We consider the dichotomy conjecture for consistent query answering under primary key constraints. It states that, for every fixed Boolean conjunctive query q, testing whether q is certain (i.e. whether it evaluates to true over all repairs of a given inconsistent database) is either polynomial time or coNP-complete. This conjecture has been verified for self-join-free and path queries. We propose a simple inflationary fixpoint algorithm for consistent query answering which, for a given database, naively computes a set $\Delta$ of subsets of facts of the database of size at most k, where k is the size of the query q. The algorithm runs in polynomial time and can be formally defined as: (1) Initialize $\Delta$ with all sets $S$ of at most $k$ facts such that $S\models q$. (2) Add any set $S$ of at most k facts to $\Delta$ if there exists a block $B$ (i.e., a maximal set of facts sharing the same key) such that for every fact $a \in B$ there is a set $S' \subseteq S \cup \{a\}$ such that $S'\in \Delta$. For an input database $D$, the algorithm answers "q is certain" iff $\Delta$ eventually contains the empty set. The algorithm correctly computes certainty when the query q falls in the polynomial time cases of the known dichotomies for self-join-free queries and path queries. For arbitrary Boolean conjunctive queries, the algorithm is an under-approximation: the query is guaranteed to be certain if the algorithm claims so. However, there are polynomial time certain queries (with self-joins) which are not identified as such by the algorithm.

@article{FigueiraPSS23,
  title = {A Simple Algorithm for Consistent Query Answering under Primary Keys},
  author = {Diego Figueira and Anantha Padmanabha and Luc Segoufin and Cristina Sirangelo},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 1},
  year = {2025},
  doi = {10.46298/lmcs-21(1:18)2025}
}