Fashionable AI brokers more and more depend on search infrastructure to execute advanced, neuro-symbolic reasoning workflows. These workflows usually compile into deeply nested, non-monotonic Boolean queries over textual content fields. Nevertheless, customary question analysis methods over inverted indices face extreme theoretical limits when dealing with these buildings. Stateful iterator fashions (Doc-at-a-Time) are structurally bounded by NC^1 components analysis, struggling a worst-case O(2^|Q|) exponential blowup in question complexity when unrolling re-convergent logic. Conversely, recursive materialization fashions (Time period-at-a-Time) incur an Ω(|U|) house complexity penalty (the Common Scan) when evaluating logical negation over the doc universe.
On this paper, we set up the theoretical boundaries of executing advanced logic natively over an inverted index. We formalize a retrieval language (L_R) based mostly on Directed Acyclic Graphs (DAGs) and show that its analysis downside is strictly P-Full. To make analysis tractable, we introduce ComputePN, a deterministic, sparsity-aware analysis algorithm. By decoupling logical negation from universe-scale materialization through a novel Constructive-Detrimental twin illustration, and using native DAG memoization, ComputePN strictly bounds analysis time to O(|Q| · |U_active|). This strategy efficiently evaluates P-Full queries natively over the index, avoiding each the combinatorial tree-expansion bottleneck and the common scan penalty, laying the formal basis for computational retrieval.
