# The P-Completeness of Inverted Index Traversal: On the Complexity of Evaluating Boolean Query DAGs

[Apple](https://yomu.fyi/company/apple) · Aug 19, 2026

**Type:** Explainer

## Summary

Modern artificial intelligence agents increasingly rely on search infrastructure to run complex, neuro-symbolic reasoning workflows that compile into deeply nested, non-monotonic Boolean queries. Standard inverted index evaluation methods struggle with these query structures, facing either exponential query complexity under Document-at-a-Time models or universe-scale space penalties under Term-at-a-Time models during logical negation. To address these theoretical limitations, the author formalizes a DAG-based retrieval language named L\_R and proves its query evaluation problem is strictly P-Complete. The paper introduces ComputePN, a deterministic and sparsity-aware algorithm that employs a Positive-Negative dual representation alongside native DAG memoization. This mechanism bounds evaluation time to O(|Q| · |U\_active|), effectively evaluating P-Complete queries natively over the index while bypassing combinatorial tree expansion and universal scan overhead.

## Context

Modern AI workflows often generate complex, deeply nested Boolean queries over text fields. Evaluating these queries natively over inverted indices causes Document-at-a-Time iterators to suffer an exponential worst-case blowup of O(2^|Q|) when unrolling re-convergent logic, while Term-at-a-Time models face an Omega(|U|) space penalty during negation.

## Approach / What changed

The author formalizes a DAG-based retrieval language, L\_R, proves its evaluation is P-Complete, and introduces ComputePN. ComputePN is a deterministic, sparsity-aware evaluation algorithm that utilizes native DAG memoization and decouples logical negation from full universe materialization via a Positive-Negative dual representation.

## Takeaways

- The query evaluation problem for the DAG-based retrieval language L\_R is strictly P-Complete.
- Document-at-a-Time evaluation suffers an O(2^|Q|) exponential complexity blowup on re-convergent logic, whereas Term-at-a-Time incurs an Omega(|U|) space penalty on negation.
- ComputePN uses a Positive-Negative dual representation and DAG memoization to bound evaluation time to O(|Q| · |U\_active|), avoiding both tree-expansion bottlenecks and universal scans.

**Tags:** [Architecture](https://yomu.fyi/topic/architecture), [Performance](https://yomu.fyi/topic/performance), [Search](https://yomu.fyi/topic/search)

[Read original post](https://machinelearning.apple.com/research/the-p-completeness-of-inverted-index-traversal)