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

A paper by Amir Aavani, published in August 2026, proves that evaluating Boolean query DAGs over inverted indexes is P-Complete, establishing theoretical limits for AI agents' search workflows. The paper introduces ComputePN, a deterministic algorithm that bounds evaluation time to O(|Q| · |U_active|) by decoupling negation from universe-scale materialization, avoiding exponential blowup and universal scan penalties.

content type paper /research/ published August 2026 The P-Completeness of Inverted Index Traversal: On the Complexity of Evaluating Boolean Query DAGs AuthorsAmir Aavani The P-Completeness of Inverted Index Traversal: On the Complexity of Evaluating Boolean Query DAGs AuthorsAmir Aavani Modern AI agents increasingly rely on search infrastructure to execute complex, neuro-symbolic reasoning workflows. These workflows often compile into deeply nested, non-monotonic Boolean queries over text fields. However, standard query evaluation strategies over inverted indices face severe theoretical limits when handling these structures. Stateful iterator models Document-at-a-Time are structurally bounded by NC^1 formula evaluation, suffering a worst-case O 2^|Q| exponential blowup in query complexity when unrolling re-convergent logic. Conversely, 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. In this paper, we establish the theoretical boundaries of executing complex logic natively over an inverted index. We formalize a retrieval language L R based on Directed Acyclic Graphs DAGs and prove that its evaluation problem is strictly P-Complete. To make evaluation tractable, we introduce ComputePN, a deterministic, sparsity-aware evaluation algorithm. By decoupling logical negation from universe-scale materialization via a novel Positive-Negative dual representation, and utilizing native DAG memoization, ComputePN strictly bounds evaluation time to O |Q| · |U active| . This approach successfully evaluates P-Complete queries natively over the index, avoiding both the combinatorial tree-expansion bottleneck and the universal scan penalty, laying the formal foundation for computational retrieval. Scalable Private Search with Wally October 16, 2024 research area Privacy /research/?domain=Privacy This paper presents Wally, a private search system that supports efficient semantic and keyword search queries against large databases. When sufficiently many clients are making queries, Wally’s performance is significantly better than previous systems. In previous private search systems, for each client query, the server must perform at least one expensive cryptographic operation per database entry. As a result, performance degraded… Synthetic Query Generation using Large Language Models for Virtual Assistants June 17, 2024 research area Knowledge Bases and Search /research/?domain=Knowledge%20Bases%20and%20Search , research area Speech and Natural Language Processing /research/?domain=Speech%20and%20Natural%20Language%20Processing conference SIGIR /research/?event=SIGIR This paper was accepted in the Industry Track at SIGIR 2024. Virtual Assistants VAs are important Information Retrieval platforms that help users accomplish various tasks through spoken commands. The speech recognition system speech-to-text uses query priors, trained solely on text, to distinguish between phonetically confusing alternatives. Hence, the generation of synthetic queries that are similar to existing VA usage can greatly improve…