Inconsistency-tolerant logics

Transcription

1 Inconsistency-tolerant logics CS 157 Computational Logic Autumn 2010

2 Inconsistent logical theories T 1 = { p(a), p(a) } T 2 = { x(p(x) q(x)), p(a), q(a) } Definition: A theory T is inconsistent if T has no models (no interpretation satisfies T) 11/15/10 2

3 Causes of inconsistency Errors Contradiction within software documentation Uncertainty and approximation Delay in update Semantic disagreement Price of a PS3 in the US vs. Canada Fundamental disagreements Birth year of Julius Caesar: 200 BC vs. 202 BC Fundamental inconsistency of the world 11/15/10 3

4 Trivialization Recall: T φ if and only if every model of T is a model of φ. Upshot: if T has no models, then the condition trivially holds for any sentence φ. Classical logic does not distinguish between inconsistent theories. Exercise: prove q from the premises p, p, using the Mendelson system. 11/15/10 4

5 Problems with inconsistency Wikipedia Mike plays Jack Bauer in 24 Microsoft knowledge base Steve Balmer hates Windows Wikipedia GDP of the US is greater than GDP of Canada We would like to distinguish between different sorts of consequences 11/15/10 5

6 Inconsistency-tolerant logics How can we preserve classical logic but prevent trivialization? Change the model theory e.g. many-valued logics Change the proof theory Change the rules of inference Change what constitutes a proof e.g. require that a proof must be self consistent 11/15/10 6

7 Relational Calculus Syntax

8 Database, Queries Database instance A finite set of ground atoms e.g., D = { p(a,a), p(b,c), q(a), r(a,d) } Query A relational calculus formula e.g., Q 1 = p(a,b) e.g., Q 2 (x) = y p(x,y) e.g., Q 3 (x) = z y ( p(x,y) r(y,z))

9 Database semantics D = { p(a,a), p(b,c), q(a), q(b) } D p(a,a)? D q(c)? D x p(x,x)? yes no yes D x (q(x) p(x,x))? no D x y (q(x) p(x,y))? yes

10 Database constraints Constraint set A finite, consistent set of relational calculus sentences e.g., C = { x ( p(x,x) q(x)), x q(x) }

11 Trivialization D = { p(a), p(b) } Constraints: { p(a) p(b) } If D C has no models, then D C φ for any φ But there is a very natural model to use: D D p(a) D p(c) Some absurd conclusions remain D p(a) p(b) Julius Caesar was born in 200 BC and in 202 BC

12 Min-change Repairs Given: D : database e.g.,{ p(a), p(b) } Ω : set of constraints e.g.,{ p(a) p(b) } A repair is a database D* that is consistent with Ω. e.g., { p(a) }, { p(b) }, { }, { p(a), p(c) },... A minimal change repair is a repair that is minimally different from D. minimal according to the relationship A \leq B if and only if (A D) (D A) \subseteq (B D) (D B)

13 Exercise Given: D : database e.g.,{ p(a), p(b) } Ω : set of constraints e.g.,{ p(a) p(b) } Which are the minimal change repairs? { p(a), p(b) } { p(a) } { p(b) } { } { p(a), p(c) }

14 Consistent query answers CQA: answers that arise no matter how the conflicts are resolved (in a minimal-change fashion) [ABC] D CQA Q(t) iff Ω for all minimal-change repairs D* of D w.r.t. to Ω, D* Q(t)

15 CQA Conflict p(a) p(b) Constraint: p(a) p(b) p(a)? NO {p(b)} p(b)? NO {p(a)} p(a) p(b)? NO {p(b)} p(a) p(b)? YES {p(a)}, {p(b)}

16 More lenient semantics In some situations, we would like to see plausible though non-guaranteed answers e.g., a student checking out whom his office mates might be Possible answers from repairs: answers that arise from some minimal-change repair

17 Possible answers Constraint: every person must be assigned a desk Data: Charlie is missing a desk assignment Minimal-change repairs: {desk(charles,desk1)}, {desk(charles,desk2)}, {desk(charles,desk3)}, Possible answers: desk1, desk2, desk3,

18 Relax not repair Instead of repairing a database to be consistent with constraints, we can relax a database to be consistent with constraints.

19 Incomplete Database An incomplete database is a theory K K consists of ground literals Pos(K) Neg(K) = {} e.g., K = { p(a), q(a), p(c), q(b) } Intuitively, Pos(K) gives the known true atoms Neg(K) gives the known false atoms The rest are unknown K d Q iff D d Q for all D Instances (K) d

20 Relaxations of a database A relaxation of D w.r.t. Ω is an incomplete database K, where Pos(K) D Neg(K) D = K d Ω Let the set of relaxations of D w.r.t. to Ω be denoted Re d (D,Ω)

21 Example D = {p(a),p(b)} Ω = { p(a) p(b)} d = {a,b,c} Re d (D,Ω) = { { p(a), p(c)}, {p(b), p(c)}, { p(a) }, {p(b)}, { p(c) }, { } }

22 Answers with Consistent Support ACS: answers that are supported by a consistent relaxation of the database. [KG] D CS Ω Q(t) iff there exists K Re d (D,Ω) st K d Q(t) Where d = adom(d,q)

