 0, respectively.5

According to our result, this choice is not optimal: instead one obtains a tighter bound

by also setting p0

2=p, which no longer corresponds to a model-based lower bound.

4Fink and Olteanu [2011] describe their approach for approximating DNF expressions only. However, the

idea of model-based bounds applies equally well to arbitrary Boolean expressions, including those in CNF.

5The details here are more involved and are given in detail in Sect.5.2.

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Oblivious Bounds on the Probability of Boolean Functions 1:5

Thus, model-based lower bounds for conjunctive expressions are not optimal and can

always be improved by using dissociation.

Our dual result states the following for the case when the two formulas are con-

nected with disjunction ∨instead of conjunction ∧: (1) the dissociation is an upper

bound iff p0

1≥pand p0

2≥p, and (2) it is a lower bound iff (1 −p0

1)(1 −p0

2)≥1−p. We

see that model-based approximation gives an optimal lower bound for disjunctions,

because (1 −p0

1)(1 −p0

2)=1·(1 −p)=1−p, however non-optimal upper bounds. Exam-

ple 7.2 illustrates this asymmetry and the possible improvement through dissociation

with a detailed simulation-based example.

Bounds for hard probabilistic queries. Query evaluation on probabilistic

databases reduces to the problem of computing the probability of its lineage expres-

sion which is a a monotone, k-partite Boolean DNF where kis ﬁxed by the number of

joins in the query. Computing the probability of the lineage is known to be #P-hard for

some queries [Dalvi and Suciu 2007], hence we are interested in approximating these

probabilities by computing dissociated Boolean expressions for the lineage. We have

previously shown in [Gatterbauer et al. 2010] that every query plan for a query corre-

sponds to one possible dissociation for its lineage expression. The results in this paper

show how to best set the probabilities for the dissociated expressions in order to obtain

both upper bounds and lower bounds. We further show that all the computation can be

pushed inside a standard relational database engine with the help of SQL queries that

use User-Deﬁned-Aggregates and views that replace the probabilities of input tuples

with their optimal symmetric lower bounds. We illustrate this approach in Sect. 6 and

validate it on TPC-H data in Sect.7.5.

Main contributions. (1) We introduce an algebraic framework for approximating

the probability of Boolean functions by treating multiple occurrences of variables as

independent and assigning them new individual probabilities. We call this approach

dissociation; (2) we determine the optimal upper and lower bounds for conjunctive and

disjunctive dissociations under the assumption of oblivious value assignments; (3) we

show how existing relaxation-based and model-based approximations ﬁt into the larger

design space of dissociations, and show that for every model-based bound there is at

least one dissociation-based bound which is as good or tighter; (4) we apply our gen-

eral framework to both upper and lower bound hard probabilistic conjunctive queries

without self-joins in guaranteed PTIME by translating the query into a sequence of

standard SQL queries; and (5) we illustrate and evaluate with several detailed ex-

amples the application of this technique. Note that this paper does not address the

algorithmic complexities in determining alternative dissociations, in general.

Outline. Section 2 starts with some notational background, and Sect. 3 formally

deﬁnes dissociation. Section 4 contains our main results on optimal oblivious bounds.

Section 5 formalizes the connection between relaxation, model-based bounds and disso-

ciation, and shows how both previous approaches can be uniﬁed under the framework

of dissociation. Section 6 applies our framework to derive upper and lower bounds for

hard probabilistic queries with standard relational database management systems.

Section 7 gives detailed illustrations on the application of dissociation and oblivious

bounds. Finally, Sect.8 relates to previous work before Sect.9 concludes.

2. GENERAL NOTATIONS AND CONVENTIONS

We use [m]as short notation for {1, . . . , m}, use the bar sign for the complement of an

event or probability (e.g., ¯x=¬x, and ¯p= 1 −p), and use a bold notation for sets (e.g.,

s⊆[m]) or vectors (e.g., x=hx1, . . . , xmi) alike. We assume a set xof independent

Boolean random variables, and assign to each variable xia primitive event which is

true with probability pi=P[xi]. We do not formally distinguish between the variable

xiand the event xithat it is true. By default, all primitive events are assumed to be

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:6 W. Gatterbauer and D. Suciu

k1=2

k2=2

(a) f

1,1

1,2

d1=2

d2=1

(b) f0

Fig. 2. Example 3.2. (a): Bipartite primal graph for CNF representing f. (b): A dissociation f0where vari-

able x1appearing k1= 2 times in fis dissociated into (replaced by) d1= 2 fresh variables in f0.

independent (e.g., P[x1∧x2] = p1p2). We are interested in bounding the probability

P[f]of a Boolean function f, i.e. the probability that the function is true if each of the

variables is independently true or false with given probabilities. When no confusion

arises, we blur the distinction between a Boolean expression ϕand the Boolean func-

tion fϕit represents (cf. [Crama and Hammer 2011, Sect. 1.2]) and write P[ϕ]instead

of P[fϕ]. We also use the words formula and expression interchangeably. We write f(x)

to indicate that xis the set of primitive events appearing in the function f, and f[x1/x]

to indicate that x1is substituted for xin f. We often omit the operator ∧and denote

conjunction by mere juxtaposition instead.

3. DISSOCIATION OF BOOLEAN FUNCTIONS AND EXPRESSIONS

We deﬁne here dissociation formally. Let f(x,y)and f0(x0,y)be two Boolean functions,

where x,x0,yare three disjoint sets of variables. Denote |x|=m,|x0|=m0, and |y|=n.

We restrict fand f0to be positive in xand x0, respectively [Crama and Hammer 2011,

Def. 1.24].

Deﬁnition 3.1 (Dissociation). We call a function f0a dissociation of fif there exists

a substitution θ:x0→xs.t. f0[θ] = f.

Example 3.2 (CNF Dissociation). Consider two functions fand f0given by CNF

expressions

f= (x1∨y1)(x1∨y2)(x2∨y1)(x2∨y3)

f0= (x0

1,1∨y1)(x0

1,2∨y2)(x0

2∨y1)(x0

2∨y3)

Then f0is a dissociation of fas f0[θ] = ffor the substitution θ=

{(x0

1,1, x1),(x0

1,2, x1),(x0

2, x2)}. Figure 2 shows the CNF expressions’ primal graphs.6

In practice, to ﬁnd a dissociation for a function f(x,y), one proceeds like this: Choose

any expression ϕ(x,y)for fand thus f=fϕ. Replace the kidistinct occurrences of

variables xiin ϕwith difresh variables x0

i,1, x0

i,2, . . . , x0

i,di, with di≤ki. The resulting

expression ϕ0represents a function f0that is a dissociation of f. Notice that we may

obtain different dissociations by deciding for which occurrences of xito use distinct

fresh variables, and for which occurrences to use the same variable. We may further

obtain more dissociations by starting with different, equivalent expressions ϕfor the

function f. In fact, we may construct inﬁnitely many dissociations this way. We also

note that every dissociation of fcan be obtained through the process outlined here.

Indeed, let f0(x0,y)be a dissociation of f(x,y)according to Deﬁnition 3.1, and let θbe

the substitution for which f0[θ] = f. Then, if ϕ0is any expression representing f0, the

expression ϕ=ϕ0[θ]represents f. We can thus apply the described dissociation process

to a certain expression ϕand obtain f0.

6The primal graph of a CNF (DNF) has one node for each variable and one edge for each pair of variables

that co-occur in some clause (conjunct). This concept originates in constraint satisfaction and it is also

varyingly called co-occurrence graph or variable interaction graph [Crama and Hammer 2011].

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:7

Example 3.3 (Alternative Dissociations). Consider the two expressions:

ϕ= (x∨y1)(x∨y2)(x∨y3)(y4∨y5)

ψ=xy4∨xy5∨y1y2y3y4∨y1y2y3y5

Both are equivalent (ϕ≡ψ) and thus represent the same Boolean function (fϕ=fψ).

Yet each leads to a quite different dissociation in the variable x:

ϕ0= (x0

1∨y1)(x0

2∨y2)(x0

3∨y3)(y4∨y5)

ψ0=x0

1y4∨x0

2y5∨y1y2y3y4∨y1y2y3y5

Here, ϕ0and ψ0represent different functions (fϕ06=fψ0) and are both dissociations of

ffor the substitutions θ1={(x0

1, x),(x0

2, x),(x0

3, x)}and θ2={(x0

1, x),(x0

2, x)}, respec-

tively. 

Example 3.4 (More alternative Dissociations). Consider the AND-function

f(x, y) = xy. It can be represented by the expressions xxy, or xxxy, etc., leading

to the dissociations x0

1x0

2y, or x0

1x0

2x0

3y, etc. For even more dissociations, represent

fusing the expression (x∨x)y∨xy, which can dissociate to (x0

1∨x0

2)y∨x0

3y, or

(x0

1∨x0

2)y∨x0

1y, etc. Note that several occurrences of a variable can be replaced by the

same new variables in the dissociated expression. 

4. OBLIVIOUS BOUNDS FOR DISSOCIATED EVENT EXPRESSIONS

Throughout this section, we ﬁx two Boolean functions f(x,y)and f0(x0,y)such that f0

is a dissociation of f. We are given the probabilities p=P[x]and q=P[y]. Our goal

is to ﬁnd probabilities p0=P[x0]of the dissociated variables so that P[f0]is an upper

or lower bound for P[f]. We ﬁrst deﬁne oblivious bounds (Sect.4.1), then characterize

them, in general, through valuations (Sect. 4.2) and, in particular, for conjunctive and

disjunctive dissociations (Sect. 4.3), then derive optimal bounds (Sect. 4.4), and end

with illustrated examples for CNF and DNF dissociations (Sect.4.5).

4.1. Deﬁnition of Oblivious Bounds

We use the subscript notation Pp,q[f]and Pp0,q[f0]to emphasize that the proba-

bility space is deﬁned by the probabilities p=hp1, p2, . . .i,q=hq1, q2, . . .i, and

p0=hp0

1, p0

2, . . .i, respectively. Given pand q, our goal is thus to ﬁnd p0such that

Pp0,q[f0]≥Pp,q[f]or Pp0,q[f0]≤Pp,q[f].

Deﬁnition 4.1 (Oblivious Bounds). Let f0be a dissociation of fand p=P[x]. We

call p0an oblivious upper bound for pand dissociation f0of fiff ∀q:Pp0,q[f0]≥Pp,q[f].

Similarly, p0is an oblivious lower bound iff ∀q:Pp0,q[f0]≤Pp,q[f].

In other words, p0is an oblivious upper bound if the probability of the dissociated

function f0is bigger than that of ffor every choice of q. Put differently, the probabilities

of x0depend only on the probabilities of xand not on those of y.

An immediate upper bound is given by p0=1, since fis monotone and f0[1/x0] =

f[1/x]. Similarly, p=0is a na¨

ıve lower bound. This proves that the set of upper and

lower bounds is never empty. Our next goal is to characterize all oblivious bounds and

to then ﬁnd optimal ones.

4.2. Characterization of Oblivious Bounds through Valuations

We will give a necessary and sufﬁcient characterization of oblivious bounds, but ﬁrst

we need to introduce some notations. If f(x,y)is a Boolean function, let ν:y→ {0,1}

be a truth assignment or valuation for y. We use νfor the vector hν(y1), . . . , ν(yn)i,

and denote with f[ν]the Boolean function obtained after applying the substitution ν.

Note that f[ν]depends on variables xonly. Furthermore, let gbe nBoolean functions,

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1:8 W. Gatterbauer and D. Suciu

(a) g=hg1, g2i

⌫=h0,1i

⌫=h1,0i

⌫=h1,1i

⌫=h0,0i

(b) gνfor different valuations ν

Fig. 3. Example 4.2. Illustration of the valuation notation with Karnaugh maps. (a): Boolean functions

g1=z1z2and g2=z1z3. (b): Boolean functions gνfor all 4 possible valuations ν. For example, gν=z1¯z2z3

for ν=h0,1i.

over variables z. We denote with gνthe Boolean function gν=Vjgν

j, where gν

j= ¯gjif

ν(yj)=0and gν

j=gjif ν(yj) = 1.

Example 4.2 (Valuation Notation). Assume g=hg1, g2iwith g1=z1z2and g2=

z1z3, and ν=h0,1i. Then gν=¬(z1z2)∧z1z3=z1¯z2z3. Figure 3 illustrates our notation

for this simple example with the help of Karnaugh maps. We encourage the reader to

take a moment and carefully study the correspondences between g,ν, and gν.

Then, any function f(x,y)admits the following expansion by the y-variables:

f(x,y) = _

f[ν]∧yν(3)

Note that any two expressions in the expansion above are logically contradictory, a

property called determinism by Darwiche and Marquis [2002], and that it can be seen

as the result of applying Shannon’s expansion to all variables of y.

Example 4.3 (Valuation Notation continued). Consider the function f= (x∨y1)(x∨

y2). For the example valuation ν=h0,1i, we have f[ν] = (x∨0)(x∨1) = xand yν= ¯y1y2.

Equation 3 gives us an alternative way to write fas disjunction over all 22valuations

of yas f=x(¯y1¯y2)∨x(y1¯y2)∨x(¯y1y2)∨y1y2.

The following proposition is a necessary and sufﬁcient condition for oblivious upper

and lower bounds, based on valuations.

PROPOSITION 4.4 (OBLIVIOUS BOUNDS AND VALUATIONS).Fix two Boolean

functions f(x,y),f0(x0,y)s.t. f0is a dissociation of f, and let pand p0denote the

probabilities of the variables xand x0, respectively. Then p0is an oblivious upper bound

iff Pp0[f0[ν]] ≥Pp[f[ν]] for every valuation νfor y. The proposition holds similarly for

oblivious lower bounds.

