A dichotomy in the Complexity of Counting Database Repairs
A dichotomy in the Complexity of Counting Database 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 ]P The class ]P contains all function problems which consist of counting the number of accepting computation paths of a non-deterministic polynomial-time Turing machine.
The complexity class FP The complexity class FP contains all counting problems in ]P which can be solved in deterministic polynomial time. SAT is in NP : Are there any variable assignments that satisfy a given Boolean formula ? ]SAT is in ]P : How many variable assignments satisfy a given Boolean formula ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
3 / 13
Complexity classes The complexity class ]P The class ]P contains all function problems which consist of counting the number of accepting computation paths of a non-deterministic polynomial-time Turing machine.
The complexity class FP The complexity class FP contains all counting problems in ]P which can be solved in deterministic polynomial time. SAT is in NP : Are there any variable assignments that satisfy a given Boolean formula ? ]SAT is in ]P : How many variable assignments satisfy a given Boolean formula ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
3 / 13
Complexity classes The complexity class ]P The class ]P contains all function problems which consist of counting the number of accepting computation paths of a non-deterministic polynomial-time Turing machine.
The complexity class FP The complexity class FP contains all counting problems in ]P which can be solved in deterministic polynomial time. SAT is in NP : Are there any variable assignments that satisfy a given Boolean formula ? ]SAT is in ]P : How many variable assignments satisfy a given Boolean formula ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
3 / 13
The Complexity Dichotomy Effective FP-]P dichotomy A class C of counting problems exhibits an effective FP-]P-dichotomy if all problems in C are either in FP or ]P-hard under polynomial-time Turing reduction and it is decidable whether a given problem in C is in FP or ]P-hard.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
4 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
D. Maslowski (UMONS)
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
DBDBD 2014
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf
Year
Town
EDBT
2016
Brussels
S
Town Mons Brussels Belgrade
Country Belgium Belgium Belgium
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
3 × 2 repairs
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
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
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
R
S
Conf
Year
Town
EDBT
2016
Belgrade
Town Mons Brussels
Country Belgium Belgium
Belgrade
Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
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
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
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
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
q1 = ∃x(R(EDBT, 2016, x) ∧ S(x, Belgium)) q2 = ∃x∃y (R(EDBT, 2016, x) ∧ S(x, y ) ∧ T (y , Europe)) q1 is true in 5/6 repairs, q2 in 6/6 repairs. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
R
Conf
Year
Town
EDBT
2016
Belgrade
S
Town Mons Brussels
Country Belgium Belgium
Belgrade
Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
q1 = ∃x(R(EDBT, 2016, x) ∧ S(x, Belgium)) q2 = ∃x∃y (R(EDBT, 2016, x) ∧ S(x, y ) ∧ T (y , Europe)) q1 is true in 5/6 repairs, q2 in 6/6 repairs. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
q1 = ∃x(R(EDBT, 2016, x) ∧ S(x, Belgium)) q2 = ∃x∃y (R(EDBT, 2016, x) ∧ S(x, y ) ∧ T (y , Europe)) q1 is true in 5/6 repairs, q2 in 6/6 repairs. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Block-Independent-Disjoint Probabilistic Database R
R
conf ICDT KDD
rank A A
conf
rank
ICDT ICDT KDD KDD
A A A B
frequency biennial annual
frequency biennial annual annual annual R
Possible world w1 with P(w1 ) = 0.3 × 0.5 = 0.15
conf KDD
P 0.3 0.6 0.5 0.5 rank B
frequency annual
Possible world w2 with 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 given Block-Independent-Disjoint Database, compute P(q). The class of problems PROBABID (q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
10 / 13
Block-Independent-Disjoint Probabilistic Database R
R
conf ICDT KDD
rank A A
conf
rank
ICDT ICDT KDD KDD
A A A B
frequency biennial annual
frequency biennial annual annual annual R
Possible world w1 with P(w1 ) = 0.3 × 0.5 = 0.15
conf KDD
P 0.3 0.6 0.5 0.5 rank B
frequency annual
Possible world w2 with 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 given Block-Independent-Disjoint Database, compute P(q). The class of problems PROBABID (q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
10 / 13
Uncertain Databases vs Probabilistic Databases R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade T
P 1/3 1/3 1/3
Country Belgium Serbia Tunisia
S
Town Mons Brussels Belgrade Belgrade Cont P Europe 1 Europe 1 Africa 1
Country Belgium Belgium Belgium Serbia
P 1 1 1/2 1/2
Difference about the models Uniform probability for tuples in a same block. Probabilities in a block sum up to 1.