23 ACS Conflict p(a) p(b) Constraint: p(a) p(b) p(a)? YES <{p(a)},{}> p(b)? YES <{p(b)},{}> p(a) p(b)? NO p(a) p(b)? YES <{p(a)},{}>

24 More complex queries p a 1 b 1 c 2 d 3 q a b r c d Constraint: xyz ( p(x,z) p(y,z) x=y) x [q(x) p(x,1)]? ACS: NO

25 Data Complexity Both ACS and CQA are NP-Hard problems [KG] [CM] Reduction from monotone 3-SAT ACS is tractable for * * queries. [KG] CQA is tractable for * queries [KG] Some subclasses of * queries [ABC, CM, FM] with suitable restrictions on the class of constraints

26 Query rewriting How can we use existing database infrastructure to support these query semantics? Solution: Query rewriting Algorithm ACS-Rewrite[Q,Ω] Rewrite Q ( * *) into Q' so that evaluating Q' on a standard DBMS gives ACS answers to Q. [KG] Similar rewriting algorithms have been devised for CQA [ABC, CM, FM, KG]

27 ACS: Ground conjunctive queries Q = p(a,1) p(b,1) C = { x1 x2 y ( p(x1,y) p(x2,y) x1=x2) } Q contradicts C. Answer: NO Rewrite: Q' = False

28 ACS: qf conjunctive queries Q(x1,y1,x2,y2) = p(x1,y1) p(x2,y2) C = { x1 x2 y ( p(x1,y) p(x2,y) x1=x2) } Q(x1,y1,x2,y2) is consistent w.r.t. C if and only if (x1=x2 y1 y2) Rewrite: Q' = Q(x1,x2,y1,y2) (x1=x2 y1 y2)

29 ACS: * queries Q = x,y ( q(x,y) r(x,y)) C ={ x q(x,x) } q(x,y) r(x,y) q(x,x) r(x,x) Rewrite: Q' = Q x r(x,x)

30 Ongoing work Identify other tractable classes Recursive queries and aggregates Other answer semantics preferentially-defeasible data? coarser granularity? Semantics that consider history?

31 References [ABC] M. Arenas, L. Bertossi, and J. Chomicki. Consistent query answers in inconsistent databases. Symposium on principles of database systems, Pages 68 79, [CM] J. Chomicki and J. Marcinkowski. Minimal-change integrity maintenance using tuple deletions. Inf. Comput.,197(1-2):90 121, [FM] A. Fuxman and R. J. Miller. First-order query rewriting for inconsistent databases. J. Comput. Syst. Sci., 73(4): , [KG] E. Kao and M. Genesereth. Answers with consistent support complexity and rewriting. Working paper, 2010.

32 Issues Constraint Language Expressiveness (Skolems, aggregates) Evaluation and Rewriting Techniques Managing Inconsistency Inconsistencies and interdependencies Automatic update in the presence of inconsistencies Queries in the presence of inconsistencies Dynamic Constraints Update Preferences Workflow Analysis, Use, Design Privacy and Security Promoting Convergence 11/15/10 32

33 Embrace cqa Technical def of semantics Issue, solution. NP hard. Symbols and font fix for formula. Distinguish formulae.

34 CQA Conflict person.desk(alice,desk1) person.desk(bob,desk1) Who is assigned to Desk1? person.desk(alice, Desk1)? NO person.desk(bob, Desk1)? NO person.desk(alice, Desk1) person.desk(bob, Desk1)? NO person.desk(alice, Desk1) person.desk(bob, Desk1)? YES

35 More complex queries Select an area on a map and ask: which workgroups have all their members assigned to a desk in the selected area?

36 More complex queries P [p.group(p,groupa) p.desk(p,desk1)]? ACS: NO P [p.group(p,groupb) (p.desk(p,desk2) p.desk(p,desk3))]? ACS: YES

37 Issues and responses Both ACS and CQA are NP-Hard problems [KG] Response: Identify tractable subclasses of queries ACS: * * [KG] CQA: * [KG] a subset of * [ABC, CM, FM] Neither is supported by a standard DBMS. Solution: Query rewriting Rewrite Q into Q', then evaluate Q' on a standard DBMS to obtain ACS/CQA answers to Q

38 Relational Calculus Semantics

39 Semantics for Incomplete Database Formally, the semantics of an incomplete database <P,N> is given by its set of possible instances. Relative to domain d, Instances d (<P,N>) = {D base(d) P D and D N= } <P,N> d Q iff D d Q for all D Instances (<P,N>). d

40 Decentralized Office assignment A building is divided into spaces. Each space as a space czar who is tasked with assigning people to offices within that space. The building manager, department manager, and department chair also exercise authority over the assignment of offices. Individual professors request specific assignments for his or her affiliates. 11/15/10 40

41 Desk Assignment Example Alice Desk1 desk(alice,desk1) Bob Desk1 desk(bob,desk1)

42 Desk Assignment Example Conflict: Data is inconsistent with the following constraint desk(alice,desk1) desk(bob,desk1) x1, x2, y (person.desk(x1,y) x1 = x2) Alice \neq Bob person.desk(x2,y)

43 Alert the users to the conflict The users can work to resolve the conflict Current snapshot...

44 Queries Users would like to query the eventual, consistent data But all we have is the current, inconsistent, snapshot. Current snapshot...