PROOF. Remember that any two events in Eq. 3 are disjoint. The total probability

theorem thus allows us to sum over the probabilities of all conjuncts:

Pp,q[f(x,y)] = X

Pp[f[ν]] ·Pq[yν]

Pp0,q[f0(x,y)] = X

Pp0[f0[ν]] ·Pq[yν]

The “if” direction follows immediately. For the “only if” direction, assume that p0is an

oblivious upper bound. By deﬁnition, Pp0,q[f0]≥Pp,q[f]for every q. Fix any valuation

ν:y→ {0,1}, and deﬁne the following probabilities q:qi= 0 when ν(yi) = 0, and

qi= 1 when ν(yi)=1. It is easy to see that Pp,q[f] = Pp[f[ν]], and similarly, Pp0,q[f0] =

Pp0[f0[ν]], which proves Pp0[f0[ν]] ≥Pp[f[ν]].

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A consequence of choosing p0obliviously is that it remains a bound even if we allow

the variables yto be arbitrarily correlated. More precisely:

COROLLARY 4.5 (OBLIVIOUS BOUNDS AND CORRELATIONS).Let f0(x0,y)be a

dissociation of f(x,y), let p0be an oblivious upper bound for p, and let g=

hg1, . . . , g|y|ibe Boolean functions in some variables zwith probabilities r=P[z]. Then:

Pp0,r[f0x0,g(z)]≥Pp,r[fx,g(z)]. The result for oblivious lower bounds is similar.

The intuition is that, by substituting the variables ywith functions gin f(x,y), we

make ycorrelated. The corollary thus says that an oblivious upper bound remains an

upper bound even if the variables yare correlated. This follows from folklore that any

correlation between the variables ycan be captured by general Boolean functions g.

For completeness, we include the proof in Appendix B.

PROOF OF COROLLARY 4.5. We derive the probabilities of fand f0from Eq.3:

Pp,r[f(x,g)] = X

Pp[f[ν]] ·Pr[gν]

Pp0,r[f0(x,g)] = X

Pp0[f0[ν]] ·Pr[gν]

The proof follows now immediately from Prop. 4.4.

4.3. Oblivious Bounds for Unary Conjunctive and Disjunctive Dissociations

A dissociation f0(x0,y)of f(x,y)is called unary if |x|= 1, in which case we write the

function as f(x, y). We next focus on unary dissociations, and establish a necessary

and sufﬁcient condition for probabilities to be oblivious upper or lower bounds for the

important classes of conjunctive and disjunctive dissociations. The criterion also ex-

tends as a sufﬁcient condition to non-unary dissociations, since these can be obtained

as a sequence of unary dissociations.7

Deﬁnition 4.6 (Conjunctive and Disjunctive Dissociation). Let f0(x0,y)be a

Boolean function in variables x0,y. We say that the variables x0are conjunctive in f0

if f0(x0,y) = Vj∈[d]fj(x0

j,y),d=|x0|. We say that a dissociation f0(x0,y)of f(x, y)is

conjunctive if x0are conjunctive in f0. Similarly, we say that x0are disjunctive in f0if

f0(x0,y) = Wj∈[d]fj(x0

j,y), and a dissociation is disjunctive if x0is disjunctive in f0.

Thus in a conjunctive dissociation, each dissociated variable x0

joccurs in exactly one

Boolean function fjand these functions are combined by ∧to obtain f0. In practice,

we start with fwritten as a conjunction, then replace xwith a fresh variable in each

conjunct:

f(x, y) = ^fj(x, y)

f0(x0,y) = ^fj(x0

j,y)

Disjunctive dissociations are similar.

Note that if x0is conjunctive in f0(x0,y), then for any substitution ν:y→ {0,1},f0[ν]

is either 0, 1, or a conjunction of variables in x0:f0[ν] = Vj∈sx0

j, for some set s⊆[d],

where d=|x0|. Similarly, if x0is disjunctive, then f0[ν]is 0, 1, or Wj∈sx0

j.8

7Necessity does not always extend to non-unary dissociations. The reason is that an oblivious dissociation

for xmay set the probability of a fresh variable by examining all variables x, while a in sequence of oblivious

dissociations each new probability P[x0

i,j ]may depend only on the variable xicurrently being dissociated.

8Note that for s=∅:f0[ν] = Vj∈sx0

j= 1 and f0[ν] = Wj∈sx0

j= 0.

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We need one more deﬁnition before we state the main result in our paper.

Deﬁnition 4.7 (Cover). Let x0be conjunctive in f0(x0,y). We say that f0covers the

set s⊆[d]if there exists a substitution νs.t. f0[ν] = Vj∈sx0

j. Similarly, if x0is disjunc-

tive, then we say that f0covers sif there exists νs.t. f0[ν] = Wj∈sx0

THEOREM 4.8 (OBLIVIOUS BOUNDS).Let f0(x0,y)be a conjunctive dissociation of

f(x, y), and let p=P[x],p0=P[x0]be probabilities of xand x0, respectively. Then:

(1)If p0

j≤pfor all j, then p0is an oblivious lower bound for p, i.e. ∀q:Pp0,q[f0]≤

Pp,q[f]. Conversely, if p0is an oblivious lower bound for pand f0covers all sin-

gleton sets {j}with j∈[d], then p0

j≤pfor all j.

(2)If Qjp0

j≥p, then p0is an oblivious upper bound for p, i.e. ∀q:Pp0,q[f0]≥Pp,q[f].

Conversely, if p0is an oblivious upper bound for pand f0covers the set [d], then

Qjp0

j≥p.

Similarly, the dual result holds for disjunctive dissociations f0(x0,y)of f(x, y):

(3)If p0

j≥pfor all j, then p0is an oblivious upper bound for p. Conversely, if p0is

an oblivious upper bound for pand f0covers all singleton sets {j},j∈[d], then

j≥p.

(4)If Qj(1 −p0

j)≥1−p, then p0is an oblivious lower bound for p. Conversely, if p0is

an oblivious lower bound for pand f0covers the set [d], then Qj(1 −p0

j)≥1−p.

PROOF. We make repeated use of Prop. 4.4. We give here the proof for conjunctive

dissociations only; the proof for disjunctive dissociations is dual and similar.

(1) We need to check Pp0[f0[ν]] ≤Pp[f[ν]] for every ν. Since the dissociation is unary,

f[ν]can be only 0,1, or x, while f0[ν]is 0,1, or Vj∈sx0

jfor some set s⊆[d].

Case 1: f[ν]=0. We will show that f0[ν]=0, which implies Pp0[f0[ν]] =

Pp[f[ν]] = 0. Recall that, by deﬁnition, f0(x0,y)becomes f(x, y)if we

substitute xfor all variables x0

j. Therefore, f0[ν][x/x0

1, . . . , x/x0

d]=0,

which implies f0[ν] = 0 because f0is monotone in the variables x0.

Case 2: f[ν]=1. Then Pp0[f0[ν]] ≤Pp[f[ν]] holds trivially.

Case 3: f[ν] = x. Then Pp[f[ν]] = p, while Pp0[f0[ν]] = Qj∈sp0

j. We prove that

s6=∅: this implies our claim, because Qj∈sp0

j≤p0

j≤p, for any choice

of j∈s. Suppose otherwise, that s=∅, hence f0[ν] = 1. Substituting

all variables x0with xtransforms f0to f. This implies f[ν]=1, which

contradicts f[ν] = x.

For the converse, assume that p0is an oblivious lower bound. Since f0covers {j},

there exists a substitution νs.t. f0[ν] = x0

j, and therefore f[ν] = x. By Prop. 4.4

we have p0

j=Pp0[f0[ν]] ≤Pp[f[ν]] = p, proving the claim.

(2) Here we need to check Pp0[f0[ν]] ≥Pp[f[ν]] for every ν. The cases when f[ν]is

either 0 or 1 are similar to the cases above, so we only consider the case when

f[ν] = x. Then f0[ν] = Vj∈sx0

jand Pp0[f0[ν]] = Qj∈sp0

j≥Qjpj≥p=Pp[f[ν]].

For the converse, assume p0is an oblivious upper bound, and let νbe the

substitution for which f0[ν] = Vjx0

j(which exists since f0is covers [d]). Then

Pp0[f0[ν]] ≥Pp[f[ν]] implies p≤Qjpj.

4.4. Optimal Oblivious Bounds for Unary Conjunctive and Disjunctive Dissociations

We are naturally interested in the “best possible” oblivious bounds. Call a dissociation

f0non-degenerate if it covers all singleton sets {j},j∈[d]and the complete set [d].

Theorem 4.8 then implies:

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COROLLARY 4.9 (OPTIMAL OBLIVIOUS BOUNDS).If f0is a conjunctive dissocia-

tion of fand f0is non-degenerate, then the optimal oblivious lower bound is p0

1=p0

. . . =p, while the optimal oblivious upper bounds are obtained whenever p0

1p0

2·· · =p.

Similarly, if f0is a disjunctive dissociation of fand f0is non-degenerate, then the op-

timal oblivious upper bound is p0

1=p0

2=. . . =p, while the optimal oblivious lower

bounds are obtained whenever (1 −p0

1)·(1 −p0

2)·· · = 1 −p.

Notice that while optimal lower bounds for conjunctive dissociations and optimal

upper bounds for disjunctive dissociations are uniquely deﬁned with p0

j=p, there are

inﬁnitely many optimal bounds for the other directions (see Fig.1). Let us call bounds

symmetric if all dissociated variable have the same probability. Then optimal symmet-

ric upper bounds for conjunctive dissociations are p0

j=d

√p, and optimal symmetric

lower bounds for disjunctive dissociations p0

j= 1 −d

√1−p.

We give two examples of degenerate dissociations. First, the dissociation f0=

(x0

1y1∨y3)∧(x0

2y2∨y3)does not cover either {1}nor {2}: no matter how we substi-

tute y1, y2, y3, we can never transform f0to x0

1. For example, f0[1/y1,0/y2,0/y3]=0and

f0[1/y1,0/y2,1/y3] = 1. But f0does cover the set {1,2}because f0[1/y1,1/y2,0/y3] =

x1x2. Second, the dissociation f0= (x0

1y1∨y2)∧(x0

2y2∨y1)covers both {1}and {2}, but

does not cover the entire set {1,2}. In these cases the oblivious upper or lower bounds

in Theorem 4.8 still hold, but are not necessarily optimal.

However, most cases of practical interest result in dissociations that are non-

degenerate, in which case the bounds in Theorem 4.8 are tight. We explain this here.

Consider the original function, pre-dissociation, written in a conjunctive form:

f(x, y) =g0∧^

j∈[d]

(x∨gj) = g0∧^

j∈[d]

fj(4)

where each gjis a Boolean function in the variables y, and where we denoted fj=x∨gj.

For example, if fis a CNF expression, then each fjis a clause containing x, and g0is

the conjunction of all clauses that do not contain x. Alternatively, we may start with a

CNF expression, and group the clauses containing xin equivalence classes, such that

each fjrepresents one equivalence class. For example, starting with four clauses, we

group into two functions f= [(x∨y1)(x∨y2)]∧[(x∨y3)(x∨y4)] = (x∨y1y2)∧(x∨y3y4) =

f1∧f2. Our only assumption about Eq.4 is that it is non-redundant, meaning that none

of the expressions g0or fjmay be dropped. Then we prove:

PROPOSITION 4.10 (NON-DEGENERATE DISSOCIATION).Suppose the function f

in Eq. 4 is non-redundant. Deﬁne f0=g0∧Vj(x0

j∨gj). Then f0covers every single-

ton set {j}. Moreover, if the implication g0⇒Wjgjdoes not hold, then f0also covers the

set [d]. Hence f0is non-degenerate. A similar result holds for disjunctive dissociations

if the dual implication g0⇐Vjgjdoes not hold.

PROOF. We give here the proof for conjunctive dissociations only; the proof for dis-

junctive dissociations follows from duality. We ﬁrst prove that f0covers any singleton

set {j}, for j∈[d]. We claim that the following logical implication does not hold:

g0∧^

i6=j

gi⇒gj(5)

Indeed, suppose the implication holds for some j. Then the following implication also

holds: g0∧Vi6=j(x∨gi)⇒(x∨gj), since for x= 0 it is the implication above, while

for x= 1 it is a tautology. Therefore, the function fjis redundant in Eq. 4, which

contradicts our assumption. Hence, the implication Eq. 5 does not hold. Let νbe any

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Valuation CNF dissociation DNF dissociation

#ν f0

c[ν]fc[ν]P[f0

c[ν]] ≥P[fc[ν]] f0

d[ν]fd[ν]P[f0

d[ν]]≥P[fd[ν]]

1.h0,0,0,0i0 0 0 ≥0 0 0 0 ≥0

2.h1,0,0,0i0 0 0 ≥0x0

1x p0

1≥p

3.h0,1,0,0i0 0 0 ≥0x0

2x p0

2≥p

4.h0,0,1,0i0 0 0 ≥0x0

3x p0

3≥p

5.h0,0,0,1ix0

1x0

2x0

3x p0

1p0

2p0

3≥p1 1 1 ≥1

6.h1,1,0,0i0 0 0 ≥0x0

1∨x0

2x¯p0

1¯p0

2≥¯p

7.h1,0,1,0i0 0 0 ≥0x0

1∨x0

3x¯p0

1¯p0

3≥¯p

8.h0,1,1,0i0 0 0 ≥0x0

2∨x0

3x¯p0

2¯p0

3≥¯p

9.h1,0,0,1ix0

2x0

3x p0

2p0

3≥p1 1 1 ≥1

10.h0,1,0,1ix0

1x0

3x p0

1p0

3≥p1 1 1 ≥1

11.h0,0,1,1ix0

1x0

2x p0

1p0

2≥p1 1 1 ≥1

12.h1,1,1,0i0 0 0 ≥0x0

1∨x0

2∨x0

3x¯p0

1¯p0

2¯p0

3≥¯p

13.h1,1,0,1ix0

3x p0

3≥p1 1 1 ≥1

14.h1,0,1,1ix0

2x p0

2≥p1 1 1 ≥1

15.h0,1,1,1ix0

1x p0

1≥p1 1 1 ≥1

16.h1,1,1,1i1 1 1 ≥1 1 1 1 ≥1

(a) Comparing 24valuations for determining oblivious bounds.

