Consistent Query Answering
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- Giles Chambers
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1 Consistent Query Answering Opportunities and Limitations Jan Chomicki Dept. CSE University at Buffalo State University of New York chomicki 1
2 Integrity constraints Integrity constraints describe valid database instances. Examples: functional dependencies (FDs): every employee has a single salary. denial constraints: no employee can make more than her manager. inclusion dependencies (INDs): managers have to be employees. The constraints are formulated in first-order logic: n, s, m, s, m. [Emp(n, s, m) Emp(m, s, m ) s > s ]. An inconsistent database violates the constraints. 2
3 Integrity constraints Integrity constraints describe valid database instances. Examples: functional dependencies (FDs): every employee has a single salary. denial constraints: no employee can make more than her manager. inclusion dependencies (INDs): managers have to be employees. The constraints are formulated in first-order logic: n, s, m, s, m. [Emp(n, s, m) Emp(m, s, m ) s > s ]. An inconsistent database violates the constraints. 3
4 Integrity constraints Integrity constraints describe valid database instances. Examples: functional dependencies (FDs): every employee has a single salary. denial constraints: no employee can make more than her manager. inclusion dependencies (INDs): managers have to be employees. The constraints are formulated in first-order logic: n, s, m, s, m. [Emp(n, s, m) Emp(m, s, m ) s > s ]. An inconsistent database violates the constraints. 4
5 Traditional view Integrity constraints are always enforced. Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary This instance cannot arise but... consider data integration. 5
6 Traditional view Integrity constraints are always enforced. Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary This instance cannot arise but... consider data integration. 6
7 Traditional view Integrity constraints are always enforced. Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary This instance cannot arise but... consider data integration. 7
8 Ignoring inconsistency SELECT * FROM Emp WHERE Salary < 25M B. Gates Redmond, WA 20M A. Grove Santa Clara, CA 10M The result is not fully reliable. 8
9 Ignoring inconsistency SELECT * FROM Emp WHERE Salary < 25M B. Gates Redmond, WA 20M A. Grove Santa Clara, CA 10M The result is not fully reliable. 9
10 Quarantining inconsistency The facts involved in an inconsistency are not used in the derivation of query answers [Bry, IICIS 97]. SELECT * FROM Emp WHERE Salary < 25M A. Grove Santa Clara, CA 10M But what about SELECT EmpName FROM Emp WHERE Salary > 10K A. Grove Partial information cannot be obtained. 10
11 Quarantining inconsistency The facts involved in an inconsistency are not used in the derivation of query answers [Bry, IICIS 97]. SELECT * FROM Emp WHERE Salary < 25M A. Grove Santa Clara, CA 10M But what about SELECT EmpName FROM Emp WHERE Salary > 10K A. Grove Partial information cannot be obtained. 11
12 Quarantining inconsistency The facts involved in an inconsistency are not used in the derivation of query answers [Bry, IICIS 97]. SELECT * FROM Emp WHERE Salary < 25M A. Grove Santa Clara, CA 10M But what about SELECT EmpName FROM Emp WHERE Salary > 10K A. Grove Partial information cannot be obtained. 12
13 Quarantining inconsistency The facts involved in an inconsistency are not used in the derivation of query answers [Bry, IICIS 97]. SELECT * FROM Emp WHERE Salary < 25M A. Grove Santa Clara, CA 10M But what about SELECT EmpName FROM Emp WHERE Salary > 10K A. Grove Partial information cannot be obtained. 13
14 Constraints with exceptions Weaken the constraints [Borgida, TODS 85], without affecting query evaluation. Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Constraint with exception: ( x, y, z, y, z ) [Emp(x, y, z) Emp(x, y, z ) y y x B.Gates] 14
