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Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational INTRODUCTION Databases and Logic Programs D-RELATIONS (INCOMPLETE- NESS) DEFAULT RELATIONS Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) (NONMONO- TONIC REASONING) Georgia State University, Atlanta, GA OA-TABLES (INCOMPLETE- NESS) BACKGROUND 1 July 2009 (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Outline of Dissertation Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar INTRODUCTION Sunderraman) INCOMPLETENESS IN RELATIONAL DATABASES INTRODUCTION D-RELATIONS D-RELATIONS (INCOMPLETE- OA-TABLES NESS) DEFAULT NEGATION AND NONMONOTONIC REASONING RELATIONS (NONMONO- DEFAULT RELATIONS TONIC NEGATION IN EXTENDED LOGIC PROGRAMS REASONING) OA-TABLES INCONSISTENCY IN RELATIONAL DATABASES (INCOMPLETE- NESS) SOURCE-AWARE REPAIRS BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Outline Inconsistency and Incompleteness in Relational Databases and 1 INTRODUCTION Logic Programs Navin Viswanath (advised by 2 D-RELATIONS (INCOMPLETENESS) Dr.Rajshekhar Sunderraman) INTRODUCTION 3 DEFAULT RELATIONS (NONMONOTONIC REASONING) D-RELATIONS (INCOMPLETE- NESS) 4 OA-TABLES (INCOMPLETENESS) DEFAULT RELATIONS (NONMONO- 5 BACKGROUND (LOGIC PROGRAMMING) TONIC REASONING) OA-TABLES (INCOMPLETE- 6 INCONSISTENCY IN EXTENDED LOGIC PROGRAMS NESS) BACKGROUND (LOGIC PRO- 7 INCONSISTENCY IN RELATIONAL DATABASES GRAMMING) INCONSISTENCY IN EXTENDED 8 CONCLUSIONS LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Introduction Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Relational databases may represent incomplete information INTRODUCTION Incomplete information has been studied extensively since the D-RELATIONS (INCOMPLETE- introduction of the relational model NESS) An application area plagued by the incompleteness problem : DEFAULT RELATIONS sensor databases (NONMONO- TONIC REASONING) Records information about locations of moving objects, physical OA-TABLES quantities like temperature etc. (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Monotonicity of First Order Logic Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Close relationship between relational databases and ﬁrst order INTRODUCTION logic : a query on a relational database is a formula in ﬁrst order D-RELATIONS (INCOMPLETE- logic NESS) First order logic is monotonic DEFAULT RELATIONS (NONMONO- Σ β TONIC REASONING) Σ ∪ {α} β OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra CWA Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Relational databases operate under the Closed World Assumption (CWA) of Reiter, a nonmonotonic form of reasoning INTRODUCTION According to the CWA, if sentence P cannot be proved from the D-RELATIONS (INCOMPLETE- Horn database DB, assume ¬P NESS) DEFAULT In the presence of indeﬁnite information (non-Horn clauses), RELATIONS (NONMONO- CWA is not appropriate TONIC REASONING) Let DB = {P(a) ∨ P(b)}. DB P(a) and DB P(a). OA-TABLES (INCOMPLETE- But ¬P(a) ∧ ¬P(b) is inconsistent with DB NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Motivation Inconsistency and Incompleteness in Relational parts supply Databases and suppliers Logic Programs PNUM PNAME SNUM PNUM Navin Viswanath SNUM SNAME (advised by p1 nut s1 p1 Dr.Rajshekhar s1 jones Sunderraman) p2 cam s1 p3 s2 smith p3 bolt s2 p2 INTRODUCTION s3 blake D-RELATIONS p4 wheel s3 p4 (INCOMPLETE- NESS) Query: “Find all suppliers who do not supply part p1” DEFAULT RELATIONS If there is a known list of suppliers, then the answer for the (NONMONO- TONIC query would be {s2, s3} REASONING) OA-TABLES null values complicates the problem further (INCOMPLETE- NESS) If (s3,null) is part of the supply relation we are uncertain BACKGROUND whether to include s3 as part of the answer or not (LOGIC PRO- GRAMMING) Similar problem occurs when one allows disjunctive information INCONSISTENCY IN EXTENDED (such as (s3,p1) OR (s3,p2)) LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Open World Assumption Inconsistency and Incompleteness in Relational Databases and Logic Programs It is sometimes important to explicitly include negative Navin Viswanath (advised by information Dr.Rajshekhar Sunderraman) In a medical database, a doctor may be more comfortable INTRODUCTION knowing that a patient does not show symptoms of a disease by D-RELATIONS knowing it explicitly rather than inferring it, say, by the CWA (INCOMPLETE- NESS) An open world is associated with a ﬁrst order theory DEFAULT RELATIONS Negative data is explicitly represented in the database (NONMONO- TONIC REASONING) When the database complies with this assumption concerning OA-TABLES negative data, the database is said to satisfy the open world (INCOMPLETE- NESS) assumption BACKGROUND (LOGIC PRO- Under the OWA, we “admit” that our knowledge of the world is GRAMMING) incomplete INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra d-relations Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) A d-relation R, over a scheme Σ, consists of two components, < R + , R − > where R + ⊆ 2τ (Σ) and R − ⊆ 2τ (Σ) INTRODUCTION D-RELATIONS R + , the positive component, is a set of tuple sets. Each tuple (INCOMPLETE- NESS) set represents a disjunctive positive fact DEFAULT RELATIONS R − , the negative component, is also a set of tuple sets. Each (NONMONO- TONIC tuple set in R − represents a disjunctive negative fact REASONING) In the case where the tuple set is singleton, we have a deﬁnite OA-TABLES (INCOMPLETE- fact NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra An example of a d-relation Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar supply Sunderraman) SNUM PNUM INTRODUCTION {(s1,p1),(s1,p4)} D-RELATIONS (INCOMPLETE- {(s1,p3)} NESS) {(s2,p2)} DEFAULT RELATIONS {(s3,p1),(s3,p4)} (NONMONO- TONIC REASONING) {(s1,p2),(s1,p3)} OA-TABLES {(s2,p3)} (INCOMPLETE- NESS) {(s3,p2),(s3,p3)} BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Precise generalizations of Operators Inconsistency and Incompleteness in Relational Databases and Logic Programs REP Navin Viswanath R −→ U (advised by Dr.Rajshekhar Θ ↓ ˙ ↓ Θ Sunderraman) REP R −→ U INTRODUCTION D-RELATIONS (INCOMPLETE- NESS) Theorem DEFAULT RELATIONS 1 ˙ repΣ (R ∪S) = S(∪)(repΣ (R), repΣ (S)). (NONMONO- TONIC REASONING) 2 ˙ repΣ (R ∩S) = S(∩)(repΣ (R), repΣ (S)). OA-TABLES (INCOMPLETE- 3 repΣ1 (σF (R)) = S(σF )(repΣ1 (R)). ˙ NESS) 4 repΣ1 (π∆ (R)) = S(π∆ )(repΣ1 (R)). ˙ BACKGROUND (LOGIC PRO- GRAMMING) 5 repΣ1 ∪Σ2 (R S) = S( ˙ )(repΣ1 (R), repΣ2 (S)). INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Query Example Inconsistency and Incompleteness in Relational Databases and Logic Programs supply Navin Viswanath {(s1,p1),(s1,p2)} (advised by Dr.Rajshekhar {(s2,p2)} Sunderraman) {(s3,p1)} Find all suppliers who do not supply ‘p1’ INTRODUCTION {(s2,p1),(s1,p1)} D-RELATIONS (INCOMPLETE- {(s3,p2)} NESS) DEFAULT Ans = −(πS (σp=‘p1 (supply ))) RELATIONS (NONMONO- TONIC σp=‘p1 (supply ) πS (σp=‘p1 (supply )) Ans REASONING) {(s3,p1)} OA-TABLES (INCOMPLETE- {(s1,p2)} {s3} {s1,s2} NESS) {(s2,p2)} BACKGROUND {s1,s2} {s3} (LOGIC PRO- GRAMMING) {(s3,p2)} INCONSISTENCY {(s2,p1),(s1,p1)} IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Default Relations Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) + − − A default relation on scheme Σ is a triple < Re , Re , Rd > INTRODUCTION + − − D-RELATIONS where Re ,Re and Rd are any subsets of τ (Σ) (INCOMPLETE- + NESS) Re is the set of facts for which R is known to hold − DEFAULT RELATIONS Re is the set of facts for which R is known not to hold (NONMONO- − TONIC Rd is the set of facts for which R is not known to hold REASONING) OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Query Example Inconsistency and Incompleteness in Relational Databases and Logic Programs Patient Navin Viswanath pname symptom (advised by Dr.Rajshekhar Tom Forgetfulness Disease Sunderraman) Jack Headache dname symptom INTRODUCTION Tom Nausea Cold Headache D-RELATIONS (INCOMPLETE- Jack Nausea Alzheimer’s Forgetfulness NESS) Jack Forgetfulness Jaundice Nausea DEFAULT RELATIONS Ann Forgetfulness Alzheimer’s Headache (NONMONO- TONIC Ann Sneezing Jaundice Forgetfulness REASONING) OA-TABLES Ann Headache ∅ (INCOMPLETE- NESS) Ann Nausea BACKGROUND ∅ (LOGIC PRO- GRAMMING) Figure: An instance of a hospital database INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Inconsistency and Incompleteness in Relational Databases and Temp Logic Programs pname symptom dname Navin Viswanath Which patients (advised by Tom Forgetfulness Alzheimer’s Dr.Rajshekhar suffer from Sunderraman) Jack Headache Cold Tom Nausea {Alzheimer’s,Cold,Jaundice} Alzheimer’s INTRODUCTION Jack Nausea {Alzheimer’s,Cold,Jaundice} disease? D-RELATIONS Jack Forgetfulness {Alzheimer’s,Cold,Jaundice} (INCOMPLETE- Ann Forgetfulness {Alzheimer’s,Cold,Jaundice} Answer NESS) Ann Nausea {Alzheimer’s,Cold,Jaundice} pname DEFAULT RELATIONS Ann Headache {Alzheimer’s,Cold,Jaundice} Tom (NONMONO- TONIC Ann Sneezing {Alzheimer’s,Cold,Jaundice} Ann REASONING) Jack Headache Alzheimer’s Jack OA-TABLES (INCOMPLETE- Tom Headache Alzheimer’s NESS) Tom Forgetfulness Jaundice Figure: The result of BACKGROUND Tom Sneezing Alzheimer’s, Cold, Jaundice the query (LOGIC PRO- GRAMMING) Jack Sneezing Alzheimer’s, Cold, Jaundice INCONSISTENCY Tom Forgetfulness Cold IN EXTENDED Tom Headache Cold LOGIC PROGRAMS Tom Headache Jaundice INCONSISTENCY Jack Headache Jaundice IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Disjunctive Databases Inconsistency and Incompleteness in Relational Databases and Logic Programs travel Navin Viswanath (advised by src dest time Dr.Rajshekhar Sunderraman) c1 c2 4 INTRODUCTION c2 c3 {3,4} D-RELATIONS {c1,c3} c4 2 (INCOMPLETE- NESS) DEFAULT RELATIONS Set valued attributes used to denote ”disjunctions” (NONMONO- TONIC Information contained in the above database: REASONING) travel(c1, c2, 4) OA-TABLES (INCOMPLETE- travel(c2, c3, 3) ∨ travel(c2, c3, 4) NESS) BACKGROUND ¬travel(c2, c3, 3) ∨ ¬travel(c2, c3, 4) (LOGIC PRO- GRAMMING) travel(c1, c4, 2) ∨ travel(c3, c4, 2) INCONSISTENCY ¬travel(c1, c4, 2) ∨ ¬travel(c3, c4, 2) IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra oa-Tables Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath oa-table scheme: R =< A1 , . . . , An >, a list of attribute names. (advised by Dr.Rajshekhar oa-table T over the scheme R is deﬁned as follows: Sunderraman) dom(A1 )×dom(A2 )×,...,×dom(An ) INTRODUCTION T ⊆ 22 D-RELATIONS (INCOMPLETE- NESS) oa-table T consists of oa-tuples; T = {w1 , w2 , . . . , wn } DEFAULT RELATIONS oa-tuple w consists of tuple-sets; w = {η1 , η2 , . . . , ηm } (NONMONO- TONIC each ηi is a possible world part. REASONING) tuple-set η consists of tuples; η = {t1 , t2 , . . . , tk } OA-TABLES (INCOMPLETE- NESS) NEG (T ) = ∪n (P(atoms(wi )) − wi ), is the set of impossible i=1 BACKGROUND world parts. (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Example of oa-table Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar travel(src,dest,time) Sunderraman) (c1,c2,4) INTRODUCTION (c2,c3,3) D-RELATIONS (INCOMPLETE- (c2,c3,4) NESS) (c1,c4,2) DEFAULT RELATIONS (c3,c4,2) (NONMONO- TONIC REASONING) (c1,c4,2) OA-TABLES (c3,c4,2) (INCOMPLETE- NESS) BACKGROUND NEG (travel) = {{(c2, c3, 3), (c2, c3, 4)}} (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Inconsistencies in oa-tables Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by T1 T2 Dr.Rajshekhar Sunderraman) a b INTRODUCTION a c D-RELATIONS (INCOMPLETE- b a NESS) DEFAULT a b RELATIONS (NONMONO- b d TONIC REASONING) e OA-TABLES f (INCOMPLETE- NESS) BACKGROUND Both oa-tables are inconsistent. For this discussion, we restrict all (LOGIC PRO- GRAMMING) oa-tables to be consistent. Handling inconsistency is a separate issue. INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra COMPACT Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar T Sunderraman) a COMPACT (T ) INTRODUCTION b a D-RELATIONS (INCOMPLETE- c b NESS) a c DEFAULT RELATIONS b d (NONMONO- TONIC d e REASONING) OA-TABLES e f (INCOMPLETE- NESS) f BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra REDUCE Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar T REDUCE (T ) Sunderraman) a INTRODUCTION a D-RELATIONS b (INCOMPLETE- a NESS) c DEFAULT a RELATIONS a (NONMONO- e TONIC d REASONING) a OA-TABLES (INCOMPLETE- e NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra REP, the Information Content of oa-tables Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar ΓR = {T |T is a oa-table over R} Sunderraman) ΣR = {U|U is a set of relations over R} INTRODUCTION D-RELATIONS (INCOMPLETE- Let T = {w1 , w2 , . . . , wn }. Then, NESS) REP(T ) = M(REDUCE (T )) DEFAULT RELATIONS M(T ) : ΓR → ΣR is deﬁned as: (NONMONO- TONIC REASONING) OA-TABLES M(T ) = {η1 ∪ η2 . . . ∪ ηn | (∀i, 1 ≤ i ≤ n)(ηi ∈ wi )∧ (INCOMPLETE- NESS) ¬(∃u ∈ NEG (T ))(u ⊆ η1 ∪η2 . . .∪ηn )} BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Examples of REP Inconsistency and Incompleteness in Relational Databases and Logic Programs T1 Navin Viswanath (advised by a a Dr.Rajshekhar Sunderraman) b M(T1 ) = , b c b INTRODUCTION D-RELATIONS c (INCOMPLETE- NESS) DEFAULT T2 RELATIONS (NONMONO- a TONIC REASONING) b a a OA-TABLES a M(T2 ) = , , b (INCOMPLETE- b c NESS) b BACKGROUND (LOGIC PRO- b GRAMMING) c INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Selection Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath ˙ σF (T ) = REDUCE (T ) where, (advised by Dr.Rajshekhar T = {{η1 , η2 , . . . , ηm } | (∃{η1 , η2 , . . . , ηm } ∈ T )( Sunderraman) (∀i, 1 ≤ i ≤ m)(ηi = σF (ηi )))} INTRODUCTION Drop tuples that do not satisfy the selection condition. D-RELATIONS T ˙ σ1=‘a1 (T ) (INCOMPLETE- NESS) a1 b1 DEFAULT a1 b2 RELATIONS a1 b1 (NONMONO- a1 b3 TONIC a1 b2 REASONING) a1 b4 a1 b3 OA-TABLES a2 b5 (INCOMPLETE- a1 b4 NESS) a2 b3 BACKGROUND (LOGIC PRO- a2 b4 GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Projection Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath ˙ πA (T ) = REDUCE (T ) where, (advised by Dr.Rajshekhar T = {{η1 , η2 , . . . , ηm } | (∃{η1 , η2 , . . . , ηm } ∈ T )( Sunderraman) (∀i, 1 ≤ i ≤ m)(ηi = π(ηi )))} INTRODUCTION Project tuples in each tuple-set of each oa-tuple. D-RELATIONS T ˙ π1 (T ) (INCOMPLETE- NESS) a1 b1 DEFAULT a1 b2 a1 RELATIONS (NONMONO- a1 b3 a1 TONIC REASONING) a1 b4 a1 OA-TABLES (INCOMPLETE- a2 b5 a2 NESS) a2 b3 a2 BACKGROUND (LOGIC PRO- a2 b4 GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Cartesian Product Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by ˙ T1 ×T2 = REDUCE (T ) where, Dr.Rajshekhar Sunderraman) T = {{η11 , η12 , . . . , ηmn } | (∃{η11 , η12 , . . . , η1m } ∈ T1 ) INTRODUCTION (∃{η21 , η22 , . . . , η2n } ∈ T2 )( D-RELATIONS (∀i, 1 ≤ i ≤ m)(∀j, 1 ≤ j ≤ n) (INCOMPLETE- NESS) (ηij = η1i × η2j ))} DEFAULT RELATIONS (NONMONO- TONIC REASONING) OA-TABLES For each pair of oa-tuples from T1 and T2 , take the Cartesian (INCOMPLETE- NESS) product of all ”possible” world combinations. BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Cartesian Product - Example Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar T1 T2 ˙ T1 ×T2 Sunderraman) c f INTRODUCTION a c g D-RELATIONS b a f c e d f (INCOMPLETE- e NESS) c a e a g d e d g f DEFAULT RELATIONS d b e b f c e c f (NONMONO- g TONIC c b g d e d f REASONING) d c g OA-TABLES (INCOMPLETE- d g NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Union/Diﬀerence Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar ˙ T1 ∪T2 = REDUCE (T1 ∪ T2 ) Sunderraman) INTRODUCTION D-RELATIONS (INCOMPLETE- NESS) ˙ T1 −T2 = REDUCE (T ) where, DEFAULT RELATIONS T = {{η11 , η12 , . . . , ηmn } | (∃{η11 , η12 , . . . , η1m } ∈ T1 ) (NONMONO- TONIC (∃{η21 , η22 , . . . , η2n } ∈ T2 )( REASONING) (∀i, 1 ≤ i ≤ m)(∀j, 1 ≤ j ≤ n) OA-TABLES (INCOMPLETE- (ηij = η1i − η2j ))} NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Correctness of Algebra Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Theorem Dr.Rajshekhar Sunderraman) 1 ˙ REP(σF (T )) = σF (REP(T )) for any reduced oa-table T . INTRODUCTION 2 ˙ REP(πA (T )) = πA (REP(T )) for any reduced oa-table T and D-RELATIONS list of attributes A. (INCOMPLETE- NESS) 3 ˙ REP(T1 ×T2 ) = REP(T1 ) × REP(T2 ) for any reduced oa-tables DEFAULT RELATIONS T1 and T2 . (NONMONO- TONIC 4 ˙ REP(T1 ∪T2 ) = REP(T1 ) ∪ REP(T2 ) for any reduced and REASONING) OA-TABLES compatible oa-tables T1 and T2 . (INCOMPLETE- NESS) 5 ˙ REP(T1 −T2 ) = REP(T1 ) − REP(T2 ) for any reduced and BACKGROUND compatible oa-tables T1 and T2 . (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Query Example Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath Patient Disease (advised by Dr.Rajshekhar cold noserun Sunderraman) cold sneeze Query: Who has the ﬂu? cold noserun P ÷ π2 (σ1= ﬂu (D)) ˙ ˙ INTRODUCTION tom fever jim cold sneeze D-RELATIONS jim fever (INCOMPLETE- ﬂu headache NESS) jim sneeze ﬂu fever Query: Who has the cold? DEFAULT don sneeze RELATIONS ﬂu sneeze P ÷ π2 (σ1= cold (D)) ˙ ˙ (NONMONO- don sneeze TONIC ﬂu fever jim REASONING) don fever ﬂu headache OA-TABLES don (INCOMPLETE- ﬂu sneeze NESS) ﬂu fever BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra I-tables, E-tables, C-tables Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) INTRODUCTION D-RELATIONS C-tables (Imielinski and Lipski) (INCOMPLETE- NESS) I-tables (Liu and Sunderraman) DEFAULT E-tables (Liu and Zhang) RELATIONS (NONMONO- TONIC REASONING) OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra GCWA Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Minker generalized the CWA to represent such information. Minker’s Generalized Closed World Assumption (GCWA) is INTRODUCTION D-RELATIONS based on the idea of minimal models. (INCOMPLETE- NESS) The minimal models of {P(a) ∨ P(b)} are {P(a)} and {P(b)}. DEFAULT RELATIONS According to the GCWA, the true sentences are ones that (NONMONO- TONIC appear in every minimal model and the false sentences are the REASONING) ones that appear in no minimal model. OA-TABLES (INCOMPLETE- The GCWA interprets disjunctions exclusively. NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra DDR Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Another related concept is Ross and Topor’s Disjunctive Sunderraman) Database Rule (DDR). INTRODUCTION closed set: set of ground atoms that can be assumed false. D-RELATIONS (INCOMPLETE- S is a closed set of DB if, for every element A ∈ S, and for NESS) every instance of a ground clause C such that A is in the head DEFAULT RELATIONS of C , there exists an atom B in the body of C such that B ∈ S. (NONMONO- TONIC REASONING) greatest closed set gcs(DB) exists and represents the negative OA-TABLES information that can be inferred from the database. (INCOMPLETE- NESS) DDR interprets disjunctions inclusively. BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Drawbacks of GCWA and DDR Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by GCWA interprets disjunctions exclusively and DDR interprets Dr.Rajshekhar Sunderraman) disjunctions inclusively. INTRODUCTION Consider DB = {P(a) ∨ P(b), P(a)}. The only minimal model D-RELATIONS of DB is {P(a)} and GCWA assigns truth value false to P(b). (INCOMPLETE- NESS) Consider D = {BG (john, A) ∨ BG (john, B), BG (john, A)} DEFAULT RELATIONS where BG denotes the bloodgroup relation. (NONMONO- TONIC DDRconclusions : {BG (john, A), BG (john, B)} and REASONING) gcs(D ) = ∅. OA-TABLES (INCOMPLETE- NESS) The disjunction in the bloodgroup relation is obviously exclusive BACKGROUND and DDR fails to negate BG (john, B). (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Possible World Semantics Inconsistency and Incompleteness in Relational Databases and Logic Programs Possible World Semantics (PWS) was introduced simultaneously Navin Viswanath by Sakama and Chan to overcome the drawbacks of GCWA and (advised by Dr.Rajshekhar DDR. Sunderraman) Consider the database INTRODUCTION D1 = {D ←; A ∨ B ← D; C ← A, B; ¬(A ∧ B)}. D-RELATIONS (INCOMPLETE- NESS) The possible worlds of D1 are {D, A} and {D, B}. DEFAULT Notice the introduction of the negative clause ¬(A ∧ B) RELATIONS (NONMONO- TONIC This permits the interpretation of the disjunction A ∨ B as REASONING) exclusive. OA-TABLES (INCOMPLETE- Atoms that appear in every possible world are true, those in no NESS) BACKGROUND possible world are false and those in some possible world are (LOGIC PRO- GRAMMING) possibly true. INCONSISTENCY By the PWS, D is true, C is false and A and B are possibly true. IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) DDR would ignore the negative clause and treat the disjunction INTRODUCTION as inclusive. D-RELATIONS (INCOMPLETE- TD1 ↑ ω(∅) = {A, B, C , D} and gcs(D1 ) = ∅. NESS) DEFAULT However, PWS needs the introduction of a negative clause to RELATIONS (NONMONO- understand the correct nature of the disjunction. TONIC REASONING) oa-tables implements the PWS semantics at the EDB level. OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Deﬁnite Logic Programs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath A term is a constant, a variable or a complex term of the form (advised by Dr.Rajshekhar f (t1 , . . . , tn ) where t1 , . . . , tn are terms and f is a function Sunderraman) symbol INTRODUCTION Atom : A formula of the form p(t1 , . . . , tn ) where p is a D-RELATIONS (INCOMPLETE- predicate symbol NESS) DEFAULT A deﬁnite logic program is a set of rules of the form RELATIONS (NONMONO- A ← B1 , . . . Bn , where A, B1 , . . . Bn are atoms (rules are called TONIC REASONING) Horn clauses) OA-TABLES (INCOMPLETE- Deﬁnite logic programs have a unique least model NESS) Declarative semantics given by the TP operator of van Emden BACKGROUND (LOGIC PRO- and Kowalski GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra The TP operator Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by The TP operator computes the least model of the deﬁnite logic Dr.Rajshekhar Sunderraman) program P INTRODUCTION The least model of P is the least ﬁxpoint of TP D-RELATIONS (INCOMPLETE- NESS) DEFAULT TP ↑ 0 = ∅ RELATIONS (NONMONO- TONIC TP ↑ i + 1 = TP (TP ↑ i) REASONING) ∞ OA-TABLES (INCOMPLETE- TP ↑ ω = Tp ↑ i. NESS) i=0 BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra TP Example Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath The “odd” program (advised by Dr.Rajshekhar Sunderraman) odd(s(0)) ← . INTRODUCTION odd(s(s(X ))) ← odd(X ). D-RELATIONS (INCOMPLETE- NESS) DEFAULT RELATIONS (NONMONO- TP ↑ 0 = ∅ TONIC REASONING) TP ↑ 1 = {odd(s(0))} OA-TABLES (INCOMPLETE- . . NESS) . BACKGROUND (LOGIC PRO- TP ↑ ω = {odd(s n (0)) | n ∈ {1, 3, 5, . . .}} GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra General Logic Programs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) A general logic program is a set of rules of the form INTRODUCTION A ← B1 , . . . Bn , ∼ C1 , . . . , ∼ Cm where A, B1 , . . . Bn , C1 , . . . Cm D-RELATIONS are atoms (INCOMPLETE- NESS) ∼ denotes a form of negation called “negation by failure” (used DEFAULT RELATIONS in systems like Prolog) (NONMONO- TONIC We assume ∼ A when atom A cannot be proved from the REASONING) OA-TABLES program (similar in spirit to CWA) (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra General Logic Program Semantics Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar General logic programs may have several minimal models Sunderraman) Finding the “intended” model is a diﬃcult problem INTRODUCTION D-RELATIONS a ←∼ b has two minimal models {a} and {b} (INCOMPLETE- NESS) The intended model is {a} DEFAULT RELATIONS Two of the most popular semantics : stable model (NONMONO- TONIC semantics(two-valued) and well-founded semantics(three-valued) REASONING) OA-TABLES Most semantics agree on a large class of general logic programs (INCOMPLETE- NESS) - the “stratiﬁed” programs (no recursion through negation) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Stable Model Semantics Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Deﬁnition Dr.Rajshekhar Sunderraman) Let Π be a general logic program. For any set S of atoms, let ΠS be the deﬁnite logic program obtained from Π by deleting INTRODUCTION 1 each rule that has a formula ∼ L in its body with L ∈ S, and D-RELATIONS (INCOMPLETE- 2 all formulas of the form ∼ L from the bodies of the remaining NESS) rules. DEFAULT RELATIONS (NONMONO- ΠS does not contain ∼, so that its model is already deﬁned. If TONIC REASONING) this model coincides with S, then we say that S is a stable OA-TABLES model of Π (INCOMPLETE- NESS) BACKGROUND A general logic program can have several stable models (when it is (LOGIC PRO- GRAMMING) not stratiﬁed) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Example : Stable models Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Consider program P: Dr.Rajshekhar Sunderraman) a ← ∼b INTRODUCTION D-RELATIONS b ← ∼a (INCOMPLETE- NESS) Let S1 = {a} DEFAULT RELATIONS (NONMONO- Then P S1 : TONIC REASONING) a←∼b OA-TABLES (INCOMPLETE- b ←∼ a NESS) BACKGROUND Similarly, S2 = {b} is also a stable model (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Extended Logic Programs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Literal : An atom p(t1 , . . . , tn ) or its negation ¬p(t1 , . . . , tn ) Sunderraman) An extended logic program is a set of rules of the form INTRODUCTION A ← B1 , . . . Bn , ∼ C1 , . . . , ∼ Cm where A, B1 , . . . Bn , C1 , . . . Cm D-RELATIONS (INCOMPLETE- are literals NESS) ¬ denotes explicit negation DEFAULT RELATIONS (NONMONO- ¬ of a literal L is accepted only if it can be proven from the TONIC REASONING) program (like proving L) OA-TABLES ∼ of a literal L is accepted as “failure to prove” L (INCOMPLETE- NESS) ¬ may be thought of as a “stronger” form of negation than ∼ BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Answer set semantics Inconsistency and Incompleteness in Relational Databases and Logic Programs Consider program P: Navin Viswanath (advised by Dr.Rajshekhar a ← ∼ ¬a Sunderraman) ¬a ← ∼a INTRODUCTION b ← ∼ ¬b D-RELATIONS (INCOMPLETE- NESS) Prime transformation to obtain a general logic program DEFAULT RELATIONS (NONMONO- TONIC a ← ∼a REASONING) OA-TABLES a ← ∼a (INCOMPLETE- NESS) b ← ∼b BACKGROUND (LOGIC PRO- GRAMMING) Find stable models of transformed program and reverse INCONSISTENCY transformations to get answer sets IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Example: Answer set semantics Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Let S1 = {a, b} INTRODUCTION Then P S1 : D-RELATIONS (INCOMPLETE- a←∼a NESS) a ←∼ a DEFAULT RELATIONS b←∼b (NONMONO- TONIC REASONING) Similarly, S2 = {a , b} is also a stable model. After reversing, OA-TABLES {¬a, b} is an answer set (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Motivation Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar A feature of extended logic programs : querying the Sunderraman) incompleteness INTRODUCTION D-RELATIONS (INCOMPLETE- NESS) Eligible(X ) ← HighGPA(X ). (1) DEFAULT RELATIONS (NONMONO- Eligible(X ) ← Minority (X ), FairGPA(X ). (2) TONIC REASONING) ¬Eligible(X ) ← ¬FairGPA(X ). (3) OA-TABLES (INCOMPLETE- Interview (X ) ← ∼ Eligible(X ), ∼ ¬Eligible(X ). (4) NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Contradictory Extended Logic Programs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Extended logic programs operate under the OWA Dr.Rajshekhar Sunderraman) A contradictory program INTRODUCTION D-RELATIONS a←. (INCOMPLETE- NESS) ¬a ← . DEFAULT RELATIONS (NONMONO- Is the following program contradictory? TONIC REASONING) OA-TABLES ¬a ← . (INCOMPLETE- NESS) a ← ∼ b. BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Contradictory Extended Logic Programs Inconsistency and