Upper bounds

1x0

2x0

1x0

2x0

Lower bounds

(b) Non-trivial valuations f0

c[x]

1_x0

2_x0

1_x0

2_x0

d[x]

Fig. 4. Example 4.11 (CNF fc) and Example 4.12 (DNF fd). (a): Determining oblivious bounds by ensuring

that bounds hold for all valuations. (b), (c): Partial order of implication (⇒) for the non-trivial valuations

fc[ν]0and fd[ν]0, e.g.: from x0

1x0

2⇒x0

1it follows that p0

1p0

2≥p⇒p0

1≥p. Note that fc6=fd.

assignment that causes Eq. 5 to fail: thus, for all j∈ {0, . . . , d},j6=i,gi[ν] = 1 and

gj[ν] = 0. Therefore f0[ν] = xj, proving that it covers {j}.

Next, assume that g0⇒Wjgjdoes not hold. We prove that f0covers [d]. Let νbe any

substitution that causes the implication to fail: g0[ν]=1and gj[ν]=0for j∈[d]. Then

f0[ν] = Vj∈[d]x0

4.5. Illustrated Examples for Optimal Oblivious Bounds

We next give two examples that illustrate optimal oblivious bounds for conjunctive and

disjunctive dissociations in some detail.

Example 4.11 (CNF Dissociation). Consider the function fcgiven by an CNF ex-

pression and its dissociation f0

fc= (x∨y1)(x∨y2)(x∨y3)y4

c= (x0

1∨y1)(x0

2∨y2)(x0

3∨y3)y4

There are 24= 16 valuations for y=hy1, y2, y3, y4i. Probabilities p0=hp0

1, p0

2, p0

3iare

thus an oblivious upper bound exactly if they satisfy the 16 inequalities given under

“CNF dissociation” in Fig. 4a. For valuations with ν4= 0 (and thus fc[ν] = 0) or all

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Oblivious Bounds on the Probability of Boolean Functions 1:13

νj= 1 (and thus f0

c[ν]=1) the inequalities trivially hold. For the remaining 7 non-

trivial inequalities, p0

1p0

2p0

3≥pimplies all others. Figure 4b shows the partial order

between the non-trivial valuations, with x0

1x0

2x0

3implying all others. Since fcand f0

are positive in xand x0, respectively, it follows that optimal oblivious upper bounds

are given by p0

1p0

2p0

3=p, e.g., by setting p0

i=3

√pfor symmetric bounds.

Oblivious lower bounds are given by the 16 inequalities after inverting the inequality

sign. Here we see that the three inequalities p0

j≤ptogether imply the others. Hence,

oblivious lower bounds are those that satisfy all three inequalities. The only optimal

oblivious upper bounds are then given by p0

j=p.

Example 4.12 (DNF Dissociation). Consider the function fdgiven by an DNF ex-

pression and its dissociation f0

fd=xy1∨xy2∨xy3∨y4

d=x0

1y1∨x0

2y2∨x0

3y3∨y4

An oblivious upper bound p0=hp0

1, p0

2, p0

3imust thus satisfy the 16 inequalities9given

under “DNF dissociation” in Fig. 4a. For valuations with ν4= 1 (and thus f0

d[ν] =

1) or νj= 0 (and thus fd[ν]=0) the inequalities trivially hold. For the remaining

inequalities we see that the elements of set {x0

1, x0

2, x0

3}together imply all others, and

that x0

1∨x0

2∨x0

3is implied by all others (Fig. 4c shows the partial order between the

non-trivial valuations). Thus, an oblivious upper bound must satisfy p0

j≥p, and the

optimal one is given by p0

j=p. Analogously, an oblivious lower bound must satisfy

¯p0

1¯p0

2¯p0

3≥¯p. Optimal ones are given for ¯p0

1¯p0

2¯p0

3= ¯p, e.g., by setting p0

j= 1 −3

√¯p.

5. RELAXATION AND MODEL-BASED BOUNDS AS DISSOCIATION

This section formalizes the connection between relaxation, model-based bounds and

dissociation that was outlined in the introduction. In other words, we show how both

previous approaches can be uniﬁed under the framework of dissociation.

5.1. Relaxation & Compensation

The following proposition shows relaxation & compensation as conjunctive dissociation

and was brought to our attention by Choi and Darwiche [2011].

PROPOSITION 5.1 (COMPENSATION AND CONJUNCTIVE DISSOCIATION).Let f1,

f2be two monotone Boolean functions which share only one single variable x. Let f

be their conjunction, and f0be the dissociation of fon x, i.e.

f=f1∧f2

f0=f1[x0

1/x]∧f2[x0

2/x]

Then P[f] = P[f0]for P[x0

1] = P[x]and P[x0

2] = P[x|f1].

PROOF OF PROP. 5.1. First, note that P[f] = P[f1]P[f2|f1]. On the other hand,

P[f0] = P[f0

1]P[f0

2]as f0

1and f0

2are independent after dissociating on the only shared

variable x. We also have P[f1] = P[f0

1]since P[x] = P[x0

1]. It remains to be shown that

9Remember that the probability of a disjunction of two independent events is P[x0

1∨x0

2]=1−¯p0

1¯p0

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1:14 W. Gatterbauer and D. Suciu

P[f0

2] = P[f2|f1]. Indeed:

P[f0

2] = P[x0

2·f0

2[1/x0

2]∨¯x0

2·f0

2[0/x0

2]]

=P[x0

2]·P[f0

2[1/x0

2]] + P[¯x0

2]·P[f0

2[0/x0

2]]

=P[x|f1]·P[f2[1/x]] + P[¯x|f1]·P[f2[0/x]]

=P[x|f1]·P[f2[1/x]|f1] + P[¯x|f1]·P[f2[0/x]|f1]

=P[f2|f1]

which proves the claim.

Note that compensation is not oblivious, since the probability p0

2depends on the other

variables occurring in ϕ1. Further note that, in general, ϕ1, ϕ2have more than one

variable in common, and in this case we have P[ϕ0]6=P[ϕ]for the same compensation.

Thus in general, compensation is applied as a heuristics, and it is then not known

whether it provides an upper or lower bound.

The dual result for disjunctions holds by replacing f1with its negation ¯

f1in P[x0

2] =

P[x|¯

f1]. This result is not immediately obvious from the previous one and has, to our

best knowledge, not been stated or applied anywhere before.

PROPOSITION 5.2 (“DISJUNCTIVE COMPENSATION”).Let f1,f2be two monotone

Boolean functions which share only one single variable x. Let fbe their disjunction,

and f0be the dissociation of fon x, i.e. f=f1∨f2, and f0=f1[x0

1/x]∨f2[x0

2/x]. Then

P[f] = P[f0]for P[x0

1] = P[x]and P[x0

2] = P[x|¯

f1].

PROOF OF PROP. 5.2. Let g=¯

f,g1=¯

f1, and g2=¯

f2. Then f=f1∨f2is equivalent

to g=g1∧g2. From Prop. 5.1, we know that P[g] = P[g0], and thus P[f] = P[f0], for

P[x0

1] = P[x]and P[x0

2] = P[x|g1] = P[x|¯

f1].

5.2. Model-based Approximation

The following proposition shows that all model-based bounds can be derived by re-

peated dissociation. However, not all dissociation-bounds can be explained as models

since dissociation is in its essence an algebraic and not a model-based technique (dis-

sociation creates more variables and thus changes the probability space). Therefore,

dissociation can improve any existing model-based approximation approach. Exam-

ple 7.2 will illustrate this with a detailed simulation-based example.

PROPOSITION 5.3 (MODEL-BASED BOUNDS AS DISSOCIATIONS).Let f,fUbe two

monotone Boolean functions over the same set of variables, and for which the logical im-

plication f⇒fUholds. Then: (a) there exists a sequence of optimal conjunctive dissoci-

ations that transform fto fU, and (b) there exists a sequence of non-optimal disjunctive

dissociations that transform fto fU. The dual result holds for the logical implication

fL⇒f: (c) there exists a sequence of optimal disjunctive dissociations that transform

fto fL, and (d) there exists a sequence of non-optimal conjunctive dissociations that

transform fto fL.

PROOF OF PROP. 5.3. We focus here on the implication f⇒fU. The proposition for

the results fL⇒fthen follows from duality.

(a) The implication f⇒fUholds iff there exists a positive function f2such that

f=fU∧f2. Pick a set of variables xs.t. f2[1/x] = 1, and dissociate fon xinto

f0=fU[x0

1/x]∧f2[x0

2/x]. By setting the probabilities of the dissociated variables to

1=pand p0

2=1, the bounds become optimal (p0

1p0

2=p). Further more, fUremains

unchanged (except for the renaming of xto x0

1), whereas f2becomes true. Hence, we

get f0=fU. Thus, all model-based upper bounds can be obtained by conjunctive disso-

ciation and choosing optimal oblivious bounds at each dissociation step.

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Oblivious Bounds on the Probability of Boolean Functions 1:15

(b) The implication f⇒fUalso holds iff there exists a positive function fdsuch that

fU=f∨fd. Let mbe the positive minterm or elementary conjunction involving all

variables of f. The function fdcan be then written as DNF fd=c1∨c2∨. . . , with

products ci⊆m. Since fis monotone, we know m⇒f, and thus also mfd⇒f. We can

therefore write f=f∨mfdor as

f=f∨mc1∨mc2∨. . .

Let xibe the set of all variables in mthat do not occur in ciand denote with mithe

conjunction of xi. Then then each mcican instead be written as miciand thus:

f=f∨m1c1∨m2c2∨. . .

WLOG, we now separate one particular conjunct miciand dissociate on the set xi

f0=f∨m1c1∨m2c2∨. . .

| {z }

[x0

1/xi]∨mici

|{z}

[x0

2/xi]

By setting the probabilities of the dissociated variables to the non-optimal upper

bounds p0

1=pand p0

2=1,f1remains unchanged (except for the renaming of xito

1), whereas f2becomes ci. Hence, we get f0=f∨m1c1∨m2c2∨ ··· ∨ ci. We can

now repeat the same process for all conjuncts mciand receive after a ﬁnite number of

dissociation steps

f00 =f∨(c1∨c2∨. . . ) = f∨fd

Hence f00 =fU. Thus, all model-based upper bounds can be obtained by disjunctive

dissociation and choosing non-optimal bounds at each dissociation step.

6. QUERY-CENTRIC DISSOCIATION BOUNDS FOR PROBABILISTIC QUERIES

Our previous work [Gatterbauer et al. 2010] has shown how to upper bound the prob-

ability of conjunctive queries without self-joins by issuing a sequence of SQL state-

ments over a standard relational DBMS. This section illustrates such dissociation-

based upper bounds and also complements them with new lower bounds. We use the

Boolean query Q:−R(X), S(X, Y ), T (Y), for which the probability computation prob-

lem is known to be #P-hard, over the following database instance D:

R A

x11

x22

S A B

z11 1

z22 1

z32 2

T B

y11

y22

Thus, relation Shas three tuples (1,1),(2,1),(2,2), and both Rand Thave two tuples

(1) and (2). Each tuple is annotated with a Boolean variable x1, x2, z1, . . . , y2, which rep-

resents the independent event that the corresponding tuple is present in the database.

The lineage expression ϕis then the DNF that states which tuples need to be present

in order for the Boolean query Qto be true:

ϕ=x1z1y1∨x2z2y1∨x2z3y2

Calculating P[ϕ]for a general database instance is #P-hard. However, if we treat each

occurrence of a variable xiin ϕas different (in other words, we dissociate ϕeagerly on

all tuples xifrom table R), then we get a read-once expression

ϕ0=x1z1y1∨x0

2,1z2y1∨x0

2,2z3y2

= (x0

1z1∨x0

2,1z2)y1∨x0

2,2z3y2

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:16 W. Gatterbauer and D. Suciu

select IOR(Q3.P) as P

from

(select T.B, T.P*Q2.P as P

from T,

(select Q1.B, IOR(Q1.P) as P

from (select S.A, S.B, S.P*R.P as P

from R, S

where R.A = S.A) as Q1

group by Q1.B) as Q2

where T.B = Q2.B) as Q3

(a) Query PR

create view VR as

select R.A,

1-power(1-R.P,1e0/count(*)) as P

from R, S, T

where R.A=S.A

and S.B=T.B

group by R.A, R.P

(b) View VRfor lower bound with PR

Fig. 5. (a): SQL query corresponding to plan PRfor deriving an upper bound for the hard probabilistic

Boolean query Q:−R(X), S(X, Y ), T (Y). Table Rneeds to be replaced with the view VRfrom (b) for deriving

a lower bound. IOR is a user-deﬁned aggregate explained in the text and stated in Appendix C.