15 Constraints with exceptions Weaken the constraints [Borgida, TODS 85], without affecting query evaluation. Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Constraint with exception: ( x, y, z, y, z ) [Emp(x, y, z) Emp(x, y, z ) y y x B.Gates] 15
16 Data cleaning Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary Repair: EmpName Address Salary B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Some information is lost. 16
17 Data cleaning Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary Repair: EmpName Address Salary B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Some information is lost. 17
18 Data cleaning Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary Repair: EmpName Address Salary B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Some information is lost. 18
19 CQA: a query-driven approach Consider all repairs: possible databases that result from fixing the original database in a minimal way. Return all the answers that belong to the result of query evaluation in every repair (consistent answers). 19
20 Emp EmpName Address Salary B. Gates Redmond, WA 20M B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M Functional dependency: EmpN ame Address Salary Repairs: B. Gates Redmond, WA 30M A. Grove Santa Clara, CA 10M B. Gates Redmond, WA 20M A. Grove Santa Clara, CA 10M 20
21 SELECT * FROM Emp WHERE Salary < 25M A. Grove Santa Clara, CA 10M But SELECT EmpName FROM Emp WHERE Salary > 10K B. Gates A. Grove 21
22 SELECT * FROM Emp WHERE Salary < 25M A. Grove Santa Clara, CA 10M But SELECT EmpName FROM Emp WHERE Salary > 10K B. Gates A. Grove 22
23 Inconsistent databases There are many situations when users want/need to live with inconsistent databases: integration of heterogeneous databases with overlapping information not enough information to resolve inconsistencies preservation of all data (even erroneous) the consistency of the database will be restored by executing further transactions inconsistency wrt soft integrity constraints (those that we hope to see satisfied but do not/cannot check) process 23
24 Research goals Formal definition of reliable ( consistent ) information in an inconsistent database. Computational mechanisms for obtaining consistent information. Computational complexity analysis: tractable vs. intractable classes of queries and integrity constraints trade-off: complexity vs. expressiveness. Implementation: preferably using DBMS technology. 24
25 Plan of the talk 1. repairs and consistent query answers 2. computing consistent query answers to relational algebra/calculus/sql queries 3. extensions and new directions: other notions of repair probabilistic databases 4. further active research directions 5. related work 6. lessons of CQA research 7. the CQA community 25
26 Constraint classes Universal:. A 1 A n B 1 B m. Denial:. A 1 A n. Functional dependencies (FDs): X Y. Inclusion dependencies (INDs): P[X] R[Y ]. 26
27 Arenas, Bertossi, Ch. [PODS 99]. Consistent query answers Repair: a database that satisfies the integrity constraints difference from the given database is minimal (the set of inserted/deleted facts is minimal under set inclusion). A tuple (a 1,...,a n ) is a consistent query answer to a query Q(x 1,...,x n ) in a database r if it is an element of the result of Q in every repair of r. 27
28 A logical aside Belief revision: semantically: repairing revising the database with integrity constraints consistent query answers counterfactual inference. Logical inconsistency: inconsistent database: database facts together with integrity constraints form an inconsistent set of formulas trivialization of reasoning does not occur because constraints are not used in relational query evaluation. 28
29 Computational issues There are too many repairs to evaluate the query in each of them. A B a 1 b 1 a 1 b 1 a 2 b 2 a 2 b 2 a n a n b n b n Under the functional dependency A B, this instance has 2 n repairs. 29
30 Computing consistent query answers Query rewriting: given a query Q and a set of integrity constraints, construct a query Q such that for every database instance r the set of answers to Q in r = the set of consistent answers to Q in r. Representing all repairs: given a set of integrity constraints and a database instance r: 1. construct a space-efficient representation of all repairs of r 2. use this representation to answer (many) queries. Specifying repairs as logic programs. 30