Difference about the dichotomy results There exists queries q such that ]CERTAINTY(q) is in P and PROBABID (q) is ]P-hard. No result exist about queries with self-joins in BID databases model. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
11 / 13
Uncertain Databases vs Probabilistic Databases R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade T
P 1/3 1/3 1/3
Country Belgium Serbia Tunisia
S
Town Mons Brussels Belgrade Belgrade Cont P Europe 1 Europe 1 Africa 1
Country Belgium Belgium Belgium Serbia
P 1 1 1/2 1/2
Difference about the models Uniform probability for tuples in a same block. Probabilities in a block sum up to 1.
Difference about the dichotomy results There exists queries q such that ]CERTAINTY(q) is in P and PROBABID (q) is ]P-hard. No result exist about queries with self-joins in BID databases model. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
11 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
13 / 13
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 ]P The class ]P contains all function problems which consist of counting the number of accepting computation paths of a non-deterministic polynomial-time Turing machine.
The complexity class FP The complexity class FP contains all counting problems in ]P which can be solved in deterministic polynomial time. SAT is in NP : Are there any variable assignments that satisfy a given Boolean formula ? ]SAT is in ]P : How many variable assignments satisfy a given Boolean formula ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
3 / 13
Complexity classes The complexity class ]P The class ]P contains all function problems which consist of counting the number of accepting computation paths of a non-deterministic polynomial-time Turing machine.
The complexity class FP The complexity class FP contains all counting problems in ]P which can be solved in deterministic polynomial time. SAT is in NP : Are there any variable assignments that satisfy a given Boolean formula ? ]SAT is in ]P : How many variable assignments satisfy a given Boolean formula ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
3 / 13
Complexity classes The complexity class ]P The class ]P contains all function problems which consist of counting the number of accepting computation paths of a non-deterministic polynomial-time Turing machine.
The complexity class FP The complexity class FP contains all counting problems in ]P which can be solved in deterministic polynomial time. SAT is in NP : Are there any variable assignments that satisfy a given Boolean formula ? ]SAT is in ]P : How many variable assignments satisfy a given Boolean formula ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
3 / 13
The Complexity Dichotomy Effective FP-]P dichotomy A class C of counting problems exhibits an effective FP-]P-dichotomy if all problems in C are either in FP or ]P-hard under polynomial-time Turing reduction and it is decidable whether a given problem in C is in FP or ]P-hard.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
4 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
D. Maslowski (UMONS)
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
DBDBD 2014
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf
Year
Town
EDBT
2016
Brussels
S
Town Mons Brussels Belgrade
Country Belgium Belgium Belgium
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Uncertain database Definition A database in which primary keys need not be satisfied.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
3 × 2 repairs
Repair (or possible world) A maximal subset of tuples that satisfy primary keys.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
5 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
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
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
R
S
Conf
Year
Town
EDBT
2016
Belgrade
Town Mons Brussels
Country Belgium Belgium
Belgrade
Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
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
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
Certainty Certain Boolean query A Boolean query is certain if it evaluates to true on each repair.
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
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
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
6 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem CERTAINTY(q) Definition For 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 Wijsen in [Wij12]. A dichotomy P/co-NP-complete of CERTAINTY(q) has been studied by Pema and Kolaitis in [KP12] and by Koutris and Suciu in [KS12].