Writing pi, qi, rifor the probabilities of variables xi, yi, zi, respectively, we can calculate

P[ϕ0] = (p0

1·r1)⊗(p0

2,1·r2)·q1⊗(p0

2,2·r3·q2)

where “·” stands for multiplication and “⊗” for independent-or.10

We know from Theorem 4.8 (3) that P[ϕ0]is an upper bound to P[ϕ]by assigning

the original probabilities to the dissociated variables. Furthermore, we have shown

in [Gatterbauer et al. 2010] that P[ϕ]can be calculated with a probabilistic query plan

PR=πp

∅1p

Yπp

Y1p

XR(X), S(X, Y ), T (Y)

where the probabilistic join operator 1p. . . (in preﬁx notation) and the probabilistic

project operator with duplicate elimination πpcompute the probability assuming that

their inputs are independent [Fuhr and R¨

olleke 1997]. Thus, when the join operator

joins two tuples with probabilities p1and p2, respectively, the output has probability

p1p2. When the independent project operator eliminates kduplicate records with prob-

abilities p1, . . . , pk, respectively, the output has probability 1−¯p1· ·· ¯pk. This connection

between read-once formulas and query plans was ﬁrst observed by Olteanu and Huang

[2008]. We write here PRto emphasize that this plan dissociates tuples in table R.11

Figure 5a shows the corresponding SQL query assuming that each of the input tables

has one additional attribute Pfor the probability of a tuple. The query deploys a user-

deﬁned aggregate (UDA) IOR that calculates the independent-or for the probabilities of

the tuples grouped together, i.e. IOR(p1, p2, . . . , pn)=1−¯p1¯p2·· · ¯pn. Appendix C states

the UDA deﬁnition for PostgreSQL.

We also know from Theorem 4.8 (4) that P[ϕ0]is a lower bound to P[ϕ]by assigning

new probabilities 1−2

√1−p2to x0

2,1and x0

2,2(or more generally, any probabilities p0

2,1

and p0

2,2with ¯p0

2,1·¯p0

2,2≥¯p2). Because of the connection between the read-once expres-

sion ϕ0and the query plan PR, we can calculate the lower bound by using the same

SQL query from Fig. 5a after exchanging the table Rwith a view VR(Fig. 5b); VRis

basically a copy of Rthat replaces the probability piof a tuple xiappearing in ϕwith

1−di

√1−piwhere diis the number of times that xiappears in the lineage of Q. The

view joins tables R,Sand T, groups the original input tuples xifrom R, and assigns

each xithe new probability 1−di

√1−picalculated as 1-power(1-T.P,1e0/count(*)).

10The independent-or combines two probabilities as if calculating the disjunction between two independent

events. It is deﬁned as p1⊗p2:= 1 −¯p1¯p2.

11Note we use the notation PRfor both the probabilistic query plan and the corresponding SQL query.

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Oblivious Bounds on the Probability of Boolean Functions 1:17

Alternatively to ϕ0, if we treat each occurrence of a variable yjin ϕas different (in

other words, we dissociate ϕeagerly on all tuples yjfrom table T), then we get another

read-once expression

ϕ00 =x1z1y0

1,1∨x2z2y0

1,2∨x2z3y2

=x1z1y0

1,1∨x2(z2y0

1,2∨z3y2)

P[ϕ00]is an upper bound to P[ϕ]by assigning the original probabilities to the dissociated

variables. In turn, P[ϕ00]can be calculated with another probabilistic query plan that

dissociates all tuples from table Tinstead of R:

PT=πp

∅1p

XR(X), πp

X1p

YS(X, Y ), T (Y)

Similarly to before, P[ϕ00]is a lower bound to P[ϕ]by assigning new probabilities 1−

√1−q1to y0

1,1and y0

1,2. And we can calculate this lower bound with the same query PT

after exchanging Twith a view VTthat replaces the probability qjof a tuple yjwith

1−dj

p1−qjwhere djis the number of times that yjappears in the lineage of Q.

Note that both query plans will calculate upper and lower bounds to query Qover

any database instance D. In fact, all possible query plans give upper bounds to the

true query probability. And as we have illustrated here, by replacing the input tables

with appropriate views, we can use the same query plans to derive lower bounds. We

refer the reader to [Gatterbauer et al. 2010] where we develop the theory of the par-

tial dissociation order among all possible query plans and give a sound and complete

algorithm that returns a set of query plans which are guaranteed to give the tightest

bounds possible in a query-centric way for any conjunctive query without self-joins. For

our example hard query Q, plans PRand PTare the best possible plans. We further

refer to [Gatterbauer and Suciu 2013] for more details and an extensive discussion on

how to speed up the resulting multi-query evaluation.

Also note that these upper and lower bounds can be derived with the help of any

standard relational database, even cloud-based databases which commonly do not al-

low users to deﬁne their own UDAs.12 To our best knowledge, this is the ﬁrst technique

that can upper and lower bound hard probabilistic queries without any modiﬁcations

to the database engine nor performing any calculations outside the database.

7. ILLUSTRATIONS OF OBLIVIOUS BOUNDS

In this section, we study the quality of oblivious bounds across varying scenarios:

We study the bounds as function of correlation between non-dissociated variables

(Sect. 7.1), compare dissociation-based with model-based approximations (Sect. 7.2),

illustrate a fundamental asymmetry between optimal upper and lower bounds

(Sect. 7.3), show that increasing the number of simultaneous dissociations does not

necessarily worsen the bounds (Sect. 7.4), and apply our framework to approximate

hard probabilistic queries over TPC-H data with a standard relational database man-

agement system (Sect.7.5).

7.1. Oblivious Bounds as Function of Correlation between Variables

Example 7.1 (Oblivious Bounds and Correlations). Here we dissociate the DNF

ϕd=xA ∨xB and the analogous CNF ϕc= (x∨A)(x∨B)on xand study the er-

12The UDA IOR can be expressed with standard SQL aggregates, e.g., “IOR(Q3.P)” can be evaluated with

“1-exp(sum(log(case Q3.P when 1 then 1E-307 else 1-Q3.P end)))” on Microsoft SQL Azure.

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1:18 W. Gatterbauer and D. Suciu

−101

Correlation ρ(A, B)

0.5

P[ϕc] and bounds

q=0.8

q=0.5

q=0.2

(a) CNF for p= 0.2

−101

Correlation ρ(A, B)

0.5

P[ϕc] and bounds

q=0.8

q=0.5

q=0.2

(b) CNF for p= 0.5

−101

Correlation ρ(A, B)

0.5

P[ϕc] and bounds

q=0.8

q=0.5

q=0.2

−101

Correlation ρ(A, B)

0.5

P[ϕd] and bounds

q=0.8

q=0.5

q=0.2

(d) DNF for p= 0.8

−101

Correlation ρ(A, B)

0.5

P[ϕd] and bounds

q=0.8

q=0.5

q=0.2

(e) DNF for p= 0.5

−101

Correlation ρ(A, B)

0.5

P[ϕd] and bounds

q=0.8

q=0.5

q=0.2

(f) DNF for p= 0.2

Fig. 6. Example 7.1. Probabilities of CNF ϕc= (x∨A)(x∨B)and DNF ϕd=xA ∨xB together with their

symmetric optimal upper and lower oblivious bounds (borders of shaded areas) as function of the correlation

ρ(A, B)between Aand B, and parameters p=P[x]and q=P[A] = P[B]. For every choice of p, there are

some Aand Bfor which the upper or lower bound becomes tight.

ror of the optimal oblivious bounds as function of the correlation between Aand B.13

Clearly, the bounds also depend on the probabilities of the variables x,A, and B. Let

p=P[x]and assume Aand Bhave the same probability q=P[A] = P[B]. We set

p0=P[x0

1] = P[x0

2]according to the optimal symmetric bounds from Corollary 4.9.

In a few steps, one can calculate the probabilities as

P[ϕd] = 2pq −pP[AB]

P[ϕ0

d] = 2p0q−p02P[AB]

P[ϕc] = p+ (1 −p)P[AB]

P[ϕ0

c] = 2p0q+p02(1 −2q) + (1 −p02)P[AB]

Results: Figure 6 shows the probabilities of the expressions P[ϕ](full lines) and those

of their dissociations P[ϕ0](border of shaded areas) for various values of p,qand as

function of the correlation ρ(A, B).14 For example, Fig.6d shows the case for DNF when

P[x]is p= 0.8and A, B have the same probability qof either 0.8,0.5, or 0.2. When A, B

13Note that this simpliﬁed example also illustrates the more general case ψd=xA ∨xB ∨Cwhen Cis

independent of Aand B, and thus P[ψd] = P[ϕd](1 −P[C]) + P[C]. As a consequence, the graphs in Fig.6 for

P[C]6= 0 would be vertically compressed and the bounds tighter in absolute terms.

14The correlation ρ(A, B )between Boolean events Aand Bis deﬁned as ρ(A, B) = cov(A,B)

√var(A)var(B)with

covariance cov(A, B) = P[AB]−P[A]P[B]and variance var(A) = P[A]−(P[A])2[Feller 1968]. Notice that

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Oblivious Bounds on the Probability of Boolean Functions 1:19

are not correlated at all (ρ= 0), then the upper bound is a better approximation when

qis small, and the lower bound is a better approximation when qis large. On the other

hand, if A, B are not correlated, then there is no need to dissociate the two instances of

xas one can then compute P[(x∨A)(x∨B)] simply as p+¯pP[A]P[B]. The more interesting

case is when A, B are positively correlated (P[AB]≥P[A]P[B], e.g., positive Boolean

functions of other independent variables z, such as the provenance for probabilistic

conjunctive queries). The right of the vertical dashed line of Fig. 6d shows that, in

this case, dissociation offers very good upper and lower bounds, especially when the

formula has a low probability. The graph also shows the effect of dissociation when

A, B are negatively correlated (left of dashed line). Notice that the correlation cannot

always be −1(e.g., two events, each with probability >0.5, can never be disjunct). The

graphs also illustrate why these bounds are obliviously optimal, i.e. without knowledge

of A, B: for every choice of p, there are some A, B for which the upper or lower bound

becomes tight. 

7.2. Oblivious Bounds versus Model-based Approximations

Example 7.2 (Disjunctive Dissociation and Models). This example compares the

approximation of our dissociation-based approach with the model-based approach by

Fink and Olteanu [2011] and illustrates how dissociation-based bounds are tighter, in

general, than model-based approximations. For this purpose, we consider again the

hard Boolean query Q:−R(X), Sd(X, Y ), T (Y)over the database Dfrom Sect. 6. We

now only assume that the table Sis deterministic, as indicated by the superscript din

Sd. The query-equivalent lineage formula is then

ϕ=x1y1∨x2y1∨x2y2

for which Fig.7a shows the bipartite primal graph. We use this instance as its primal

graph forms a P4, which is the simplest 2-partite lineage that is not read-once.15 In

order to compare the approximation quality, we need to limit ourselves to an example

which is tractable enough so we can generate the whole probability space. In practice,

we allow each variable to have any of 11 discrete probabilities D={0,0.1,0.2,...,1}

and consider all 114= 14641 possible probability assignments ν:hp1, p2, q1, q2i → D4

with p=P[x]and q=P[y]. For each ν, we calculate both the absolute error δ∗=P[ϕ∗]−

P[ϕ]and the relative error ε∗=δ∗

P[ϕ], where P[ϕ∗]stands for any of the approximations,

and the exact probability P[ϕ]is calculated by the Shannon expansion on y1as ϕ≡

y1(x1∨x2)∨ ¬y1(x2y2)and thus Pp,q[ϕ] = (1 −(1 −p1)(1 −p2))q1+ (1 −q1)p2q2.

Models: We use the model-based approach by Fink and Olteanu [2011] to approxi-

mate ϕwith lowest upper bound (LUB) formulas ϕUi and greatest lower bound (GLB)

formulas ϕLi, for which ϕLi ⇒ϕand ϕ⇒ϕU i, and neither of the upper (lower)

bounds implies another upper (lower) bound. Among all models considered, we focus

on only read-once formulas. Given the lineage ϕ, the 4 LUBs are ϕU1=x1y1∨x2,

ϕU2=y1∨x2y2,ϕU3= (x1∨x2)y1∨y2, and ϕU4=x1∨x2(y1∨y2). The 3 GLBs are

ϕL1= (x1∨x2)y1,ϕL2=x1(y1∨y2), and ϕL3=x1y1∨x2y2. For each ν, we choose

mini(P[ϕUi]) and maxi(P[ϕLi]) as the best upper and lower model-based bounds, re-

spectively.

ρ(A, B) = P[AB ]−q2

q−q2and, hence: P[AB] = ρ(A, B )·(q−q2)+q2. Further, P[AB]=0(i.e. disjointness between

Aand B) is not possible for q > 0.5, and from P[A∨B]≤1, one can derive P[AB]≥2q−1. In turn, ρ=−1

is not possible for q < 0.5, and it must hold P[AB]≥0. From both together, one can derive the condition

ρmin(q) = max(−q

1−q,−1+q2−2q

q−q2)which gives the minimum possible value for ρ, and which marks the left

starting point of the graphs in Fig.6 as function of q.

15 A path Pnis a graph with vertices {x1,...,xn}and edges {x1x2, x2x3,...,xn1xn}.

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1:20 W. Gatterbauer and D. Suciu

(a) ϕ

2,1

2,2

(b) ϕ0

1,2

1,1

0p1

oblivious

lower

bounds

oblivious

upper

bounds

(d) p0for DNF

average worst

absolute

mδm

U1.54%15.8%

δm

L−0.98%−10.5%

dδd

U0.35%3.7%

δd

L−0.94%−8.7%

relative

mεm

U4.55%289.3%

εm

L−2.23%−28.9%

dεd

U0.73%6.7%

εd

L−2.17%−28.9%

(e) Errors

0 500

000

500

000

−30%

−20%

−10%

10%

20%

30%

40%

50%

Relative errors 

Mode l-based m

U,m

Dissoc iation d

U,d

(f) Top 15% of 14641 data points

sorted by decreasing εd

000

000 6000

−30%

−20%

−10%

10%

20%

30%

40%

50%

Relative errors 

Mode l-based m

U,m

Dissoc iation d

U,d

(g) Top 50% of 14641 data points

sorted by decreasing ε

Fig. 7. Example 7.2. (a)-(c): Bipartite primal graphs for DNF ϕand two dissociations. Notice that the primal

graphs of ϕ0

1and ϕ0

2are forests and thus correspond to read-once expressions. (d): For a given disjunctive

dissociation d, there is only one optimal oblivious upper bound but inﬁnitely many optimal lower bounds.

