Semantics for Active Integrity Constraints Using Approximation Fixpoint Theory
Bart Bogaerts, Luís Cruz-Filipe
(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:
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
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
Example: Logic programming
- Given:
- a logic program P
- Immediate consequence operator TP
- Fitting's four-valued immediate consequence operator ΨP
Contributions
Contributions
- We define an approximator for AICs
- Sheds new light on the relationship with, e.g., logic programming
- Induces
new semantics for AICs
- AFT-well-founded repair
- Kripke-Kleene repair
- grounded repairs (earlier work)
- stable repairs
- 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
AFT-well-founded repair computation:
(unfoundedness reduction)
(rule application)
Example
AFT-well-founded repair computation:
(rule application)
(unfoundedness reduction)
(rule application)
Example
AFT-well-founded repair computation:
(rule application)
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
- Grounded Fixpoints and Active Integrity Constraints. Luìs Cruz-Filipe
- Bart Bogaerts, Luís Cruz-Filipe. Semantics for Active Integrity Constraints Using Approximation Fixpoint Theory
- Bart Bogaerts, Luís Cruz-Filipe. Fixpoint Semantics for Active Integrity Constraints
Semantics for Active Integrity Constraints Using Approximation Fixpoint Theory
By krr
Semantics for Active Integrity Constraints Using Approximation Fixpoint Theory
IJCAI'17
- 1,782