Fixpoint Semantics for Active Integrity Constraints: Extended Abstract

Bart Bogaerts, Luís Cruz-Filipe
(Vrije Universiteit Brussel, KU Leuven, University of Southern Denmark)

Background

Databases

  • Modern databases: integrity constraints
  • In practice: they get violated
  • How to repair the database? 
  • Many solutions have been proposed
  • We focus on Active Integrity Constraints (AICs)

Databases

  • Assumption: relational database
  • Fixed set of atoms At
  • A database is a subset of At

Active Integrity Constraints

AICs

Set of rules of the form:

l_1 \wedge l_2\wedge \cdots \wedge l_n \supset \alpha
l1∧l2∧⋯∧ln⊃αl_1 \wedge l_2\wedge \cdots \wedge l_n \supset \alpha

where:

  • Each li is a literal (an atom or its negation)
  • α is an action of the form +a or -a

AICs

  • Specify constraints on a database
  • And also how to repair them if violated

AICs

  • Joint behavior of such rules can be complex
    • Hard to determine good repairs
  • Various semantics exist
  • However, often unsatisfying 
    • Unwieldy
    • Do not respect stratification
    • Do not respect inertia
    • Too restrictive

Goals

Observation

  • AIC intuitions similar to NMR fields:
    • Logic programming
    • Default logic
    • Autoepistemic logic
    • Abstract Argumentation
    • ...
  • E.g., minimality of change
  • E.g., law of inertia

Goal

  • Transfer expertise between NMR and AICs...
  • ... in a principled way

Contributions

Contributions

  • We apply Approximation Fixpoint Theory to AICs
  • Result: a new family of semantics
    • Closely related to semantics of NMR domains
    • Foundations to transfer expertise
    • Some with very interesting properties

Approximation Fixpoint Theory

Approximation Fixpoint Theory (AFT)

  • Abstract, algebraic framework
  • Based on Lattice theory
  • Given: lattice operator and approximating (bilattice) operator:
    • Kripke-Kleene fixpoint
    • Well-founded fixpoint
    • Supported fixpoints
    • (partial) Grounded fixpoints
    • (partial) Stable fixpoints

Approximation Fixpoint Theory (AFT)

  • Used to formalize semantics of various NMR domains
    • Logic programming
    • Abstract argumentation
    • ...
  • Unifies paradigms
  • Simplifies proofs
  • Enables transfer of results
  • Provides confidence

Example: Logic programming

  • Given:
    • a logic program P
    • Immediate consequence operator TP
    • Fitting's four-valued immediate consequence operator ΨP
  • ΨP is an approximator of TP
  • All fixpoints coincide with equally named semantics

    • Contributions

    Contributions

    • We define a semantic operator for AICs
    • Based on first principles:
      • Inertia (change nothing unless a rule dictates it)
      • Cancellation (+a and -a cancel each other out)
      • Completeness (if a body is satisfied, the head must be derived)
    • Induces new semantics for AICs
      • grounded repairs

    Grounded repairs

    • Intuitiely:
      • No self-supporting (sets of) updates
      • All changes are grounded in the database
    • A set of update actions is grounded if whenever we remove some of them, at least one of them is rederived.

    Example

    \begin{array}{l} a \wedge \neg b \supset -a \\ \neg a \wedge b \supset -b \\ a \wedge \neg c \supset +c \end{array} \quad DB = \{a,b\}
    a∧¬b⊃−a¬a∧b⊃−ba∧¬c⊃+cDB={a,b}\begin{array}{l} a \wedge \neg b \supset -a \\ \neg a \wedge b \supset -b \\ a \wedge \neg c \supset +c \end{array} \quad DB = \{a,b\}
    \{-a,-b\} \text{ not grounded}
    {−a,−b} not grounded\{-a,-b\} \text{ not grounded}
    \{+c\} \text{ grounded}
    {+c} grounded\{+c\} \text{ grounded}

    Contributions

    • We define an approximator for AICs
    • Induces new semantics for AICs
      • AFT-well-founded repair
      • Kripke-Kleene repair
      • partial grounded repairs
      • (partial) stable repairs
    • Sheds new light on the relationship with, e.g., logic programming
    • Complexity analysis: same as for equally-named logic-programming semantics
    • Study relationship with existing semantics