We evaluate P[ϕ0]for three of the latter (two of which coincide with models m) and keep the maximum as

the best oblivious lower bound. (e): In comparison, dissociation gives substantially better upper bounds than

model-based bounds (0.73% vs. 4.55% average and 6.7% vs. 289.3% worst-case relative errors), yet lower

bounds are only slightly better. (f): Relative errors of 4 approximations for individual data points sorted by

the dissociation error for upper bounds and for lower bounds separately; this is why the dissociation errors

show up as smooth curves (red) while the model based errors are unsorted and thus ragged (blue). (g): Here

we sorted the errors for each approximation individually; this is why all curves are smooth.

Dissociation: Analogously, we consider the possible dissociations into read-once for-

mulas. For our given ϕ, those are ϕ0

1= (x1∨x0

2,1)y1∨x0

2,2y2and ϕ0

2=x1y0

1,1∨x2(y0

1,2∨y2),

with Fig. 7b and Fig. 7c illustrating the dissociated read-once primal graphs.16 From

Corollary 4.9, we know that Pp0,q[ϕ0

1]≥Pp,q[ϕ]for the only optimal oblivious upper

bounds p0

2,1=p0

2,2=p2and Pp0,q[ϕ0

1]≤Pp,q[ϕ]for any p0

2with ¯p0

2,1¯p0

2,2= ¯p2. In

particular, we choose 3 alternative optimal oblivious lower bounds p0

2∈ {hp2,0i,h1−

√1−p2,1−√1−p2i,h0, p2i} (see Fig. 7d). Analogously Pp,q0[ϕ0

2]≥Pp,q[ϕ]for q0

1,1=

1,2=q1and Pp,q0[ϕ0

2]≤Pp,q[ϕ]for q0

1∈ {hq1,0i,h1−√1−q1,1−√1−q1i,h0, q1i}. For

each ν, we choose the minimum among the 2 upper bounds and the maximum among

the 6 lower bounds as the best upper and lower dissociation-based bounds, respectively.

Results: Figures 7e-g show that dissociation-based bounds are always better or equal

to model-based bounds. The reason is that all model-based bounds are a special case of

oblivious dissociation bounds. Furthermore, dissociation gives far better upper bounds,

but only slighter better lower bounds. The reason is illustrated in Fig.7d: the single

dissociation-based upper bound p0=hp, pialways dominates the two model-based up-

per bounds, whereas the two model-based lower bounds are special cases of inﬁnitely

many optimal oblivious lower dissociation-based bounds. As extreme case, it is there-

fore possible for a model-based lower bound to coincide with the best among all optimal

16Note that we consider here dissociation on both x- and y-variables, thus do not treat them as distinct.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:21

(a) ψ

2,1

2,2

(b) ψ0

1,2

1,1

0p1

oblivious

lower

bounds

oblivious

upper

bounds

(d) p0for CNF

average worst

absolute

DNF

δd

U0.35%3.7%

δd

L−0.94%−8.7%

CNF

δc

U0.94%8.7%

δc

L−0.35%−3.7%

relative

DNF

εd

U0.73%6.7%

εd

L−2.17%−28.9%

CNF

εc

U2.69%54.5%

εc

L−1.09%−28.2%

(e) Errors

0 500

000

500

000

−30%

−20%

−10%

10%

20%

30%

40%

50%

Relative errors 

DNF dissoci ation d

U,d

CNF dissociati on c

U,c

(f) Top 15% of 14641 data points

sorted by decreasing εd

000

000 6000

−30%

−20%

−10%

10%

20%

30%

40%

50%

Relative errors 

DNF dissoci ation d

U,d

CNF dissociati on c

U,c

(g) Top 50% of 14641 data points

sorted by decreasing ε

Fig. 8. Example 7.3. (a)-(c): Bipartite primal graphs for CNF ψand two dissociations. (d): For a given con-

junctive dissociation c, we use the only optimal oblivious lower bound and three of inﬁnitely many optimal

oblivious upper bounds. (e): In comparison, disjunctive dissociation gives better upper bounds than conjunc-

tive dissociation (0.73% vs. 2.69% average and 6.7% vs. 54.5% worst-case relative errors), and v.v. for lower

bounds. (f): Relative errors of 4 approximations for individual data points sorted by the disjunctive dissoci-

ation error εdfor upper bounds and for lower bounds separately; this is why the DNF dissociation errors

show up as smooth curves (red) while the CNF dissociation errors are unsorted and thus ragged (gray). (g):

Here we sorted the errors for each approximation individually; this is why all curves are smooth.

oblivious lower dissociation bounds. For our example, we evaluate three oblivious lower

bounds, two of which coincide with models. 

7.3. Conjunctive versus Disjunctive Dissociation

Example 7.3 (Disjunctive and Conjunctive Dissociation). This example illustrates

an interesting asymmetry: optimal upper bounds for disjunctive dissociations and op-

timal lower bounds for conjunctive dissociations are not only unique but also better, on

average, than optimal upper bounds for conjunctive dissociations and optimal lower

bounds for disjunctive dissociations, respectively (see Figure 1). We show this by com-

paring the approximation of a function by either dissociating a conjunctive or a dis-

junctive expression for the same function.

We re-use the setup from Example 7.2 where we had a function expressed by a dis-

junctive expression ϕ. Our DNF ϕcan be written as CNF ψ= (x1∨x2)(y1∨x2)(y1∨y2)

with fϕ=fψ17, and two conjunctive dissociations ψ0

1= (x1∨x0

2,1)(y1∨x0

2,2)(y1∨y2)

and ψ0

2= (x1∨x2)(y0

1,1∨x2)(y0

1,2∨y2)(Figures 8a-c shows the primal graphs).

Again from Corollary 4.9, we know that Pp0,q[ϕ0

1]≤Pp,q[ϕ]for the only optimal

oblivious lower bounds p0

2,1=p0

2,2=p2and Pp0,q[ϕ0

1]≥Pp,q[ϕ]for any p0

2with

2,1p0

2,2=p2. In particular, we choose 3 alternative optimal oblivious lower bounds

2∈ {hp2,1i,h√p2,√p2i,h1, p2i} (see Fig. 7d). Analogously Pp,q0[ϕ0

2]≤Pp,q[ϕ]for

17Notice that this transformation from DNF to CNF is hard, in general. We not do not focus on algorithmic

aspects in this paper, but rather show the potential of this new approach.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:22 W. Gatterbauer and D. Suciu

(a) ϕn

1,1

1,2

n1,2

(b) ϕ0

100101102103

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pr ob ab ilitie s

P[ϕn]

p= 0.1

100101102103

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pr ob ab ilitie s

ma x P[ϕ0

n]≈0.531 2

P[ϕn] = 0 .5

(d) P[ϕn] = const

Fig. 9. Example 7.4. (a), (b): Primal graphs for path PnDNF ϕnand its dissociation ϕ0

n. (c), (d): P[ϕn]to-

gether with their symmetric optimal upper and lower oblivious bounds (borders of shaded areas) as function

of n. (c) keeps p= 0.1constant, whereas (d) varies pso as to keep P[ϕ]=0.5constant for increasing n. The

upper bounds approximate the probability of the DNF very well and even become tight for n→ ∞.

1,1=q0

1,2=q1and Pp,q0[ϕ0

2]≥Pp,q[ϕ]for q0

1∈ {hq1,1i,h√q1,√q1i,h1, q1i}. For each

ν, we choose the maximum among the 2 lower bounds and the minimum among the 6

upper bounds as the best upper and lower conjunctive dissociation-based bounds, re-

spectively. We then compare with the approximations from the DNF ϕin Example 7.2.

Results: Figures 8e-g show that optimal disjunctive upper bounds are, in general but

not consistently, better than optimal conjunctive upper bounds for the same function

(≈83.5% of those data points with different approximations are better for conjunctive

dissociations). The dual result holds for lower bounds. This duality can be best seen in

the correspondences of absolute errors between upper and lower bounds. 

7.4. Multiple Dissociations at Once

Here we investigate the inﬂuence of the primal graph and number of dissociations on

the tightness of the bounds. Both examples correspond to the lineage of the standard

unsafe query Q:−R(X), S(X, Y ), T (Y)over two different database instances.

Example 7.4 (Path Pnas Primal Graph). This example considers a DNF expres-

sion whose primal graph forms a Pn, i.e. a path of length n(see Fig.9a). Note that this

is a generalization of the path P4from Example 7.2 and corresponds to the lineage of

the same unsafe query over larger database instance with 2n−1tuples:

ϕn=x1y1∨x1y2∨x2y2∨. . . ∨xn−1yn∨xnyn

ϕ0

n=x0

1,1y1∨x0

1,2y2∨x0

2,1y2∨. . . ∨x0

n−1,2yn∨x0

nyn

Exact: In the following, we assume the probabilities of all variables to be pand use

the notation pn:= P[ϕn]and p∗

n:= P[ϕ∗

n], where ϕ∗

ncorresponds to the formula ϕn

without the last conjunct xnyn. We can then express pnas function of p∗

n,pn−1and

p∗

n−1by recursive application of Shannon’s expansion to xnand yn:

pn=P[xn]P[yn] + P[¯yn]pn−1+P[¯xn]p∗

p∗

n=P[yn]P[xn−1] + P[¯xn−1]p∗

n−1+P[¯yn]pn−1

We thus get the linear recurrence system

pn=A1pn−1+B1p∗

n−1+C1

p∗

n=A2pn−1+B2p∗

n−1+C2

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:23

(a) ϕn

1,1

n,1

n,n

1,n

(b) ϕ0

0 10 20 30 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pr ob ab ilitie s

P[ϕn]

p= 0.1

100101102103

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pr ob ab ilitie s

li mn→∞ P[ϕ0

n]≈0.5 80 3

P[ϕn] = 0 .5

(d) P[ϕn] = const

Fig. 10. Example 7.5. (a), (b): Primal graphs for complete bipartite DNF ϕnand its dissociation ϕ0

n. (c), (d):

P[ϕn]together with their symmetric optimal upper and lower oblivious bounds (borders of shaded areas) as

function of n. (d) varies pso as to keep P[ϕ]=0.5constant for increasing size n. The oblivious upper bound

is ultimately bounded despite having n2fresh variables in the dissociated DNF for increasing n.

with A1= ¯p,B1=p¯p2,C1=p2(2 −p),A2= ¯p,B2=p¯p, and C2=p2. With a few manip-

ulations, this recurrence system can be transformed into a linear non-homogenous re-

currence relation of second order pn=Apn−1+Bpn−2+Cwhere A=A1+B2= ¯p(1+p),

B=A2B1−A1B2=−p2¯p2, and C=B1C2+C1(1 −B2) = p2(p¯p2+ (2 −p)(1 −p¯p)).

Thus we can recursively calculate P[ϕn]for any probability assignment pstarting with

initial values p1=p2and p2= 3p2−2p3.

Dissociation: Figure 9b shows the primal graph for the dissociation ϕ0

n. Variables

x1to xn−1are dissociated into two variables with same probability p0, whereas xn

into one with original probability p. In other words, with increasing n, there are more

variables dissociated into two fresh ones each. The probability P[ϕ0

n]is then equal to

the probability that at least one variable xiis connected to one variable yj:

P[ϕ0

n] = 1 −(1 −pp0)1−p(1 −¯p02)n−21−p(1 −¯p¯p0)

We set p0=pfor upper bounds, and p0= 1 −√1−pfor lower bounds.

Results: Figures 9c-d shows the interesting result that the disjunctive upper bounds

become tight for increasing size of the primal graph, and thus increasing number of

dissociations. This can be best seen in Fig.9d for which pis chosen as to keep P[ϕ]=0.5

constant for varying nand we have limn→∞ P[ϕ0

n] = P[ϕn] = 0.5for upper bounds.

In contrast, the disjunctive lower bounds become weaker but still have a limit value

limn→∞ P[ϕ0

n]≈0.2929 (derived numerically). 

