Apple ML paper proves P-completeness of Boolean query evaluation over inverted indices

Apple’s Machine Learning Research group published a paper proving that evaluating complex, non-monotonic Boolean queries compiled into Directed Acyclic Graphs (DAGs) over inverted indices is strictly P-complete. The authors formalize a retrieval language (L_R) and show that the evaluation problem for this language is P-complete, establishing a theoretical upper bound on the tractability of such queries.

The paper analyzes two standard query evaluation strategies over inverted indices. Stateful iterator models (Document-at-a-Time) are shown to be structurally bounded by NC^1 formula evaluation, with a worst-case exponential blowup of O(2^|Q|) when unrolling re-convergent logic. Recursive materialization models (Term-at-a-Time) incur an Ω(|U|) space complexity penalty—the Universal Scan—when evaluating logical negation over the document universe.

To address these limitations, the authors introduce ComputePN, a deterministic, sparsity-aware evaluation algorithm. ComputePN decouples logical negation from universe-scale materialization using a Positive-Negative dual representation and leverages native DAG memoization. The algorithm strictly bounds evaluation time to O(|Q| · |U_active|), avoiding both the combinatorial tree-expansion bottleneck and the universal scan penalty.

The work lays a formal foundation for computational retrieval, particularly relevant to neuro-symbolic reasoning workflows that increasingly depend on search infrastructure to execute complex logical queries over text fields.