31 Query rewriting First-order queries transformed using semantic query optimization techniques: [Arenas, Bertossi, Ch., PODS 99]. Functional dependencies: ( x)( y)( z)( y )( z )( Emp(x, y, z) Emp(x, y, z ) y = y ) ( x)( y)( z)( y )( z )( Emp(x, y, z) Emp(x, y, z ) z = z ) Query: Emp(x, y, z). Transformed query: Emp(x, y, z) ( y )( z )( Emp(x, y, z ) y = y ) ( y )( z )( Emp(x, y, z ) z = z ). 31
32 Query rewriting First-order queries transformed using semantic query optimization techniques: [Arenas, Bertossi, Ch., PODS 99]. Functional dependencies: ( x)( y)( z)( y )( z )( Emp(x, y, z) Emp(x, y, z ) y = y ) ( x)( y)( z)( y )( z )( Emp(x, y, z) Emp(x, y, z ) z = z ) Query: Emp(x, y, z). Transformed query: Emp(x, y, z) ( y )( z )( Emp(x, y, z ) y = y ) ( y )( z )( Emp(x, y, z ) z = z ). 32
33 Query rewriting First-order queries transformed using semantic query optimization techniques: [Arenas, Bertossi, Ch., PODS 99]. Functional dependencies: ( x)( y)( z)( y )( z )( Emp(x, y, z) Emp(x, y, z ) y = y ) ( x)( y)( z)( y )( z )( Emp(x, y, z) Emp(x, y, z ) z = z ) Query: Emp(x, y, z). Transformed query: Emp(x, y, z) ( y )( z )( Emp(x, y, z ) y = y ) ( y )( z )( Emp(x, y, z ) z = z ). 33
34 Scope of query rewriting Query rewriting: queries involving conjunctions of literals (relational algebra: σ,, ) and binary universal integrity constraints [Arenas, Bertossi, Ch., PODS 99]. existentially-quantified conjunctions (π, σ, ) and single-key dependencies [Fuxman, Miller, ICDT 05]: CTree queries ( no non-key joins) extended to exclusion dependencies [Grieco et al., CIKM 05]. 34
35 SELECT Name FROM Emp WHERE Salary > 10K SELECT Name FROM Emp e1 WHERE Salary > 10K AND NOT EXISTS (SELECT * FROM EMPLOYEE e2 WHERE e2.name = e1.name AND e2.salary <= 10K) 35
36 Experimental results (ConQuer) [Fuxman, Fazli, Miller, SIGMOD 05]. The system ConQuer: back-end: DB2 UDB. query rewriting into SQL, producing unnested queries queries from the TPC-H workload databases can be annotated with consistency indicators tested for synthetic databases with 400K 8M tuples, 0 50% conflicts relatively little overhead compared to evaluating the original query using the backend 36
37 Conflict hypergraph Denial constraints only. Vertices: facts in the original instance. Edges: (minimal) sets of facts that violate some constraint. Repairs: maximal independent sets. B. Gates Redmond, WA 20M A. Grove Santa Clara, CA 10M B. Gates Redmond, WA 30M 37
38 Computing CQAs using the conflict hypergraph [Ch., Marcinkowski, I&C, 2005]. Algorithm HProver: 1. input: query Φ a disjunction of ground atoms 2. Φ = P 1 (t 1 ) P m (t m ) P m+1 (t m+1 ) P n (t n ) 3. find a repair including P 1 (t 1 ),...,P m (t m ) and excluding P m+1 (t m+1 ),...,P n (t n ) by enumerating the appropriate edges. Excluding a fact A: A is not in the original instance, or A belongs to an edge {A,B 1,...,B k } in the conflict hypergraph and B 1,...,B k belong to the repair. P (data complexity) for quantifier-free queries and denial constraints. 38
39 Hippo Q : σ,,, Upper Envelope Translation Q : σ,, Q c :,, DB Evaluation Candidates Grounding Conflict Detection Conflict Graph HProver Consistent Answers 39
40 Experimental results (Hippo) [Ch., Marcinkowski, Staworko, CIKM 04]. The system Hippo: back-end: PostgreSQL conflict hypergraph (edges) in main memory optimization can eliminate many (sometimes all) database accesses in HProver tested for synthetic databases with up to 200K tuples, 2% conflicts computing consistent query answers using the conflict hypergraph faster than evaluating transformed queries relatively little overhead compared to evaluating the original query using the backend 40