Remark All complexity results concern data complexity
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
7 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
q1 = ∃x(R(EDBT, 2016, x) ∧ S(x, Belgium)) q2 = ∃x∃y (R(EDBT, 2016, x) ∧ S(x, y ) ∧ T (y , Europe)) q1 is true in 5/6 repairs, q2 in 6/6 repairs. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
R
Conf
Year
Town
EDBT
2016
Belgrade
S
Town Mons Brussels
Country Belgium Belgium
Belgrade
Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
q1 = ∃x(R(EDBT, 2016, x) ∧ S(x, Belgium)) q2 = ∃x∃y (R(EDBT, 2016, x) ∧ S(x, y ) ∧ T (y , Europe)) q1 is true in 5/6 repairs, q2 in 6/6 repairs. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
The problem ]CERTAINTY(q) Definition For a fixed Boolean query q, the counting problem ]CERTAINTY(q) is : INPUT An uncertain database db. OUTPUT How many repairs of db satisfy q ? ?
R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade
S
Town Mons Brussels Belgrade Belgrade
Country Belgium Belgium Belgium Serbia
T
Country Belgium Serbia Tunisia
Cont Europe Europe Africa
q1 = ∃x(R(EDBT, 2016, x) ∧ S(x, Belgium)) q2 = ∃x∃y (R(EDBT, 2016, x) ∧ S(x, y ) ∧ T (y , Europe)) q1 is true in 5/6 repairs, q2 in 6/6 repairs. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
8 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Contributions Complexity result [MW13] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy
Complexity result [MW14] The class of problems ]CERTAINTY(q), where q is a Boolean conjunctive query (possibly with self-joins) in which all primary keys consist 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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
9 / 13
Block-Independent-Disjoint Probabilistic Database R
R
conf ICDT KDD
rank A A
conf
rank
ICDT ICDT KDD KDD
A A A B
frequency biennial annual
frequency biennial annual annual annual R
Possible world w1 with P(w1 ) = 0.3 × 0.5 = 0.15
conf KDD
P 0.3 0.6 0.5 0.5 rank B
frequency annual
Possible world w2 with 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 given Block-Independent-Disjoint Database, compute P(q). The class of problems PROBABID (q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
10 / 13
Block-Independent-Disjoint Probabilistic Database R
R
conf ICDT KDD
rank A A
conf
rank
ICDT ICDT KDD KDD
A A A B
frequency biennial annual
frequency biennial annual annual annual R
Possible world w1 with P(w1 ) = 0.3 × 0.5 = 0.15
conf KDD
P 0.3 0.6 0.5 0.5 rank B
frequency annual
Possible world w2 with 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 given Block-Independent-Disjoint Database, compute P(q). The class of problems PROBABID (q), where q is a Boolean conjunctive query without self-join, exhibits an effective FP-]P-dichotomy.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
10 / 13
Uncertain Databases vs Probabilistic Databases R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade T
P 1/3 1/3 1/3
Country Belgium Serbia Tunisia
S
Town Mons Brussels Belgrade Belgrade Cont P Europe 1 Europe 1 Africa 1
Country Belgium Belgium Belgium Serbia
P 1 1 1/2 1/2
Difference about the models Uniform probability for tuples in a same block. Probabilities in a block sum up to 1.
Difference about the dichotomy results There exists queries q such that ]CERTAINTY(q) is in P and PROBABID (q) is ]P-hard. No result exist about queries with self-joins in BID databases model. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
11 / 13
Uncertain Databases vs Probabilistic Databases R
Conf EDBT EDBT EDBT
Year 2016 2016 2016
Town Mons Brussels Belgrade T
P 1/3 1/3 1/3
Country Belgium Serbia Tunisia
S
Town Mons Brussels Belgrade Belgrade Cont P Europe 1 Europe 1 Africa 1
Country Belgium Belgium Belgium Serbia
P 1 1 1/2 1/2
Difference about the models Uniform probability for tuples in a same block. Probabilities in a block sum up to 1.
Difference about the dichotomy results There exists queries q such that ]CERTAINTY(q) is in P and PROBABID (q) is ]P-hard. No result exist about queries with self-joins in BID databases model. D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
11 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
Conclusion Summary Uncertain Database Model ; Conjunctive queries without self-join ; Conjunctive queries in which all primary keys consist of a single attribute ; 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 with self-joins - to BID databases ? Does the class of problems ]CERTAINTY(q), where q is an union of conjunctive queries, exhibit an effective FP-]P-dichotomy ? D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
12 / 13
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.
D. Maslowski (UMONS)
DBDBD 2014
October 17th, 2014
13 / 13