Example 7.5 (Complete bipartite graph Kn,n as Primal Graph). This example con-

siders a DNF whose primal graph forms a complete bipartite graph of size n, i.e. each

variable xiis appearing in one clause with each variable yj(see Fig. 10a). Note that

this example corresponds to lineage for the standard unsafe query over a database

instance with O(n2)tuples:

ϕn=_

(i,j)∈[n]2

xiyj

ϕ0

n=_

j∈[n]

yj_

i∈[n]

i,j 

Exact: We again assume that all variables have the same probability p=P[xi] =

P[yi].P[ϕn]is then equal to the probability that there is at least one tuple xiand at

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:24 W. Gatterbauer and D. Suciu

select distinct s nationkey

from Supplier, Partsupp, Part

where s suppkey = ps suppkey

and ps partkey = p partkey

and s suppkey <=$1

and p name like $2

(a) Deterministic query Q(a)

Supplier(s suppkey, s nationkey)

PartSupp(ps suppkey, ps partkey)

Part(p partkey, p name)

(b) Relevant TPC-H schema

Fig. 11. Example 7.6. Parameterized SQL query Q(a)and relevant portion of the TPC-H schema.

least one tuple yi:

P[ϕn] = 1−(1 −p)n2(6)

Dissociation: Figure 10b shows the primal graph for the dissociation ϕ0

n. Each vari-

able xiis dissociated into nfresh variables with same probability p0, i.e. there are n2

fresh variables in total. The probability P[ϕ0

n]is then equal to the probability that at

least one variable yiis connected to one variables xi,j :

P[ϕ0

n] = 1 −1−p1−(1 −p0)nn

We will again choose pas to keep r:=P[ϕn]constant with increasing n, and then

calculate P[ϕ0

n]as function of r. From Eq.6, we get p= 1 −n

p1−√rand then set p0=p

for upper bounds, and p0= 1 −n

√1−pfor lower bounds as each dissociated variable is

replaced by nfresh variables. It can then be shown that P[ϕ0

n]for the upper bound is

monotonically increasing for nand bounded below 1 with the limit value:

lim

n→∞

P[ϕ0

n] = 1 −(1 −√r)√r

Results: Figure 10d keeps P[ϕ]=0.5constant (by decreasing pfor increasing n) and

shows the interesting result that the optimal upper bound is itself upper bounded and

reaches a limit value, although there are more variables dissociated, and each variable

is dissociated into more fresh ones. This limit value is 0.5803 for r= 0.5. However, lower

bounds are not useful in this case. 

7.5. Dissociation with a Standard Relational Database Management System

Example 7.6 (TPC-H). Here we apply the theory of dissociation to bound hard

probabilistic queries with the help of PostgreSQL 9.2, an open-source relational

DMBS.18 We use the TPC-H DBGEN data generator19 to generate a 1GB database.

We then add a column Pto each table, and assign to each tuple either the probabil-

ity p= 0.1, or p= 0.5, or a random probability from the set {0.01,0.02,...,0.5}(“p=

rand 0.5”). Choosing tuple probabilities p≤0.5helps us avoid ﬂoating-point errors for

queries with very large lineages whose query answer probabilities would otherwise be

too close to 1.20 Our experiments use the following parameterized query (Fig.11):

Q(a) :−S(s, a),PS (s, u), P (u, n), s ≤$1, n like $2

Relations S,PS and Prepresent tables Supplier,PartSupp and Part, respectively. Vari-

able astands for attribute nationkey (“answer tuple”), sfor suppkey,ufor partkey (“unit”),

18http://www.postgresql.org/download/

19http://www.tpc.org/tpch/

20Compare Fig. 13b and Fig. 13e to observe the impact of the choice of input tuple probabilities pon the

probabilities of query answers for the same query.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:25

create view VP as

select p partkey, s nationkey,

1-power(1-P.P,1e0/count(*)) as P

from Part P, Partsupp, Supplier

where p partkey=ps partkey

and ps suppkey = s suppkey

and s suppkey <=$1

and p name like $2

group by p partkey, s nationkey, P.P

(a) View VPfor lower bounds with P∗

select s nationkey, IOR(Q3.P) as P

from

(select S.s nationkey, S.P*Q2.P as P

from Supplier S,

(select Q1.ps suppkey, s nationkey, IOR(Q1.P) as P

from

(select ps suppkey, s nationkey, PS.P*VP.P as P

from Partsupp PS, VP

where ps partkey = p partkey

and ps suppkey <=$1) as Q1

group by Q1.ps suppkey, s nationkey) as Q2

where s suppkey = Q2.ps suppkey) as Q3

group by Q3.s nationkey

(b) Query P∗

Fig. 12. Example 7.6. (a) View deﬁnition VPand (b) and query P∗

Pfor deriving the lower bounds by disso-

ciating table P. Note the inclusion of the attribute nationkey in VPfor reasons explained in the text.

and nfor name. The probabilistic version of this query asks which nations (as deter-

mined by the attribute nationkey) are most likely to have suppliers with suppkey ≤$1

that supply parts with a name like $2 when all tuples in Supplier,PartSupp, and Part are

probabilistic. Parameters $1 and $2 allow us to reduce the number of tuples that can

participate from tables Supplier and Part, respectively, and to thus study the effects of

the lineage size on the predicted dissociation bounds and running times. By default, ta-

bles Supplier,Partsupp and Part have 10k, 800k and 200k tuples, respectively, and there

are 25 different numeric attributes for nationkey. For parameter $1, we choose a value

∈ {500,1000,...,10000}which selects the respective number of tuples from table Sup-

plier. For parameter $2, we choose a value ∈ {’%’,’%red%’,’%red%green%’ }which selects

200k, 11k or 251 tuples in table Part, respectively.

Translation into SQL: Note that the lineage for each individual query answer cor-

responds to the lineage for the Boolean query Qfrom Sect. 6, which we know is hard,

in general. We thus bound the probabilities of the query answers by evaluating four

different queries that correspond to the query-centric dissociation bounds from Sect.6:

dissociating either table Supplier or table Part, and calculating either upper and lower

bounds. To get the ﬁnal upper (lower) bounds, we take the minimum (maximum) of the

two upper (lower) bounds for each answer tuple. The two query plans are as follows:

PS(a) = πp

a1p

uπp

a,u 1p

sS(s, a),PS (s, u), s ≤$1, P (u, n), n like $2

PP(a) = πp

a1p

sS(s, a), πp

s1p

uPS (s, u), s ≤$1, P (u, n), n like $2

Notice one technical detail for determining the lower bounds with plan PP: Any tuple

tfrom table Part may appear a different number of times in the lineages for different

query answers.21 Thus, for every answer athat has tin its lineage, we need to cre-

ate a distinct copy of tin the view VP, with a probability that depends only on the

number of times that tappears in the lineage for a.22 Thus, the view deﬁnition for VR

needs to include the attribute nationkey (Fig. 12a) and PPneeds to be adapted as follows

(Fig.12b):

P∗

P(a) = πp

a1p

sS(s, a), πp

s,a 1p

uPS (s, u), s ≤$1,VP (u, a)

21Lower bounds by dissociating Supplier are easier since the table includes the query answer attribute

nationkey. As consequence, any tuple from Supplier may appear in the lineage of one query answer only.

22We could actually use the total number of times the tuple appears in all lineages and still get lower bounds.

However, the resulting lower bounds would be weaker and not obliviously optimal.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:26 W. Gatterbauer and D. Suciu

6 16 15 21 17 1 11 20 3 5

0.45

0.5

0.55

0.6

0.65 Diss. on Supplier

Best of both Diss.

Part dissociated

Actual Probability

(a) $1=10000, $2= ’%red%green%’

p=rand 0.5, MaxLin = 48

11 16 9 21 4 6 8 20 1 0

1Diss. on Supplier

Best of both Diss.

Part dissociated

Actual Probability

(b) $1=10000, $2= ’%red%’

p=rand 0.5, MaxLin = 1941

15 11 0 17 16 6 21 3 4 2

0.995

0.996

0.997

0.998

0.999

1Diss. on Supplier

Best of both Diss.

Part dissociated

Actual Probability

p= 0.5, MaxLin = 48

9 18 10 4 2 1 8 7 20 14

0.4

0.41

0.42

0.43

0.44

0.45

0.46 Diss. on Supplier

Best of both Diss.

Part dissociated

Actual Probability

(d) $1=3000, $2= ’%red%’

p= 0.1, MaxLin = 611

11 4 17 9 1 0 3 8 21 10

0.81

0.82

0.83

0.84

0.85

0.86 Diss. on Supplier

Best of both Diss.

Part dissociated

Actual Probability

(e) $1=10000, $2= ’%red%’

p= 0.1, MaxLin = 1941

11 17 0 4 8 1 3 20 18 16

1Diss. on Supplier

Best of both Diss.

Part dissociated

(f) $1=10000, $2= ’%’

p= 0.1, MaxLin = 35040

Case Answer Fig. #Part %diss. #fresh #Supp. %diss. #fresh tighter bounds

(A) (2) (c) 40 7.5% 2 43 0.0% -PS

(B) (6) (a) 42 11.9% 2 53 7.5% 2 PS

(D) (11) (f) 32899 6.3% 2438 100.0% 80 PP

(g) Overview of 4 cases discussed in the text

Fig. 13. Example 7.6. Probabilities for the top 10 query answers for varying query parameters $1, $2,

and tuple probabilities p. The ranking is determined by the upper dissociation bounds (upper end of the

red interval) and is identical to the one determined by the actual probabilities (crosses), except in (c) where

tuples 6 and 21 are ﬂipped. MaxLin shows the maximal lineage among the query answers, which is too big in

(f) for exact probabilistic inference. (b): x= 0.999999999 999 = 1−10−12. (f): x= 0.999 999 9999 = 1−10−10 .

In order to speed up the resulting multi-query evaluation, we apply a determinis-

tic semi-join reduction on the input tables and then reuse intermediate query results

across all four subsequent queries. Since query optimization is not the focus of this

paper, we do not show these techniques in Fig. 12 and instead refer to [Gatterbauer

and Suciu 2013] for a detailed discussion of techniques to speed up query dissociation.

In addition, the exact SQL statements that allow the interested reader to repeat our

experiments with PostgreSQL are available on the LaPushDB project page.23

23http://LaPushDB.com/

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:27

Ground truth: To compare our bounds against ground truth, we construct the lin-

eage DNF for each query answer and use DeMorgan to write it as “dual lineage CNF”

without exponential increase in size. For example, the DNF ϕ=x1x3∨x1x2can be

written as CNF ¯ϕ= (¯x1∨¯x3)∧(¯x1∨¯x2)with P[ ¯ϕ] = 1 −P[ϕ]. The problem of eval-

uating a probabilistic CNF can further be translated into the problem of computing

the partition function of a propositional Markov random ﬁeld, for which the AI com-

munity has developed sophisticated solvers. For our experiments, we use a tool called

SampleSearch [Gogate and Domingos 2010; Gogate and Dechter 2011]24.

Quality Results: Figure 13 shows the top 10 query answers, as predicted by the up-

per dissociation bounds for varying query parameters $1 and $2, as well as varying

tuple probabilities p. The crosses show the ground truth probabilities as determined

by SampleSearch. The red intervals shows the interval between upper and lower dis-

sociation bounds. Recall that the ﬁnal dissociation interval is the intersection between

the interval from dissociation on Supplier (left of the red interval) and on Part (right of

the red interval). The graphs suggest that the upper dissociation bounds are very close

to the actual probabilities, which is reminiscent of Sect. 7.4 having shown that upper

bounds for DNF dissociations are commonly closer to the true probabilities than lower

bounds. For Fig. 13f, we have no ground truth as the lineage for the top tuple (11) has

size 32899 (i.e. the corresponding DNF has 32899 clauses), which is too big for exact

probabilistic inference. However, extrapolating from Fig. 13d and Fig. 13e to Fig. 13f,

we speculate that upper dissociation bounds give good approximations here as well.

The different interval sizes (i.e. quality of the bounds) arise from different numbers

of dissociated tuples in the respective lineages. We illustrate with 4 cases (Fig. 13g):

(A) If there is a plan that does not dissociate any tuple, then both upper and lower

bounds coincide. This scenario is also called data-safe [Jha et al. 2010]. For ex-

ample, the lineage for answer (2) in Fig. 13c has 40 unique tuples from table

Part, out of which 3 (7.5%) are dissociated into 2 fresh ones with PP. However, all

of the 43 tuples from table Supplier that appear in the lineage appear only once.

Therefore, PSdissociates no tuple when calculating the answer probability for

(2), and as a result gives us the exact value.

(B) The lineage for the top-ranked answer (6) in Fig.13a has 42 unique tuples from

table Part, out of which 5 (≈11.9%) are dissociated into 2 fresh ones with PP. In

contrast, the lineage has 53 unique tuples from table Supplier, out of which only

4 (≈7.5%) are dissociated into 2 fresh ones with PS. Intuitively, PSshould give

us therefore tighter bounds, which is conﬁrmed by the results.

show results for the same query, but for different tuple probabilities p= 0.1or

p=rand 0.5) has 1830 unique tuples from table Part, out of which ≈5.8% are

dissociated into 2 or 3 fresh ones with PP. In contrast, the lineage has only 434

unique tuples from table Supplier, out of which ≈95.6% are dissociated into into

2 to 11 fresh variables. Thus, PPgives far tighter bounds in this case.

(D) The lineage for the same answer (11) in Fig. 13f has 32899 unique tuples from

table Part, out of which ≈6.3% are dissociated into 2-4 fresh ones with PP. In

contrast, the lineage has only 438 unique tuples from table Supplier, out of which

all are dissociated into 80 fresh ones with PS(this is an artifact of the TPC-H

random database generator). Thus, the bounds for PSare very poor.

Importantly, relevance ranking of the answer tuples by upper dissociation bounds

approximates the ranking by query reliability very well. For example, for the case of

p= 0.1and $2=’%red%’ (Fig. 13d and Fig. 13e), the ranking given by the minimum

upper bounds was identical to the ranking given by the ground truth for all parameter

24http://www.hlt.utdallas.edu/~vgogate/SampleSearch.html

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1:28 W. Gatterbauer and D. Suciu

5000 10000

100

150

200

250

300

350

400

Time [msec]

Upper & lower bounds

Upper bounds only

Lineage query

Deterministic SQL

(a) $2 = ’%red%green%’

5000 10000

Time [sec]

Upper & lower bounds

Upper bounds only

Lineage query

Deterministic SQL

(b) $2 = ’%red%’

5000 10000

100

120

Time [sec]

Upper & lower bounds

Upper bounds only

Lineage query

Deterministic SQL

$2 ’%red%green%’ ’%red%’ ’%red%’ ’%’ ’%’

$1 500 10000 500 10000 500 10000

Total lineage size 42 1,004 2,218 44,152 40,000 800,000

SampleSearch [msec] 321 808 1,152 106,396 79,449 −

Upper & lower bounds [msec] 169 342 1,070 7,803 5,886 118,144

Upper bounds only [msec] 139 199 932 3,203 3,569 42,164

Lineage SQL [msec] 109 119 417 1,112 844 5,731

Deterministic SQL [msec] 124 128 444 746 649 2,032

(d)

Fig. 14. Example 7.6. Timing results for queries with varying parameters $1 and $2: For small lineage

(<10000), dissociation bounds can be calculated in a small multiple (<4) of the time needed for a standard

deterministic query. For large lineages (>10000), calculation scales linearly in the size of the total lineage.

choices $1 ∈ {500,1000,...,10000}. Figure 13c shows an example of an incorrect rank-

ings (for parameters $1=10000, $2=’%red%green%’, p= 0.5): Here tuples 6 and 21 are

ﬂipped as compared to their actual probabilities 0.99775 and 0.99777, respectively.