Incompleteness in Relational After prime transformation Databases and Logic Programs Navin Viswanath a ← . (advised by Dr.Rajshekhar Sunderraman) a ← ∼ b. INTRODUCTION Stable model contains both a and a (¬a) D-RELATIONS (INCOMPLETE- Prime transformation loses the semantic connection between a NESS) and ¬a DEFAULT RELATIONS (NONMONO- A simple solution: To every rule with L in the head, add ∼ ¬L TONIC REASONING) to the body OA-TABLES (INCOMPLETE- Problem: Even NESS) BACKGROUND (LOGIC PRO- a←. GRAMMING) ¬a ← . INCONSISTENCY IN EXTENDED LOGIC PROGRAMS is not contradictory INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Coherence Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) ¬a ← . INTRODUCTION a ← ∼ b. D-RELATIONS (INCOMPLETE- NESS) DEFAULT RELATIONS Assume we also had ¬b ← . (NONMONO- TONIC Principle of coherence :Strong negation(¬) implies weak REASONING) OA-TABLES negation(∼) (Alferes et al.) (INCOMPLETE- NESS) Simulating CWA for a predicate L in an extended logic program : BACKGROUND ¬L ←∼ L (Gelfond and Lifschitz) (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Coherence alone is too “strong” a requirement INTRODUCTION The program a ←∼ b does not entail a D-RELATIONS (INCOMPLETE- NESS) Transformation has to include both coherence and some form of DEFAULT negation by failure RELATIONS (NONMONO- Derive a literal L through a rule containing ∼ if we are willing to TONIC REASONING) accept the CWA for L (by adding ∼ ¬L to the body) OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Inconsistency and Incompleteness in Relational A ← B1 , . . . , Bn , ∼ C1 , . . . , ∼ Cm Databases and Logic Programs Navin Viswanath (advised by Deﬁnition Dr.Rajshekhar Sunderraman) When (m = 0) or (m = 2 and C1 = ¬C2 ). INTRODUCTION T (r ) = prime(r ) . D-RELATIONS When m ≥ 1 and {C1 , . . . , Cm } does not contain a pair of (INCOMPLETE- NESS) complementary literals. DEFAULT RELATIONS T (r ) is given by a pair of rules prime(r1 ) and prime(r2 ) where (NONMONO- TONIC REASONING) r1 : A ← B1 , . . . , Bn , ¬C1 , . . . , ¬Cm OA-TABLES (INCOMPLETE- r2 : A ← B1 , . . . , Bn , ∼ C1 , . . . , ∼ Cm , ∼ ¬A NESS) BACKGROUND (LOGIC PRO- When m > 2 and {C1 , . . . , Cm } contains atleast a pair of GRAMMING) complementary literals, say Ci , i.e. {Ci , ¬Ci } ⊂ {C1 , . . . , Cm }. INCONSISTENCY IN EXTENDED T (r ) is given by a pair of rules prime(r1 ) and prime(r2 ) where LOGIC PROGRAMS INCONSISTENCY r :A 1 ← B , . . . , Bn , ¬C1 , . . . , ∼ Ci , ∼ ¬Ci , . . . , ¬Cm 1 IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b OA-TABLES (INCOMPLETE- a ← ∼ b, ∼ ¬a NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b OA-TABLES (INCOMPLETE- a ← ∼ b, ∼ ¬a NESS) ¬a ← ¬b BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b a ← b OA-TABLES (INCOMPLETE- a ← ∼ b, ∼ ¬a NESS) prime ¬a ← ¬b −→ −− BACKGROUND (LOGIC PRO- ¬a ← ∼ b, ∼ a GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b a ← b OA-TABLES (INCOMPLETE- a ← ∼ b, ∼ ¬a a ← ∼ b, ∼ a NESS) prime ¬a ← ¬b −→ −− BACKGROUND (LOGIC PRO- ¬a ← ∼ b, ∼ a GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b a ← b OA-TABLES (INCOMPLETE- a ← ∼ b, ∼ ¬a a ← ∼ b, ∼ a NESS) prime ¬a ← ¬b −− −→ a ← b BACKGROUND (LOGIC PRO- ¬a ← ∼ b, ∼ a GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example I Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar a ←∼ b Sunderraman) ¬a ←∼ b INTRODUCTION D-RELATIONS (INCOMPLETE- The transformed program NESS) DEFAULT RELATIONS (NONMONO- TONIC REASONING) a ← ¬b a ← b OA-TABLES (INCOMPLETE- a ← ∼ b, ∼ ¬a a ← ∼ b, ∼ a NESS) prime ¬a ← ¬b −→ −− a ← b BACKGROUND (LOGIC PRO- ¬a ← ∼ b, ∼ a a ← ∼ b, ∼ a GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Example II Inconsistency and Incompleteness in Relational Databases and Logic Programs c ←∼ b Navin Viswanath (advised by a ←∼ b Dr.Rajshekhar Sunderraman) ¬a ←∼ b INTRODUCTION The transformed program D-RELATIONS (INCOMPLETE- NESS) c ← ¬b c ← b DEFAULT RELATIONS c ← ∼ b, ∼ ¬c c ← ∼ b, ∼ c (NONMONO- TONIC a ← ¬b a ← b REASONING) prime a ← ∼ b, ∼ ¬a −→ −− a ← ∼ b, ∼ a OA-TABLES (INCOMPLETE- ¬a ← ¬b a ← b NESS) ¬a ← ∼ b, ∼ a a ← ∼ b, ∼ a BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY c is a consequence of the transformed program IN EXTENDED LOGIC PROGRAMS Other semantics would declare the program inconsistent INCONSISTENCY although inconsistency is local to a IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Transformation Properties Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) INTRODUCTION For deﬁnite logic programs T (P) = P D-RELATIONS For general logic programs (without ¬), the transformation has (INCOMPLETE- NESS) no eﬀect on the semantics of the program DEFAULT RELATIONS For extended logic program P, if T (P) is inconsistent by a (NONMONO- TONIC semantics, say SEM, then P is inconsistent by SEM REASONING) OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Inconsistent Databases Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Functional dependency class → professor INTRODUCTION Class Professor Class Professor D-RELATIONS c1 p1 (INCOMPLETE- c1 p2 NESS) c2 p2 DEFAULT c3 p3 RELATIONS c3 p3 (NONMONO- TONIC REASONING) Figure: Two relations whose union is inconsistent w.r.t FD OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Repairs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) A repair of a database is the set of changes made to the INTRODUCTION database so that consistency is restored D-RELATIONS (INCOMPLETE- NESS) We are interested in the minimal repairs, the repairs that involve DEFAULT minimal updates to the