Consistency Axioms for RPRs

• The paper introduces a formal framework of consistency-related axioms that enforce preferred repairs based on user-specified priorities in inconsistent databases.
• It categorizes repairs into globally-optimal, Pareto-optimal, and completion-optimal families, each with distinct computational complexity and practical algorithmic consequences.
• The axiomatic framework balances minimal change with user preferences, guiding efficient conflict resolution and consistent query answering in data management.

Consistency-related axioms for RPRs (Repair Preferences) represent a formal foundation for selecting and characterizing preferred solutions among possible repairs of inconsistent relational databases, especially in the presence of user-specified priorities. These axioms articulate the essential desiderata for any mechanism that incorporates preferences during conflict resolution and have concrete algorithmic and computational consequences. The interplay between these axioms and the structure of repair families leads to a nuanced landscape of query answering and practical data management strategies.

1. Formal Axiomatics of Preferred Repairs

When designing a framework for preferred repairing, any admissible family of preferred repairs must satisfy a set of foundational axioms:

• P1: Non-emptiness. For every database instance $I$ (with constraints $F$) and priority relation $>$, there exists at least one preferred repair: $\mathrm{XRep}(I, F, >) \neq \varnothing$.
• P2: Monotonicity. Extending the priority relation (i.e., refining $>$ to $>_2 \supseteq >_1$) can only shrink (or leave unchanged) the set of preferred repairs: $$\mathrm{XRep}(I, F, >_2) \subseteq \mathrm{XRep}(I, F, >_1)$$.
• P3: Non-discrimination. If there are no user-specified preferences (i.e., $>$ is empty), the set of preferred repairs coincides with the classical repairs: $$\mathrm{XRep}(I, F, \varnothing) = \mathrm{Rep}(I, F)$$.
• P4: Categoricity. If the priority is total – every conflict is completely ordered – the preferred repair is unique: $| \mathrm{XRep}(I, F, >) | = 1$ if $>$ is total.
• P5: Conservativeness (implicit). Preferred repairs are always classical repairs: $$\mathrm{XRep}(I, F, >) \subseteq \mathrm{Rep}(I, F)$$.

These axioms constrain any preferred repair mechanism to be both informative (reflecting user preferences) and conservative with respect to classical database repairs.

2. Structural Classes of Preferred Repairs

Preferred repairs are organized into three main families, differentiated by the semantics of how preferences are "lifted" from conflicts to entire repairs:

Repair Family	Definitional Principle	Computational Complexity (Checking)
Globally-optimal	No subset $X \subseteq I'$ can be replaced by $Y \subseteq I \setminus I'$ such that $\forall x \in X, \exists y \in Y: y > x$ and (I' \ X) ∪ Y is a repair.	coNP-complete
Pareto-optimal	For no non-empty $X,Y$ as above, every $y \in Y$ is preferred to every $x \in X$.	LOGSPACE
Completion-optimal	Some total extension $>' \supseteq >$ yields $I'$ as the unique globally-optimal repair.	PTIME

A strict hierarchy holds: $$\mathrm{CRep} \subseteq \mathrm{GRep} \subseteq \mathrm{PRep}$$.

• Globally-optimal repairs (GRep): maximize "global domination" over all possible replacements, with a repair $I'$ preferred unless a strictly better configuration (given the priorities) exists.
• Pareto-optimal repairs (PRep): strengthen the improvement condition, requiring each replacing fact to be "uniformly better" than the facts replaced.
• Completion-optimal repairs (CRep): represent those repairs that would arise if the priority relation were "completed" to a total order.

In certain special cases (e.g., one key constraint or a single functional dependency), these families collapse to a singleton, but generally, multiple preferred repairs may exist.

3. Complexity and Algorithmic Consequences

The computational properties of these families differ markedly:

• Globally-optimal repair checking is coNP-complete; answering queries under GRep is $\Pi_2^P$-complete.
• Pareto-optimal repair checking is tractable (LOGSPACE), but query answering remains coNP-complete.
• Completion-optimal repair checking is in PTIME; whether it is PTIME-complete or in LOGSPACE is open, but query answering is coNP-complete.

Thus, Pareto and completion optimality admit efficient repair checking, though the query answering problem retains high complexity except in restricted scenarios (e.g., single FD and ground queries yield PTIME).

4. Preferential Conflict Resolution

Preferences are encoded using an acyclic binary "priority" relation $>$ defined on facts, operational only between "neighbors" in the conflict graph (i.e., facts sharing a conflict). This acyclicity is crucial; cycles in $>$ are not allowed under the present framework.

• Partial priorities are handled by computing repairs under various compatible total extensions, then selecting those that optimize the original priorities.
• Practical conflict resolution proceeds by iteratively removing less-preferred facts according to $>$; where no priority applies, arbitrary or completion-based choices resolve.
• Example: In salary records, the rule "prefer higher salary in case of manager conflicts" manifests as a partial priority and drives which facts are deleted in repairs.

5. Implications of the Axiomatic Framework

The axiomatic framework:

• Provides a formal standard for evaluating preference-incorporating repair methods, ensuring compatibility with both user intent and minimal change principles.
• Exposes a trade-off space: stricter adherence to preference leads to uniqueness (categoricity) but increased algorithmic difficulty.
• Guides the development and benchmarking of repair algorithms: LOGSPACE checking for Pareto-optimal repairs supports practical, scalable implementations under moderate preference structures.
• Suggests future directions in handling more complex integrity constraints (e.g., universal constraints), non-acyclic priorities, probabilistic and ranked logic-based preferences, and possible connections with other database paradigms.

6. Broader Implications and Future Directions

The theory of consistency-related axioms for repair preferences informs both the design of database management systems and theoretical work in data consistency and information integration:

• The separation between different repair families, along with their respective complexity results, enables precise matching of algorithmic strategy to application requirements.
• Algorithmic insights (LOGSPACE/PTIME checking) are directly applicable in implementations where speed and tractability are crucial (e.g., streaming data repair, automated data cleaning).
• Open research avenues include expanding the treatment to cyclic priorities, integrating probabilistic or weighted preferences, and improving consistent query answering complexity.
• The framework shapes how preferences are handled in knowledge integration, federated data systems, and logical AI systems where consistent reasoning from conflicting sources is essential.

Ultimately, these axioms define the current best practice for systematic, preference-aware data repair, balancing user requirements, tractable algorithms, and sound theory.