    Property

    All AFT-style semantics for AICs have the shifting property

    Property

    If the AFT-well-founded repair is two-valued, it is also well-founded (as defined by Cruz-Filipe et al. (2013))

    Property

    All stable repairs are justified

    If your set of AICs is unipolar, the inverse holds as well

    Property

    All justified repairs are grounded

    AFT-well-founded repair

    • Natural
    • Polynomially computable
    • Approximates classes of repairs
      • grounded
      • stable
      • justified

    Example

    \begin{array}{l} a \wedge \neg b \supset -a \\ \neg a \wedge b \supset -b \\ a \wedge \neg c \supset +c \end{array} \quad DB = \{a,b\}
    a∧¬b⊃−a¬a∧b⊃−ba∧¬c⊃+cDB={a,b}\begin{array}{l} a \wedge \neg b \supset -a \\ \neg a \wedge b \supset -b \\ a \wedge \neg c \supset +c \end{array} \quad DB = \{a,b\}

    AFT-well-founded repair computation:

    -a:u,\quad -b:u,\quad+c:u
    −a:u,−b:u,+c:u-a:u,\quad -b:u,\quad+c:u

    (unfoundedness reduction)

    -a:f,\quad -b:f,\quad+c:u
    −a:f,−b:f,+c:u-a:f,\quad -b:f,\quad+c:u

    (rule application)

    -a:f,\quad -b:f,\quad+c:t
    −a:f,−b:f,+c:t-a:f,\quad -b:f,\quad+c:t

    Example

    \begin{array}{l} \neg a \supset +a \\ \neg a \wedge \neg b \wedge \neg c \supset +c \\ a \wedge \neg b \supset +b\\ a \wedge b \wedge c \supset -b \end{array}\qquad DB = \emptyset
    ¬a⊃+a¬a∧¬b∧¬c⊃+ca∧¬b⊃+ba∧b∧c⊃−bDB=∅\begin{array}{l} \neg a \supset +a \\ \neg a \wedge \neg b \wedge \neg c \supset +c \\ a \wedge \neg b \supset +b\\ a \wedge b \wedge c \supset -b \end{array}\qquad DB = \emptyset

    AFT-well-founded repair computation:

    +a:u,\quad +b:u,\quad+c:u
    +a:u,+b:u,+c:u+a:u,\quad +b:u,\quad+c:u

    (rule application)

    +a:t,\quad +b:u,\quad+c:u
    +a:t,+b:u,+c:u+a:t,\quad +b:u,\quad+c:u

    (unfoundedness reduction)

    +a:t,\quad +b:u,\quad+c:f
    +a:t,+b:u,+c:f+a:t,\quad +b:u,\quad+c:f

    (rule application)

    +a:t,\quad +b:t,\quad+c:f
    +a:t,+b:t,+c:f+a:t,\quad +b:t,\quad+c:f

    Example

    \begin{array}{l} \neg a \supset +a \\ \neg b \wedge \neg c \supset +b \\ \neg b \wedge \neg c \supset +c \end{array} \quad DB = \emptyset
    ¬a⊃+a¬b∧¬c⊃+b¬b∧¬c⊃+cDB=∅\begin{array}{l} \neg a \supset +a \\ \neg b \wedge \neg c \supset +b \\ \neg b \wedge \neg c \supset +c \end{array} \quad DB = \emptyset

    AFT-well-founded repair computation:

    -a:u,\quad -b:u,\quad+c:u
    −a:u,−b:u,+c:u-a:u,\quad -b:u,\quad+c:u

    (rule application)

    -a:t,\quad -b:u,\quad+c:u
    −a:t,−b:u,+c:u-a:t,\quad -b:u,\quad+c:u

    fixpoint

    Conclusion

    Conclusion

    • Novel class of semantics for AICs
      • Based on intuitions from NMR
      • Intuitive behaviour
      • Complexity analysis
    • Studied relationship with existing semantics
    • New insights on relationship with, e.g., logic programming

    Read more

    • Fixpoint Semantics for Active Integrity Constraints B. Bogaerts and L. Cruz-Filipe. Artificial Intelligence Volume 255, p. 43-70, 2018.
    • Stratification in Approximation Fixpoint Theory and Its Application to Active Integrity Constraints B. Bogaerts and L. Cruz-Filipe. (work in progress)
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