orginal database RELATIONS (NONMONO- A consistent query answer is the set of tuples that is true in TONIC REASONING) every minimal repair of the database OA-TABLES (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Repairs Example Inconsistency and Incompleteness A database inconsistent w.r.t FD Class → Professor in Relational Databases and Logic Programs Class Professor Navin Viswanath (advised by c1 p1 Dr.Rajshekhar Sunderraman) c2 p2 c3 p3 INTRODUCTION D-RELATIONS c1 p2 (INCOMPLETE- NESS) Figure: The integrated database DEFAULT RELATIONS (NONMONO- TONIC REASONING) Class Professor Class Professor OA-TABLES (INCOMPLETE- c1 p1 c1 p2 NESS) BACKGROUND c2 p2 c2 p2 (LOGIC PRO- GRAMMING) c3 p3 c3 p3 INCONSISTENCY IN EXTENDED Figure: The minimal repairs LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Logic Programs for Repairs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Logic programming has been used to obtain the repairs of the database INTRODUCTION D-RELATIONS Construct a repair program such that the answer sets of the (INCOMPLETE- NESS) repair program correspond to the repairs of the database DEFAULT RELATIONS The repair program is a disjunctive logic program with two kinds (NONMONO- TONIC of negation, explicit negation and default negation REASONING) There can be an exponential number of repairs for an OA-TABLES (INCOMPLETE- inconsistent database NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Preferred Repairs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Look only for a subset of the possible repairs of the database, the “preferred repairs” INTRODUCTION D-RELATIONS Repairs may be computed based on a preference for a subset of (INCOMPLETE- NESS) the sources which we consider “reliable’ DEFAULT Requires the database to include information regarding what RELATIONS (NONMONO- data is conﬁrmed by what source TONIC REASONING) Such a data model is the Information Source Tracking OA-TABLES (INCOMPLETE- Method(IST) of Sadri NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Information Source Tracking Method Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Teaches Dr.Rajshekhar Sunderraman) Class Professor I INTRODUCTION c1 p1 110 D-RELATIONS c1 p2 100 (INCOMPLETE- NESS) c2 p2 111 DEFAULT c3 p3 1 1 -1 RELATIONS (NONMONO- TONIC Figure: An example of a table in the IST method REASONING) OA-TABLES (INCOMPLETE- NESS) Tuple (c3, p3) is in the relation if sources s1 and s2 are correct and BACKGROUND (LOGIC PRO- source s3 is incorrect GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Source-aware Repairs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Introduce propositional constants of the form si in the repair program to indicate source si is believed : called s-literals INTRODUCTION D-RELATIONS The facts in the database are introduced in the logic program as (INCOMPLETE- NESS) “conditional facts” DEFAULT teaches(c3, p3) ← s1 , s2 , ¬s3 should indicate teaches(c3, p3) is RELATIONS (NONMONO- obtained as a consistent answer(it is in every repair) if we TONIC REASONING) believe in sources s1 and s2 and do not believe in s3 OA-TABLES (INCOMPLETE- Modify answer set semantics to obtain preferred repairs NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Source-aware Answer Sets Inconsistency and Incompleteness in Relational Databases and Logic Programs Let Ssource = S ∪ slits . Navin Viswanath (advised by slits denotes the set of sources we want to believe(disbelieve). Dr.Rajshekhar Sunderraman) Deﬁnition INTRODUCTION The transformation ΠSsource of Π w.r.t Ssource is obtained by: D-RELATIONS (INCOMPLETE- NESS) 1 Deleting every rule with not L in the body with L ∈ Ssource and deleting every s-rule that: DEFAULT RELATIONS (NONMONO- 1 has ¬s in the body with s ∈ slits OR TONIC 2 does not have every literal from slits in its body REASONING) OA-TABLES 2 Deleting the negative literals from the bodies of the remaining rules (INCOMPLETE- NESS) and deleting every literal from the bodies of the remaining s-rules BACKGROUND (LOGIC PRO- Ssource is a source-aware answer set of Π if it is the least model of GRAMMING) ΠSsource . INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Source-aware Repairs Inconsistency and Incompleteness s1 s2 s3 in Relational Databases and P P P Logic Programs X Y X Y X Y Navin Viswanath (advised by a b a e b d Dr.Rajshekhar Sunderraman) c d b c c d INTRODUCTION Figure: Data collected from independent sources s1 , s2 and s3 D-RELATIONS (INCOMPLETE- NESS) DEFAULT P RELATIONS (NONMONO- X Y I TONIC REASONING) a b 1 0 0 OA-TABLES c d 1 0 0 (INCOMPLETE- NESS) a e 0 1 0 BACKGROUND b c 0 1 0 (LOGIC PRO- GRAMMING) b d 0 0 1 INCONSISTENCY c d 0 0 1 IN EXTENDED LOGIC PROGRAMS Figure: The integrated database along with source information INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Source-aware Repairs Inconsistency and Incompleteness in Relational Databases and P P P P Logic Programs X Y X Y X Y X Y Navin Viswanath (advised by a b , a b , a e , a e P c d c d c d c d Dr.Rajshekhar Sunderraman) X Y I b c b d b d b d INTRODUCTION a b 1 0 0 D-RELATIONS c d 1 0 0 Figure: The set of all minimal repairs of the (INCOMPLETE- NESS) a e 0 1 0 database DEFAULT b c 0 1 0 RELATIONS (NONMONO- b d 0 0 1 TONIC REASONING) c d 0 0 1 P P OA-TABLES X Y X Y (INCOMPLETE- Figure: The integrated a b , a b NESS) database along with source c d c d BACKGROUND information b c b d (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED Figure: The repairs of the database based on LOGIC PROGRAMS a belief in source s1 INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra The Repair Program Inconsistency and Incompleteness The repair program consists of: in Relational Databases and change program: responsible for the insertions and deletions in Logic Programs order to restore consistency Navin Viswanath (advised by persistence rules: enforce the fact that tuples in the database Dr.Rajshekhar Sunderraman) remain intact unless they violate constraints INTRODUCTION The change program: D-RELATIONS Facts: (INCOMPLETE- NESS) DEFAULT p(a, b) ← . p(a, e) ← . p(b, c) ← . RELATIONS (NONMONO- p(b, d) ← . p(c, d) ← . TONIC REASONING) OA-TABLES Triggering rule: (INCOMPLETE- NESS) BACKGROUND ¬p (X , Y ) ∨ ¬p (X , Z ) ← p(X , Y ), p(X , Z ), Y = Z . (LOGIC PRO- GRAMMING) INCONSISTENCY Stabilizing rule: IN EXTENDED LOGIC PROGRAMS ¬p (X , Z ) ← p(X , Y ), ydom(Z ), Y = Z . INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra The Repair Program Inconsistency and Incompleteness The persistence rules: in Relational Databases and s-rules: Logic Programs Navin Viswanath (advised by ps (a, b) ← s1 . ps (a, e) ← s2 . ps (b, c) ← s2 . Dr.Rajshekhar Sunderraman) ps (b, d) ← s3 . ps (c, d) ← s1 , s3 . INTRODUCTION D-RELATIONS Persistence defaults: (INCOMPLETE- NESS) DEFAULT p (X , Y ) ← ps (X , Y ). RELATIONS (NONMONO- p (X , Y ) ← p(X , Y ), not ¬p (X , Y ). TONIC REASONING) ¬p (X , Y ) ← xdom(X ), ydom(Y ), not p(X , Y ), OA-TABLES (INCOMPLETE- not p (X , Y ). NESS) BACKGROUND (LOGIC PRO- Starter rules: For 1 ≤ i ≤ 3, GRAMMING) INCONSISTENCY IN EXTENDED si ← si . ¬si ← ¬si . LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Correctness of Source-aware Repairs Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Theorem INTRODUCTION 1 For every source-aware answer set Ssource of Π, there exists a D-RELATIONS repair r of the database instance r w.r.t the integrity (INCOMPLETE- NESS) constraints IC such that r = {p(¯) | p (¯) ∈ Ssource } a a DEFAULT RELATIONS 2 For every repair r of the database instance r w.r.t the integrity (NONMONO- TONIC constraints IC that is consistent with the set of sources REASONING) believed(disbelieved), there exists a source-aware answer set OA-TABLES (INCOMPLETE- Ssource such that r = {p(¯) | p (¯) ∈ Ssource } a a NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Conclusions Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by We have presented data models capable of handling incomplete Dr.Rajshekhar Sunderraman) information under the open world assumption INTRODUCTION d-relations handle disjunctive information in an open world D-RELATIONS relational database (INCOMPLETE- NESS) default relations introduce the two-negation concept in a DEFAULT RELATIONS relational database (NONMONO- TONIC We have developed a translation technique to handle REASONING) inconsistent information in extended logic programs OA-TABLES (INCOMPLETE- NESS) We have developed a technique to compute “preferred” repairs BACKGROUND based on lineage information (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Future Work Inconsistency and Incompleteness in Relational Databases and Logic Programs Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Semantic deﬁnition of the repairs of a disjunctive database that INTRODUCTION is inconsistent D-RELATIONS (INCOMPLETE- Eﬃcient computation of the repairs NESS) Use of data models for incomplete information to represent DEFAULT RELATIONS repairs (NONMONO- TONIC REASONING) Semantic Web reasoning and its relationship to extended logic OA-TABLES programs (INCOMPLETE- NESS) BACKGROUND (LOGIC PRO- GRAMMING) INCONSISTENCY IN EXTENDED LOGIC PROGRAMS INCONSISTENCY IN RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra Publications Inconsistency Navin Viswanath. and Incompleteness Explicit and default negation relational databases and logic programs. in Relational In Proceedings of the SIGMOD Workshop on Innovative Database Research, IDAR, 2008. Databases and Logic Programs Navin Viswanath and Rajshekhar Sunderraman. Handling disjunctions in open world relational databases. Navin Viswanath In Proceedings of NAFIPS, pages 1 – 8, 2008. (advised by One of six best student papers. Dr.Rajshekhar Sunderraman) Navin Viswanath and Rajshekhar Sunderraman. Defaults in open world relational databases. INTRODUCTION In Proceedings of IPMU, pages 212 – 219, 2008. Won a grant of 250 euros. D-RELATIONS (INCOMPLETE- Navin Viswanath and Rajshekhar Sunderraman. NESS) Degrees of exclusivity in disjunctive databases. DEFAULT In Proceedings of ISMIS, pages 375 – 380, 2008. RELATIONS (NONMONO- Navin Viswanath and Rajshekhar Sunderraman. TONIC Query processing in paraconsistent databases in the presence of integrity constraints. REASONING) In Proceedings of SEKE, 2007. OA-TABLES Navin Viswanath and Rajshekhar Sunderraman. (INCOMPLETE- NESS) A paraconsistent relational data model. In Handbook of Research on Innovations in Database Technologies and Applications : Current and Future Trends, pages BACKGROUND 18 – 27. Idea Group Publishing, 2009. (LOGIC PRO- GRAMMING) Navin Viswanath and Rajshekhar Sunderraman. Source-aware repairs for inconsistent databases. INCONSISTENCY In Proceedings of DBKDA, pages 125 – 130, 2009. IN EXTENDED LOGIC PROGRAMS Navin Viswanath and Rajshekhar Sunderraman. Handling inconsistencies in extended logic programs through program transformation. INCONSISTENCY In Proceedings of IICAI, 2009. IN To appear. RELATIONAL DATABASES Navin Viswanath (advised by Dr.Rajshekhar Sunderraman) Inconsistency and Incompleteness in Relational Databases and Logic Progra