Capturing P: On the Expressive Power and Efficient Evaluation of Boolean Retrieval

Modern information retrieval is transitioning from simple document filtering to complex, neuro-symbolic reasoning workflows. However, current retrieval architectures face a fundamental efficiency dilemma when handling the rigorous logical and arithmetic constraints required by this new paradigm. Standard iterator-based engines (Document-at-a-Time) do not natively support complex, nested logic graphs; forcing them to execute such queries typically results in intractable runtime performance. Conversely, naive recursive approaches (Term-at-a-Time), while capable of supporting these structures, suffer from prohibitive memory consumption when enforcing broad logical exclusions. In this paper, we propose that a retrieval engine must be capable of ``Capturing $\mathbf{P}$'' -- evaluating any polynomial-time property directly over its index in a computationally efficient manner. We define a formal Retrieval Language ($\mathcal{L}_R$) based on Directed Acyclic Graphs (DAGs) and prove it precisely captures the complexity class $\mathbf{P}$. We introduce \texttt{ComputePN}, a novel evaluation algorithm that makes $\mathcal{L}_R$ tractable. By combining native DAG traversal with a memory-efficient ``Positive-Negative'' response mechanism, \texttt{ComputePN} ensures the efficient evaluation of any query in $\mathcal{L}_R$. This work establishes the theoretical foundation for turning the search index into a general-purpose computational engine.