A dichotomy in the Complexity of Counting Database Repairs · Complexity classes The complexity...

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A dichotomy in the Complexity of CountingDatabase Repairs

Dany Maslowski

Université de Mons, Belgium

October 17th, 2014

Outline

1 Complexity Classes and the Complexity Dichotomy

2 Uncertain Database Model

3 The problem ]CERTAINTY(q)

4 Probabilistic Database Model

5 Uncertain Databases vs Probabilistic Databases

6 Conclusion

D. Maslowski (UMONS) DBDBD 2014 October 17th, 2014 2 / 13

Complexity classes

The complexity class ]PThe class ]P contains all function problems which consist of countingthe number of accepting computation paths of a non-deterministicpolynomial-time Turing machine.

The complexity class FPThe complexity class FP contains all counting problems in ]P whichcan be solved in deterministic polynomial time.

SAT is in NP : Are there any variable assignments that satisfy a givenBoolean formula ?

]SAT is in ]P : How many variable assignments satisfy a givenBoolean formula ?

Complexity classes

The Complexity Dichotomy

Effective FP-]P dichotomyA class C of counting problems exhibits an effective FP-]P-dichotomyif all problems in C are either in FP or ]P-hard under polynomial-timeTuring reduction and it is decidable whether a given problem in C is inFP or ]P-hard.

Uncertain database

DefinitionA database in which primary keys need not be satisfied.

R Conf Year TownEDBT 2016 MonsEDBT 2016 BrusselsEDBT 2016 Belgrade

S Town CountryMons BelgiumBrussels BelgiumBelgrade BelgiumBelgrade Serbia

T Country ContBelgium EuropeSerbia EuropeTunisia Africa

3×2 repairs

Repair (or possible world)A maximal subset of tuples that satisfy primary keys.

Uncertain database

3×2 repairs

Uncertain database

3×2 repairs

Uncertain database

R Conf Year Town

EDBT 2016 Mons

EDBT 2016 Brussels

EDBT 2016 Belgrade

S Town CountryMons BelgiumBrussels BelgiumBelgrade Belgium

Belgrade Serbia

3×2 repairs

Uncertain database

3×2 repairs

Uncertain database

3×2 repairs

Certainty

Certain Boolean queryA Boolean query is certain if it evaluates to true on each repair.

q1 = ∃x(R(EDBT,2016,x)∧S(x ,Belgium))

q2 = ∃x∃y(R(EDBT,2016,x)∧S(x ,y)∧T (y ,Europe))

q1 is not certain, q2 is certain

Certainty

R Conf Year Town

EDBT 2016 MonsEDBT 2016 Brussels

EDBT 2016 Belgrade

S Town CountryMons BelgiumBrussels Belgium

Belgrade Belgium

Belgrade Serbia

Certainty

The problem CERTAINTY(q)

DefinitionFor a fixed Boolean query q, the problem CERTAINTY(q) is :

INPUT An uncertain database db.

OUTPUT Is q true in every repair of db ?

First-order expressibility of CERTAINTY(q) has been studied by Wijsenin [Wij12].

A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied byPema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].

RemarkAll complexity results concern data complexity

The problem ]CERTAINTY(q)

DefinitionFor a fixed Boolean query q, the counting problem ]CERTAINTY(q) is :

INPUT An uncertain database db.OUTPUT How many repairs of db satisfy q ? ?

q1 is true in 5/6 repairs, q2 in 6/6 repairs.

R Conf Year Town

EDBT 2016 MonsEDBT 2016 Brussels

EDBT 2016 Belgrade

S Town CountryMons BelgiumBrussels Belgium

Belgrade Belgium

Belgrade Serbia

Contributions

Complexity result [MW13]The class of problems ]CERTAINTY(q), where q is a Booleanconjunctive query without self-join, exhibits an effectiveFP-]P-dichotomy

Complexity result [MW14]The class of problems ]CERTAINTY(q), where q is a Booleanconjunctive query (possibly with self-joins) in which all primary keysconsist of a single attribute, exhibits an effective FP-]P-dichotomy.

if q = ∃x(R(EDBT,2016,x)∧S(x ,Belgium)), ]CERTAINTY(q) ∈ FP ;

if q = ∃x ,y(R(x ,y)∧R(y ,a)), ]CERTAINTY(q) is hard for ]P ;

the query ∃x ,y ,z(R(x ,y ,a)∧R(y , t ,a)) does not meet our criterions.

