Inconsistency and Incompleteness in Relational Databases and Logic by g4509244

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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

								
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