41 Specifying repairs as logic programs [Arenas, Bertossi, Ch., TPLP 03], [Greco, Greco and Zumpano, TKDE 03], [Barcelo, Bertossi, PADL 03], [Eiter et al., ICLP 03]: using logic programs with negation and disjunction repairs answer sets several different encodings implemented using main-memory LP systems (dlv, smodels) Π p 2-complete problems Scope: arbitrary universal constraints, some inclusion dependencies arbitrary first-order queries 41
42 The approach of [Arenas, Bertossi, Ch., TPLP 03]. Facts: Emp( B.Gates, Redmond WA, 20K). Emp( B.Gates, Redmond WA, 30K). Emp( A.Grove, Santa Clara CA, 10K). Rules: Emp (x, y, z) Emp (x, y, z ) Emp(x, y, z), Emp(x, y, z ), y y. Emp (x, y, z) Emp (x, y, z ) Emp(x, y, z), Emp(x, y, z ), z z. Emp (x, y, z) Emp(x, y, z), not Emp (x, y, z). Emp (x, y, z) not Emp(x, y, z), not Emp (x, y, z). 42
43 [Eiter at al., ICLP 03]. Experimental data (INFOMIX) The system INFOMIX: combines CQA with data integration (GAV) relational backend: PostgreSQL uses the disjunctive LP system dlv for repair computations optimization techniques: localization, factorization tested on legacy databases of up to 50K tuples. 43
44 Data complexity of consistent query answers Keys / primary keys) Denial Universal σ,, P P P(binary) σ,,, P P? σ,π co-npc / P co-npc co-np-hard σ,, π co-npc / P(Ctree) co-npc co-np-hard Tractable approaches: query rewriting conflict graphs Sources: [Ch., Marcinkowski, I&C 05], [Fuxman, Miller, ICDT 05], [Cali, Lembo, Rosati, PODS 03]. 44
45 Tractable/intractable queries Tractable (P): under any denial constraints: SELECT * FROM P UNION (SELECT * FROM Q EXCEPT SELECT * FROM R) 45
46 Schema: CREATE TABLE P(A PRIMARY KEY, B); CREATE TABLE Q(C PRIMARY KEY, D) Tractable (P): SELECT Q.D FROM P, Q WHERE P.B = Q.C Intractable (co-np-complete): SELECT Q.D FROM P, Q WHERE P.B = Q.D 46
47 Schema: CREATE TABLE P(A PRIMARY KEY, B); CREATE TABLE Q(C PRIMARY KEY, D) Tractable (P): SELECT Q.D FROM P, Q WHERE P.B = Q.C Intractable (co-np-complete): SELECT Q.D FROM P, Q WHERE P.B = Q.D 47
48 Schema: CREATE TABLE P(A PRIMARY KEY, B); CREATE TABLE Q(C PRIMARY KEY, D) Tractable (P): SELECT Q.D FROM P, Q WHERE P.B = Q.C Intractable (co-np-complete): SELECT Q.D FROM P, Q WHERE P.B = Q.D 48
49 Alternative frameworks Different assumptions about database completeness and correctness (in the presence of inclusion dependencies): possibly incorrect but complete: repairs by deletion only [Ch., Marcinkowski, I&C 05] possibly incorrect and incomplete: fix FDs by deletion, INDs by insertion [Cali, Lembo, Rosati, PODS 03]. 49
50 Different notions of minimal repairs: minimal cardinality changes [Lopatenko, Bertossi, 06]: tractability of incremental CQA: given a consistent database D and a fixed sequence of updates U, what is the complexity of computing CQA over U(DB)? repairing attribute values. 50
51 Attribute-based repairs Several different approaches: (A) ground and non-ground repairs [Wijsen, TODS 05] (B) project-join repairs [Wijsen, FQAS 06] (C) repairs minimizing Euclidean distance [Bertossi et al., DBPL 05] (D) repairs of minimum cost [Bohannon et al., SIGMOD 05]. Computational complexity: (A) and (B): similar to tuple based repairs (C) and (D): checking existence of a repair of cost < K NP-complete. 51
52 Attribute-based repairs Several different approaches: (A) ground and non-ground repairs [Wijsen, TODS 05] (B) project-join repairs [Wijsen, FQAS 06] (C) repairs minimizing Euclidean distance [Bertossi et al., DBPL 05] (D) repairs of minimum cost [Bohannon et al., SIGMOD 05]. Computational complexity: (A) and (B): similar to tuple based repairs (C) and (D): checking existence of a repair of cost < K NP-complete. 52
53 Project-join repairs PJ-repairs: repairs of a lossless join decomposition. EmpDept Name Dept Location John Sales Buffalo Functional dependencies: Name Dept Dept Location Mary Sales Toronto Repairs: John Sales Buffalo Mary Sales Toronto 53
54 Project-join repairs PJ-repairs: repairs of a lossless join decomposition. EmpDept Name Dept Location John Sales Buffalo Functional dependencies: Name Dept Dept Location Mary Sales Toronto Repairs: John Sales Buffalo Mary Sales Toronto 54