Contributions

Block-Independent-Disjoint Probabilistic DatabaseR conf rank frequency P

ICDT A biennial 0.3ICDT A annual 0.6KDD A annual 0.5KDD B annual 0.5

R conf rank frequencyICDT A biennialKDD A annual

R conf rank frequencyKDD B annual

Possible world w1 with Possible world w2 withP(w1) = 0.3×0.5 = 0.15 P(w2) = 0.1×0.5 = 0.05

Complexity result (Dalvi and Suciu [DS07])Let q be a Boolean query. The problem PROBABID(q) is : for a givenBlock-Independent-Disjoint Database, compute P(q).The class of problems PROBABID(q), where q is a Boolean conjunctive query withoutself-join, exhibits an effective FP-]P-dichotomy.

Block-Independent-Disjoint Probabilistic DatabaseR conf rank frequency P

ICDT A biennial 0.3ICDT A annual 0.6KDD A annual 0.5KDD B annual 0.5

R conf rank frequencyICDT A biennialKDD A annual

R conf rank frequencyKDD B annual

Possible world w1 with Possible world w2 withP(w1) = 0.3×0.5 = 0.15 P(w2) = 0.1×0.5 = 0.05

Complexity result (Dalvi and Suciu [DS07])Let q be a Boolean query. The problem PROBABID(q) is : for a givenBlock-Independent-Disjoint Database, compute P(q).The class of problems PROBABID(q), where q is a Boolean conjunctive query withoutself-join, exhibits an effective FP-]P-dichotomy.

Uncertain Databases vs Probabilistic DatabasesR Conf Year Town P

EDBT 2016 Mons 1/3EDBT 2016 Brussels 1/3EDBT 2016 Belgrade 1/3

S Town Country PMons Belgium 1Brussels Belgium 1Belgrade Belgium 1/2Belgrade Serbia 1/2

T Country Cont PBelgium Europe 1Serbia Europe 1Tunisia Africa 1

Difference about the modelsUniform probability for tuples in a same block.Probabilities in a block sum up to 1.

Difference about the dichotomy resultsThere exists queries q such that ]CERTAINTY(q) is in P andPROBABID(q) is ]P-hard.No result exist about queries with self-joins in BID databasesmodel.

Uncertain Databases vs Probabilistic DatabasesR Conf Year Town P

EDBT 2016 Mons 1/3EDBT 2016 Brussels 1/3EDBT 2016 Belgrade 1/3

S Town Country PMons Belgium 1Brussels Belgium 1Belgrade Belgium 1/2Belgrade Serbia 1/2

T Country Cont PBelgium Europe 1Serbia Europe 1Tunisia Africa 1

Difference about the modelsUniform probability for tuples in a same block.Probabilities in a block sum up to 1.

Difference about the dichotomy resultsThere exists queries q such that ]CERTAINTY(q) is in P andPROBABID(q) is ]P-hard.No result exist about queries with self-joins in BID databasesmodel.

ConclusionSummary

Uncertain Database Model ;

Conjunctive queries without self-join ;

Conjunctive queries in which all primary keys consist of a singleattribute ;

CERTAINTY(q) ]CERTAINTY(q) ;

Uncertain Database as a special case of Probabilistic Database ;

Effective FP-]P-dichotomies.

Open questions

Can we adapt our dichotomy result - that concerns queries withself-joins - to BID databases ?

Does the class of problems ]CERTAINTY(q), where q is an union ofconjunctive queries, exhibit an effective FP-]P-dichotomy ?

ConclusionSummary

Open questions

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Thank you for your attention !

Nilesh N. Dalvi and Dan Suciu.Management of probabilistic data : foundations and challenges.In Leonid Libkin, editor, PODS, pages 1–12. ACM, 2007.

Phokion G. Kolaitis and Enela Pema.A dichotomy in the complexity of consistent query answering for queries with two atoms.Inf. Process. Lett., 112(3) :77–85, 2012.

Paraschos Koutris and Dan Suciu.A dichotomy on the complexity of consistent query answering for atoms with simple keys.CoRR, abs/1212.6636, 2012.

Dany Maslowski and Jef Wijsen.A dichotomy in the complexity of counting database repairs.Journal of Computer and System Sciences, 79(6) :958 – 983, 2013.

Dany Maslowski and Jef Wijsen.Counting database repairs that satisfy conjunctive queries with self-joins.In 17th International Conference on Database Theory, pages 155–164, 2014.

Jef Wijsen.Certain conjunctive query answering in first-order logic.ACM Trans. Database Syst., 37(2) :9, 2012.

A dichotomy in the Complexity of Counting Database Repairs · Complexity classes The complexity...

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