Timing Results: Figure 14 compares the times for evaluating the deterministic query

(Fig.11a) with the times for calculating the dissociation bounds for changing parame-

ters $1 and $2. As experimental platform, we use PostgreSQL 9.2 on a 2.5 Ghz Intel

Core i5 with 16G of main memory. We run each query 5 times and take the average

execution time. Figure 14d also shows the size of the total lineage of a query (which

is the same as the number of query results for the deterministic query without pro-

jection) and the times needed by SampleSearch to evaluate the ground truth, if pos-

sible.25 Since table Supplier contains exactly 10k tuples with suppkey ∈ {1,...,10000},

any choice of $1≥10000 has no effect on the query. We show separate graphs for the

time needed to calculate the upper bounds only (which our theory and experiments

suggest give better absolute approximations) and the time for both upper and lower

bounds (lower bounds are more expensive due to the required manipulation of the in-

put tuples). We also show the times for retrieving the lineage with a lineage query. Any

probabilistic approach that evaluates the probabilities outside of the database engine

needs to issue this query to retrieve the DNF for each answer. The time needed for the

lineage query thus serves as minimum benchmark for any probabilistic approximation.

Our timing results show that, for small lineages (<10000), calculating upper and

lower bounds can be achieved in a time that is only a small multiple (<4) of the time

needed for an equivalent deterministic query. For large lineages (>10000), calculating

the bounds scales linearly with the size of the lineage (Fig. 14c), whereas determin-

25The reported times are for evaluating all answer DNFs without the overhead for the lineage query.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:29

istic query evaluation can scale in sublinear time (recall that the cardinality of the

answer set is maximal 25 across all queries since the database contains only 25 differ-

ent values for the answer attribute nationkey). Importantly, scalability for dissociation

is independent of the tractability of the data instance, e.g. maximum treewidth of the

lineage for any query answer. In contrast, SampleSearch quickly takes too long for in-

creasing lineage. For example, SampleSearch needs 108 sec for calculating the ground

truth for Fig. 13e (The maximum lineage size among all 25 query answers is 1941 in

this scenario). Upper dissociation bounds can be calculated in only 3 sec and give the

same ranking (and almost the same probabilities).

Key take-aways: Overall, our quality and timing experiments suggest that dissocia-

tion bounds (in particular, the upper bounds) can provide a good approximation of the

actual probabilities and provide a ranking of the query answers that is often identi-

cal to the ranking for their actual probabilities. These bounds can be calculated with

guaranteed polynomial scalability in the size of the data. In particular for queries with

small lineage sizes (<10000), calculating the bounds took only a small multiple (<4)

of the time needed to evaluate standard deterministic queries. 

8. RELATED WORK AND DISCUSSION

Dissociation is related to a number of recent approaches in the graphical models and

constraint satisfaction literature which approximate an intractable problem with a

tractable relaxed version after treating multiple occurrences of variables or nodes as

independent or ignoring some equivalence constraints: Choi et al. [2007] approximate

inference in Bayesian networks by “node splitting,” i.e. removing some dependencies

from the original model. Ram´

ırez and Geffner [2007] treat the problem of obtaining

a minimum cost satisfying assignment of a CNF formula by “variable renaming,” i.e.

replacing a variable that appears in many clauses by many fresh new variables that

appear in few. Pipatsrisawat and Darwiche [2007] provide lower bounds for MaxSAT

by “variable splitting,” i.e. compiling a relaxation of the original CNF. Andersen et al.

[2007] improve the relaxation for constraint satisfaction problems by “reﬁnement

through node splitting,” i.e. making explicit some interactions between variables. Choi

and Darwiche [2009] relax the maximum a posteriori (MAP) problem in probabilistic

graphical models by dropping equivalence constraints and partially compensating for

the relaxation. Our work provides a general framework for approximating the proba-

bility of Boolean functions with both upper and lower bounds. These bounds shed light

on the connection between previous relaxation-based and model-based approximations

and unify them as concrete choices in a larger design space. We thus refer to all of the

above approaches as instances of dissociation-based approximations.

Another line of work that is varyingly called “discretization,” “bucketing,” “binning,”

or “quantization” proposes relaxations by merging or partitioning states or nodes (in-

stead of splitting them), and to then perform simpliﬁed calculations over those parti-

tions: Dechter and Rish [2003] approximate a function with high arity by a collection of

smaller-arity functions with a parameterized scheme called “mini-buckets.” As exam-

ple, the sum Pif(xi)·g(xi)for non-negative functions fand gcan be upper bounded

by the summation (maxif(xi)) ·Pig(xi), i.e. all different values of f(xi)are replaced

by the single maximum value maxif(xi), which simpliﬁes the calculations. Similarly,

the sum can be lower bounded by (minif(xi)) ·Pig(xi). St-Aubin et al. [2000] use

Algebraic Decision Diagrams (ADDs) and reduce the sizes of the intermediate value

functions generated by replacing the values at the terminals of the ADD with ranges

of values. Bergman et al. [2011],[2013] construct relaxations of Multivalued Decision

Diagrams (MDDs) by merging vertices when the size of the partially constructed MDD

grows too large. Gogate and Domingos [2011] compress potentials computed during

the execution of variable elimination by “quantizing” them, i.e. replacing a number

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

1:30 W. Gatterbauer and D. Suciu

of distinct values in range of the potential by a single value. Since all of the above

approaches reduce the number of distinct values in the range of a function, we collec-

tively refer to them as quantization-based approximations. A more detailed literature

overview in this space is given by Gogate and Domingos [2013].

Note that dissociation-based approaches (that split nodes) and quantization-based

approaches (that merge nodes) are not inverse operations, but are rather two com-

plementary approaches that may be combined to yield improved methods. The in-

verse of dissociation is what we refer to as assimilation:26 Consider the Boolean for-

mula ϕ=x1y1∨x1y2∨x2y2which is a dissociation of the much simpler formula

ϕ∗=x(y1∨y2). Hence, we know from dissociation that P[ϕ∗]≤P[ϕ]for p≤min(p1, p2)

and that P[ϕ∗]≥P[ϕ]for p≥1−¯p1¯p2. Note that for quantization, we can always choose

max(p1, p2)or min(p1, p2)to get a guaranteed upper or lower bound. In contrast, for

assimilation, we may have to choose a different value to get a guaranteed bound. Also,

for the case p1=p2, assimilation will generally still be an approximation, whereas

quantization would be exact. Thus, these are two different approaches.

Existing approaches for query processing over probabilistic databases that are both

general and tractable are either: (1) simulation-based approaches that adapt general

purpose sampling methods [Jampani et al. 2008; Kennedy and Koch 2010; Re et al.

2007]; or (2) model-based approaches that approximate the original number of mod-

els with guaranteed lower or upper bounds [Olteanu et al. 2010; Fink and Olteanu

2011]. We have show in this paper that, for every model-based bound, there exists a

dissociation bound that is at least as good or better.

Our work on dissociation originated while generalizing propagation-based ranking

methods from graphs [Detwiler et al. 2009] to hypergraphs and conjunctive queries.

In [Gatterbauer et al. 2010], we applied dissociation in a query-centric way to upper

bound hard probabilistic queries, and showed the connection between propagation on

graphs and dissociation on hypergraphs (see [Gatterbauer and Suciu 2013] for all de-

tails). In this paper, we provide the theoretical underpinnings of these results in a

generalized framework with both upper and lower bounds. A previous version of this

paper was made available as [Gatterbauer and Suciu 2011].

9. OUTLOOK

We introduced dissociation as a new algebraic technique for approximating the prob-

ability of Boolean functions. We applied this technique to derive obliviously optimal

upper and lower bounds for conjunctive and disjunctive dissociations and proved that

dissociation always gives equally good or better approximations than models. We did

not address the algorithmic complexities of exploring the space of alternative disso-

ciations, but rather see our technique as a basic building block for new algorithmic

approaches. Such future approaches can apply dissociation at two conceptual levels:

(1) at the query-level, i.e. at query time and before analyzing the data, or (2) at the

data-level, i.e. while analyzing the data.

The advantage of query-centric approaches is that they can run in guaranteed poly-

nomial time,27 yet at the cost of no general approximation guarantees.28 Here, we en-

vision that the query-centric, ﬁrst-order logic-based view of operating on data by the

26We prefer the word “assimilation” as inverse of dissociation over of the more natural choice of “association”

as it implies correctly that two items are actually merged and not merely associated.

27Recall from our experiments (Fig. 14c) that query-centric dissociation scales linearly in the size of the

lineage, independent of intricacies in the data, such as the treewidth of the lineage.

28However, also recall from our experiments (Fig.13) that query-centric approaches may work well in prac-

tice. Notice here the similarity to loopy belief propagation [Frey and MacKay 1997], which is applied widely

and successfully, despite lacking general performance guarantees.

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

Oblivious Bounds on the Probability of Boolean Functions 1:31

database community can also inﬂuence neighboring communities, in particular those

working on lifted inference (see e.g. [Van den Broeck et al. 2011]).

The advantage of data-centric approaches is that exact solutions can be arbitrarily

approximated, yet at the cost of no guaranteed runtime [Roth 1996]. Here, we envision

a range of new approaches (that may combine dissociation with quantization) to com-

pile an existing intractable formula into a tractable target language, e.g., read-once

formulas or formulas with bounded treewidth. For example, one can imagine an ap-

proximation scheme that adds repeated dissociation to Shannon expansion in order to

avoid the intermediate state explosion.

ACKNOWLEDGMENTS

We would like to thank Arthur Choi and Adnan Darwiche for helpful discussions on Relaxation & Com-

pensation, and for bringing Prop. 5.1 to our attention [Choi and Darwiche 2011]. We would also like to

thank Vibhav Gogate for helpful discussions on quantization and guidance for using his tool SampleSearch,

Alexandra Meliou for suggesting the name “dissociation,” and the reviewers for their careful reading of

this manuscript and their detailed feedback. This work was partially supported by NSF grants IIS-0915054

and IIS-1115188. More information about this research, including the PostgreSQL statements to repeat the

experiments on TPC-H data, can be found on the project page: http://LaPushDB.com/.

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Oblivious Bounds on the Probability of Boolean Functions 1:33

A. NOMENCLATURE

x, y, z independent Boolean random variables

ϕ, ψ Boolean formulas, probabilistic event expressions

f, g, fϕBoolean function, represented by an expression ϕ

P[x],P[ϕ]probability of an event or expression

pi, qj, rkprobabilities pi=P[xi],qj=P[yj],rk=P[zk]

x,g,psets {x1,...,xk}or vectors hx1,...,xkiof variables, functions or probabilities

Pp,q[f]probability of function f(x,y)for p=P[x],q=P[y]

¯x, ¯ϕ, ¯pcomplements ¬x, ¬ϕ, 1−p

f0, ϕ0dissociation of a function for expression ϕ

θsubstitution θ:x0→x; deﬁnes a dissociation f0of fif f0[θ] = f

f[x0/x]substitution of x0for xin f

m, m0, n m =|x|,m0=|x0|,n=|y|

dinumber of new variables that xiis dissociated into

νvaluation or truth assignment ν:y→ {0,1}with yi=νi

f[ν], ϕ[ν]function for expression ϕwith valuation νsubstituted for y

gνgν=Vjgν

j, where gν

j= ¯gjif νj= 0 and gν

j=gjif νj=1

B. REPRESENTING COMPLEX EVENTS (DISCUSSION OF COROLLARY 4.5)

It is known from Poole’s independent choice logic [Poole 1993] that arbitrary cor-

relations between events can be composed from disjoint-independent events only. A

disjoint-independent event is represented by a non-Boolean independent random vari-

able ywhich takes either of kvalues v=hv1, . . . , vkiwith respective probabili-

ties q=hq1, . . . , qkiand Piqi= 1. Poole writes such a “disjoint declaration” as

y([v1:q1, . . . , vk:qk]).

In turn, any kdisjoint events can be represented starting from k−1independent

Boolean variables z=hz1, . . . , zk−1iand probabilities P[z] = hq1,q2

¯q1,q3

¯q1¯q2,..., qk−1

¯q1...¯qk−2i,

by assigning the disjoint-independent event variable yits value viwhenever event Ai

is true, with Aideﬁned as:

(y=v1)≡A1:−z1

(y=v2)≡A2:−¯z1z2

(y=vk−1)≡Ak1:−z1. . . ¯zk−2zk−1

(y=vk)≡Ak:−¯z1. . . ¯zk−2¯zk−1.

For example, a disjoint-independent event y(v1:1

5, v2:1

2, v3:1

5, v4:1

10 )can be represented

with three independent Boolean variables z= (z1, z2, z3)and P[z]=(1

5,5

8,2

3).