55 Project-join repairs PJ-repairs: repairs of a lossless join decomposition. EmpDept Name Dept Location John Sales Buffalo Functional dependencies: Name Dept Dept Location Mary Sales Toronto Repairs: John Sales Buffalo Mary Sales Toronto 55
56 Decomposition π Name,Dept (EmpDept) π Dept,Location (EmpDept) Name Dept Location John Sales Buffalo John Sales Toronto Mary Sales Buffalo Mary Sales Toronto PJ-repairs: John Sales Buffalo Mary Sales Buffalo John Sales Toronto Mary Sales Toronto 56
57 Decomposition π Name,Dept (EmpDept) π Dept,Location (EmpDept) Name Dept Location John Sales Buffalo John Sales Toronto Mary Sales Buffalo Mary Sales Toronto PJ-repairs: John Sales Buffalo Mary Sales Buffalo John Sales Toronto Mary Sales Toronto 57
58 Decomposition π Name,Dept (EmpDept) π Dept,Location (EmpDept) Name Dept Location John Sales Buffalo John Sales Toronto Mary Sales Buffalo Mary Sales Toronto PJ-repairs: John Sales Buffalo Mary Sales Buffalo John Sales Toronto Mary Sales Toronto 58
59 Probabilistic framework for dirty databases [Andritsos, Fuxman, Miller, ICDE 06]. Framework: potential duplicates identified and grouped into clusters worlds repairs: one tuple from each cluster world probability: product of tuple probabilities clean answers: in the query result in some (supporting) world clean answer probability: sum of the probabilities of supporting worlds 59
60 EmpSal EmpName Salary prob Functional dependency: EmpName Salary B. Gates 20M 0.7 B. Gates 30M 0.3 A. Grove 10M 0.5 A. Grove 20M
61 Query rewriting: SELECT Name FROM Emp WHERE Salary > 15M SELECT Name, SUM(e.prob) FROM Emp e WHERE Salary > 15M GROUP BY Name Evaluation of the rewritten query: B. Gates 20M 0.7 B. Gates 30M 0.3 A. Grove 10M 0.5 A. Grove 20M 0.5 B. Gates 1 A. Grove
62 Query rewriting: SELECT Name FROM Emp WHERE Salary > 15M SELECT Name, SUM(e.prob) FROM Emp e WHERE Salary > 15M GROUP BY Name Evaluation of the rewritten query: B. Gates 20M 0.7 B. Gates 30M 0.3 A. Grove 10M 0.5 A. Grove 20M 0.5 B. Gates 1 A. Grove
63 Further active topics SQL queries: aggregation: glb/lub answers grouping Data integration: GAV, LAV, GLAV,... tension between repairing and satisfying source-to-target dependencies P2P: how to isolate an inconsistent peer 63
64 Nulls: repairs with nulls? clean semantics vs. SQL conformance Priorities: preferred repairs conflict resolution XML: what is an integrity constraint? what is a repair? minimum edit distance 64
65 Related work Belief revision: revising database with integrity constraints revised theory changes with each database update emphasis on semantics (AGM postulates), not computation complexity results [Eiter, Gottlob, AI 92] do not quite transfer beyond revision: merging, arbitration Disjunctive information: repair possible world (sometimes) consistent answer certain answer (sometimes) using disjunctions to represent resolved conflicts complexity results [Imielinski et al., JCSS 95] do not quite transfer 65
66 Lessons of CQA research Need to tame the semantic explosion: different repair semantics overwhelm the potential user More focus on applications: 99% technology, 1% applications often only repairing is needed heuristics Integrate with other tools: schema matching/mapping data cleaning 66
67 The CQA community Some data: over 30 active researchers publications (since 1999) papers in major conferences (PODS, SIGMOD, ICDT, ICDE, CIKM, DBPL) and journals (TODS, TKDE, TCS, I&C, AMAI, JAL, TPLP) outreach to the AI community (qualified success) yearly workshop: IIDB (+LAAIC?) 67
68 Selected papers 1. M. Arenas, L. Bertossi, J. Chomicki, Consistent Query Answers in Inconsistent Databases. PODS L. Bertossi, J. Chomicki, Query Answering in Inconsistent Databases. In Logics for Emerging Applications of Databases, J. Chomicki, R. van der Meyden, G. Saake [eds.], Springer-Verlag, J. Chomicki and J. Marcinkowski, On the Computational Complexity of Minimal-Change Integrity Maintenance in Relational Databases. In Inconsistency Tolerance, L. Bertossi, A. Hunter, T. Schaub, editors, Springer-Verlag, L. Bertossi, Consistent Query Answering in Databases. SIGMOD Record, June
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