Apple researchers prove Boolean query evaluation over search indexes is computationally hard

Key takeaway

• Apple researchers have proven that evaluating complex Boolean queries natively over search indexes is a computationally hard (P-Complete) problem, and introduced an algorithm called ComputePN that sidesteps the worst-case performance penalties of existing approaches.

• By using a Positive-Negative dual representation and DAG memoization, ComputePN bounds query evaluation time to O(|Q| · |U_active|), making it possible for AI agents to execute sophisticated logic-based search workflows at scale without combinatorial explosion or full-database scans.

3 Key Points

Context & Analysis

FAQ

What is P-Completeness in this context?

The paper formalizes a retrieval language (L_R) based on Directed Acyclic Graphs (DAGs) and proves that the problem of evaluating complex Boolean queries expressed in this language over inverted indexes is P-Complete—meaning it is among the hardest problems solvable in polynomial time, and is believed to lack efficient solutions.

What are the limits of existing query evaluation strategies?

Document-at-a-Time models suffer worst-case O(2^|Q|) exponential blowup in query complexity due to structural limits in how they handle re-convergent logic. Term-at-a-Time models incur Ω(|U|) space complexity penalty—requiring full-database materialization—when evaluating negation over the document universe.

How does ComputePN improve performance?

ComputePN decouples logical negation from universe-scale materialization via a novel Positive-Negative dual representation and utilizes native DAG memoization, strictly bounding evaluation time to O(|Q| · |U_active|), thus avoiding both combinatorial tree-expansion and universal scan penalties.