It follows that arbitrary correlations between events can be modeled starting from

independent Boolean random variables alone. For example, two complex events Aand

Bwith P[A] = P[B] = qand varying correlation (see Sect. 7.1) can be represented as

composed events A:−z1z2∨z3∨z4and B:−¯z1z2∨z3∨z5over the primitive events z

with varying probabilities P[z]. Events Aand Bbecome identical for P[z] = (0,0, q, 0,0),

independent for P[z] = (0,0,0, q, q), and disjoint for P[z] = (0.5, q, 0,0,0) with q≤0.5.

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1:34 W. Gatterbauer and D. Suciu

C. USER-DEFINED AGGREGATE IOR (SEC. 6 AND SEC. 7.5)

Here we show the User-deﬁned Aggregate (UDA) IOR in PostgreSQL:

create or replace function ior sfunc(ﬂoat, ﬂoat) returns ﬂoat as

’select $1 * (1.0 - $2)’

language SQL;

create or replace function ior ﬁnalfunc(ﬂoat) returns ﬂoat as

’select 1.0 - $1’

language SQL;

create aggregate ior (ﬂoat)(

sfunc = ior sfunc,

stype = ﬂoat,

ﬁnalfunc = ior ﬁnalfunc,

initcond = ’1.0’);

ACM Transactions on Database Systems, Vol. V, No. N, Article 1, Publication date: January YYYY.

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Model counting is a fundamental problem in many practical applications, including query evaluation in probabilistic databases and failure-probability estimation of networks. In this work, we focus on a variant of this problem where the underlying formula is expressed in the Disjunctive Normal Form (DNF), also known as #DNF. This problem has been shown to be #P-complete, making it often intractable to solve exactly. Much research has therefore focused on obtaining approximate solutions, particularly in the form of

(\varepsilon, \delta)

approximations. The primary contribution of this paper is a new approach, called pepin, an approximate #DNF counter that significantly outperforms prior state-of-the-art approaches. Our work is based on the recent breakthrough in the context of the union of sets in the streaming model. We demonstrate the effectiveness of our approach through extensive experiments and show that it provides an affirmative answer to the challenge of efficiently computing #DNF.

Distributionally robust, skeptical inferences in supervised classification using imprecise probabilities

Thesis

Full-text available

Dec 2020

Yonatan Carlos Carranza Alarcón

Decision makers are often faced with making single hard decisions, without having any knowledge of the amount of uncertainties contained in them, and taking the risk of making damaging, if not dramatic, mistakes. In such situations, where the uncertainty is higher due to imperfect information, it may be useful to provide set-valued but more reliable decisions. This works thus focuses on making distributionally robust, skeptical inferences (or decisions) in supervised classification problems using imprecise probabilities. By distributionally robust, we mean that we consider a set of possible probability distributions, i.e. imprecise probabilities, and by skeptical we understand that we consider as valid only those inferences that are true for every distribution within this set. Specifically, we focus on extending the Gaussian discriminant analysis and multilabel classification approaches to the imprecise probabilistic setting. Regarding to Gaussian discriminant analysis, we extend it by proposing a new imprecise classifier, considering the imprecision as part of its basic axioms, based on robust Bayesian analysis and near-ignorance priors. By including an imprecise component in the model, our proposal highlights those hard instances on which the precise model makes mistakes in order to provide cautious decisions in the form of set-valued class, instead. Regarding to multi-label classification, we first focus on reducing the time complexity of making a cautious decision over its output space of exponential size by providing theoretical justifications and efficient algorithms applied to the Hamming loss. Relaxing the assumption of independence on labels, we obtain partial decisions, i.e. not classifying at all over some labels, which generalize the binary relevance approach by using imprecise marginal distributions. Secondly, we extend the classifierchains approach by proposing two different strategies to handle imprecise probabilityestimates, and a new dynamic, context-dependent label ordering which dynamically selects the labels with low uncertainty as the chain moves forwards.

Relational Diagrams and the Pattern Expressiveness of Relational Languages

Article

Apr 2025
SIGMOD REC

Comparing relational languages by their logical expressiveness is well understood. Less understood is how to compare relational languages by their ability to represent relational query patterns. Indeed, what are query patterns other than ''a certain way of writing a query''? And how can query patterns be defined across procedural and declarative languages, irrespective of their syntax? Our SIGMOD 2024 paper proposes a semantic definition of relational query patterns that uses a variant of structurepreserving mappings between the relational tables of queries. This formalism allows us to analyze the relative pattern expressiveness of relational languages. Notably, for the nondisjunctive language fragment, we show that relational calculus (RC) can express a larger class of patterns than the basic operators of relational algebra (RA). We also propose Relational Diagrams, a complete and sound diagrammatic representation of safe relational calculus. These diagrams can represent all query patterns for unions of non-disjunctive queries, in contrast to visual query representations that derive visual marks from the basic operators of algebra. Our anonymously preregistered user study shows that Relational Diagrams allow users to recognize relational patterns meaningfully faster and more accurately than they can with SQL.

Minimally Factorizing the Provenance of Self-join Free Conjunctive Queries

Article

May 2024

We consider the problem of finding the minimal-size factorization of the provenance of self-join-free conjunctive queries, i.e.,we want to find a formula that minimizes the number of variable repetitions. This problem is equivalent to solving the fundamental Boolean formula factorization problem for the restricted setting of the provenance formulas of self-join free queries. While general Boolean formula minimization is Σ p 2 -complete, we show that the problem is NP-Complete in our case. Additionally, we identify a large class of queries that can be solved in PTIME, expanding beyond the previously known tractable cases of read-once formulas and hierarchical queries. We describe connections between factorizations, Variable Elimination Orders (VEOs), and minimal query plans. We leverage these insights to create an Integer Linear Program (ILP) that can solve the minimal factorization problem exactly. We also propose a Max-Flow Min-Cut (MFMC) based algorithm that gives an efficient approximate solution. Importantly, we show that both the Linear Programming (LP) relaxation of our ILP, and our MFMC-based algorithm are always correct for all currently known PTIME cases. Thus, we present two unified algorithms (ILP and MFMC) that can both recover all known PTIME cases in PTIME, yet also solve NP-Complete cases either exactly (ILP) or approximately (MFMC), as desired.

On The Reasonable Effectiveness of Relational Diagrams: Explaining Relational Query Patterns and the Pattern Expressiveness of Relational Languages

Article

Mar 2024

Comparing relational languages by their logical expressiveness is well understood. Less well understood is how to compare relational languages by their ability to represent relational query patterns. Indeed, what are query patterns other than "a certain way of writing a query"? And how can query patterns be defined across procedural and declarative languages, irrespective of their syntax? To the best of our knowledge, we provide the first semantic definition of relational query patterns by using a variant of structure-preserving mappings between the relational tables of queries. This formalism allows us to analyze the relative pattern expressiveness of relational language fragments and create a hierarchy of languages with equal logical expressiveness yet different pattern expressiveness. Notably, for the non-disjunctive language fragment, we show that relational calculus can express a larger class of patterns than the basic operators of relational algebra. Our language-independent definition of query patterns opens novel paths for assisting database users. For example, these patterns could be leveraged to create visual query representations that faithfully represent query patterns, speed up interpretation, and provide visual feedback during query editing. As a concrete example, we propose Relational Diagrams, a complete and sound diagrammatic representation of safe relational calculus that is provably (i) unambiguous, (ii) relationally complete, and (iii) able to represent all query patterns for unions of non-disjunctive queries. Among all diagrammatic representations for relational queries that we are aware of, ours is the only one with these three properties. Furthermore, our anonymously preregistered user study shows that Relational Diagrams allow users to recognize patterns meaningfully faster and more accurately than SQL.

Probabilistic Reasoning at Scale: Trigger Graphs to the Rescue

Article

May 2023

The role of uncertainty in data management has become more prominent than ever before, especially because of the growing importance of machine learning-driven applications that produce large uncertain databases. A well-known approach to querying such databases is to blend rule-based reasoning with uncertainty. However, techniques proposed so far struggle with large databases. In this paper, we address this problem by presenting a new technique for probabilistic reasoning that exploits Trigger Graphs (TGs) -- a notion recently introduced for the non-probabilistic setting. The intuition is that TGs can effectively store a probabilistic model by avoiding an explicit materialization of the lineage and by grouping together similar derivations of the same fact. Firstly, we show how TGs can be adapted to support the possible world semantics. Then, we describe techniques for efficiently computing a probabilistic model and formally establish the correctness of our approach. We also present an extensive empirical evaluation using a prototype called LTGs. Our comparison against other leading engines shows that LTGs is not only faster, even against approximate reasoning techniques, but can also reason over probabilistic databases that existing engines cannot scale to.

Inference and Decision in Credal Occupancy Grids: Use Case on Trajectory Planning

Article

Jul 2021

Occupancy grids are common tools used in robotics to represent the robot environment, and that may be used to plan trajectories, select additional measurements to acquire, etc. However, deriving information about those occupancy grids from sensor measurements often induce a lot of uncertainty, especially for grid elements that correspond to occluded or far away area from the robot. This means that occupancy information may be quite uncertain and imprecise at some places, while being very accurate at others. Modelling finely this occupancy information is essential to decide the optimal action the robot should take, but a refined modelling of uncertainty often implies a higher computational cost, a prohibitive feature for real-time applications. In this paper, we introduce the notion of credal occupancy grids, using the very general theory of imprecise probabilities to model occupancy uncertainty. We also show how one can perform efficient, real-time inferences with such a model, and show a use-case applying the model to an autonomous vehicle trajectory planning problem.

Optimization bounds from binary decision diagrams (extended abstract)

Conference Paper

Full-text available

Sep 2014
Lect Notes Comput Sci

Bounds on the optimal value are often indispensable for the practical solution of discrete optimization problems, particularly in the branching procedures used by constraint programming (CP) and integer programming solvers. Such bounds are frequently obtained by solving a continuous relaxation of the problem, perhaps a linear programming (LP) relaxation. In this paper, we explore an alternative strategy of obtaining bounds from a discrete relaxation, namely a binary decision diagram (BDD). Such a strategy is particularly suitable for CP, because BDDs provide enhanced propagation as well [2-5].

Structured Message Passing

Article

Full-text available

Sep 2013

In this paper, we present structured message passing (SMP), a unifying framework for approximate inference algorithms that take advantage of structured representations such as algebraic decision diagrams and sparse hash tables. These representations can yield significant time and space savings over the conventional tabular representation when the message has several identical values (context-specific independence) or zeros (determinism) or both in its range. Therefore, in order to fully exploit the power of structured representations, we propose to artificially introduce context-specific independence and determinism in the messages. This yields a new class of powerful approximate inference algorithms which includes popular algorithms such as cluster-graph Belief propagation (BP), expectation propagation and particle BP as special cases. We show that our new algorithms introduce several interesting bias-variance trade-offs. We evaluate these trade-offs empirically and demonstrate that our new algorithms are more accurate and scalable than state-of-the-art techniques.

Optimization Bounds from Binary Decision Diagrams

Article

Full-text available

Jan 2013
INFORMS J COMPUT

We explore the idea of obtaining bounds on the value of an optimization problem from a discrete relaxation based on binary decision diagrams (BDDs). We show how to construct a BDD that represents a relaxation of a 0-1 optimization problem, and how to obtain a bound for a separable objective function by solving a shortest (or longest) path problem in the BDD. As a test case we apply the method to the maximum independent set problem on a graph. We find that for most problem instances, it delivers tighter bounds in less computation time, than state-of-the-art integer programming software obtains by solving a continuous relaxation augmented with cutting planes.

An Introduction to Probability Theory and its Applications.

Article

Jan 1951
J Roy Stat Soc

Approximate Confidence Computation in Probabilistic Databases.

Article

Jan 2010

This paper introduces a deterministic approximation algorithm with error guarantees for computing the probability of propositional formulas over discrete random variables. The algorithm is based on an incremental compilation of formulas into decision diagrams using three types of decompositions: Shannon expansion, independence partitioning, and product factorization. With each decomposition step, lower and upper bounds on the probability of the partially compiled formula can be quickly computed and checked against the allowed error. This algorithm can be effectively used to compute approximate confidence values of answer tuples to positive relational algebra queries on general probabilistic databases (c-tables with discrete probability distributions). We further tune our algorithm so as to capture all known tractable conjunctive queries without selfjoins on tuple-independent probabilistic databases: In this case, the algorithm requires time polynomial in the input size even for exact computation. We implemented the algorithm as an extension of the SPROUT query engine. An extensive experimental effort shows that it consistently outperforms state-of-art approximation techniques by several orders of magnitude.

SPROUT: Lazy vs. Eager Query Plans for Tuple-Independent Probabilistic Databases

Article

Jan 2009

On repetition-free Boolean functions

Article

Jan 1977

Vladimir Alexander Gurvich

An Introduction to Probability Theory and Its Applications

Article

Jan 1972
J Roy Stat Soc

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Proceedings of the International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR)

Book

Jan 2009

Relax then Compensate: On Max-ProductBeliefPropagation andMore

Article

Jan 2009

We introduce a new perspective on approximations to the maximum a posteriori (MAP) task in probabilistic graphical models, that is based on simplifying a given instance, and then tightening the approximation. First, we start with a structural relaxation of the original model. We then infer from the relaxation its deficien-cies, and compensate for them. This perspective allows us to identify two distinct classes of approximations. First, we find that max-product belief propagation can be viewed as a way to compensate for a relaxation, based on a particular idealized case for exactness. We identify a second approach to compensation that is based on a more refined idealized case, resulting in a new approximation with distinct properties. We go on to propose a new class of algorithms that, starting with a relaxation, iteratively seeks tighter approximations.

Oblivious Bounds on the Probability of Boolean Functions

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