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⊃α
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
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}
\{-a,-b\} \text{ not grounded}
{−a,−b} not grounded
\{+c\} \text{ grounded}
{+c} 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}
AFT-well-founded repair computation:
-a:u,\quad -b:u,\quad+c:u
−a:u,−b:u,+c:u
(unfoundedness reduction)
-a:f,\quad -b:f,\quad+c:u
−a:f,−b:f,+c:u
(rule application)
-a:f,\quad -b:f,\quad+c:t
−a:f,−b:f,+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=∅
AFT-well-founded repair computation:
+a:u,\quad +b:u,\quad+c:u
+a:u,+b:u,+c:u
(rule application)
+a:t,\quad +b:u,\quad+c:u
+a:t,+b:u,+c:u
(unfoundedness reduction)
+a:t,\quad +b:u,\quad+c:f
+a:t,+b:u,+c:f
(rule application)
+a:t,\quad +b:t,\quad+c:f
+a:t,+b:t,+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=∅
AFT-well-founded repair computation:
-a:u,\quad -b:u,\quad+c:u
−a:u,−b:u,+c:u
(rule application)
-a:t,\quad -b:u,\quad+c:u
−a:t,−b:u,+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)
Fixpoint Semantics for Active Integrity Constraints: Extended Abstract
By krr
Fixpoint Semantics for Active Integrity Constraints: Extended Abstract
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