Query Logic Dependency Graph

Updated 15 November 2025

Query Logic Dependency Graph is a formal structure that models dependencies among reasoning tasks, sub-queries, and logical atoms using DAGs, bipartite graphs, or hypergraphs.
It is constructed via techniques like retrieval-augmented few-shot prompting and classical logic analysis to ensure acyclic, well-defined scheduling for robust inference.
The graph framework enhances parallel query processing and model justification by enabling efficient scheduling, improved accuracy, and reduced latency.

A query logic dependency graph is a formal structure that explicitly models the dependencies among reasoning tasks, sub-queries, or logical atoms in the context of complex query processing, program semantics, or retrieval-augmented generation. This approach is foundational in multiple areas—database theory (as bipartite or hypergraph representations of conjunctive queries), logic programming (as positive dependency graphs for stable model semantics), and modern LLM-based reasoning frameworks (as DAGs for parallelized content expansion)—with the overarching aim of capturing logical interdependencies to support correctness, efficient scheduling, and robust inference.

1. Formal Definitions and Variants

A query logic dependency graph is typically encoded as a directed acyclic graph (DAG), bipartite graph, or hypergraph, depending on the application domain:

Directed Acyclic Graph (DAG): Orion (Gao et al., 28 Oct 2025) represents complex queries as $G = (V, E)$ , where $V = \{v_1, \ldots, v_n\}$ each denotes a key reasoning point, and $E \subseteq V \times V$ encodes directed edges $(v_k \to v_j)$ indicating point $j$ depends on point $k$ . Edges are annotated by dependency type: Null, Contextual, or Dependent.
Bipartite Graph: In GCQ and CCQ theory (Bonchi et al., 2018), the dependency graph $D_\phi = (G_V \cup G_E, E')$ has nodes for variables and predicate occurrences, with undirected edges denoting variable-predicate dataflow links; these graphs encode logical dataflow in conjunctive queries.
Edge-Labeled Graphs (with Conjunction Nodes): In logic program semantics (Li et al., 2021), enriched dependency graphs use conjunction nodes to represent multi-literal rule bodies and positive/negative edge types to differentiate deduction and negation.
Generalized Dependency Graphs: Positive dependency graphs admit several variants (Fandinno et al., 2022), e.g., $G^{pnn}(T)$ (positive nonnegated) and $G^{sp}(T)$ (strictly positive), designed to support different reasoning tasks and semantic theorems.

The graph must typically be acyclic to ensure well-defined scheduling and correct fixed-point semantics.

2. Construction Techniques

Retrieval-Augmented Few-Shot Prompting (Orion): Given a query $Q$ , Orion retrieves top- $k$ supporting passages and uses few-shot prompts to elicit from the LLM a JSON object specifying key points, dependency types, and “depends_on” lists. A parser constructs $G=(V,E)$ , verifies acyclicity, and annotates edges by type (Gao et al., 28 Oct 2025).
LLM-based Decomposition and Planning (PankRAG): Complex queries $Q$ are decomposed into sub-questions $V$ ; parallelizable task groups are detected via LLM prompting, sequential dependencies are identified for edge construction, and top-down passes resolve ambiguities using predecessors' answers (Li et al., 7 Jun 2025).
Classical Logic Program Analysis: For normal logic programs, dependency graphs are built by mapping rules $(H \leftarrow B_1, ..., B_m, not\ B_{m+1},...,not\ B_n)$ into edges from body atoms to heads (positive/negative), enriched with conjunction nodes for bodies with multiple literals (Li et al., 2021). Generalized graphs are assembled from strictly positive or positive nonnegated occurrences in arbitrary propositional theories (Fandinno et al., 2022).
String Diagram Parsing (GCQ): In database theory, the string diagram of a query is translated into a bipartite variable-predicate graph by recording incidence between variable wires and predicate boxes (Bonchi et al., 2018).

3. Operational Role in Reasoning and Inference

The dependency graph governs reasoning order, concurrency, and logical consistency:

Parallel and Conditional Expansion (Orion): Orion expands each key point according to the partial order of $G$ . For node $v_j$ , the model input incorporates parent point data based on edge type (contextual: point text; dependent: fully expanded content). Contextual edges enable concurrent prefill steps, while dependent edges serialize expansion (Gao et al., 28 Oct 2025).
RAG Scheduling and Disambiguation (PankRAG): Nodes with zero in-degree in $G$ (“parallelizable”) are resolved first. Progress dependencies block downstream queries until predecessor results are available. Subsequent reranking of retrieved candidates leverages the answers to predecessor sub-queries for enhanced semantic consistency (Li et al., 7 Jun 2025).
Model Justification and Semantics (Logic Programming): Edge-labeled dependency graphs drive fixpoint reasoning, loop checking, and justification graphs for ASP semantics, facilitating computation and explanation of stable, co-stable, or well-founded models (Li et al., 2021).
Query Containment and Optimization (GCQ): Bipartite dependency graphs enable interface-preserving homomorphism checks for query inclusion, underpinning efficient containment solvers in database engines (Bonchi et al., 2018).

4. Algorithmic Scheduling and Parallelism

Decomposition into a dependency graph directly enables sophisticated parallelization strategies:

Pipeline Scheduling (Orion): Key-point generation is compute-bound while content expansion is memory-bound; Orion pipelines these phases across multiple queries, maximizing GPU utilization by interleaving compute and memory intensive stages (Gao et al., 28 Oct 2025).
Wave-Based Query Resolution (PankRAG): Sub-queries in a wave (zero in-degree) are issued in parallel; as answers resolve, downstream queries “fire” per topological order. Dependency-aware reranking further synchronizes retrieval and answer generation using the graph (Li et al., 7 Jun 2025).

Empirical results showcase that explicit modeling of inter-point dependencies achieves up to 4.33× token-generation speed, 3.42× latency reduction (multi-query, pipeline scheduled), and up to +18.75% improvement in reasoning quality on multi-point logic tasks over traditional chain-of-thought or sequential approaches (Gao et al., 28 Oct 2025). Ablations confirm that dependency annotations are essential—purely parallel expansion without the graph causes 10–15% loss in reasoning quality.

5. Applications and Theoretical Significance

Efficient, High-Quality LLM Reasoning: Frameworks such as Orion realize both low latency and robust logical consistency for real-time web applications by leveraging the logic dependency DAG as a concurrency and consistency scaffold (Gao et al., 28 Oct 2025).
Retrieval-Augmented Generation (RAG): PankRAG’s dependency-aware query graph ensures relevant, context-enriched retrieval, mitigating hallucination and elevating comprehensiveness, faithfulness, and relevance metrics on standard datasets (Li et al., 7 Jun 2025).
Answer Set Programming Semantics: Generalized positive dependency graphs ( $G^{sp}$ , $G^{pnn}$ ) sharpen classical theorems: under acyclicity of $G^{sp}(T)$ , supported and pointwise stable models are guaranteed stable; loops and splitting require the richer $G^{pnn}(T)$ . Applications include automatic verification of tightness and generalization to infinitary or first-order theories (Fandinno et al., 2022).
Database Query Optimization: GCQ and CCQ dependency graphs provide a graph-theoretic view of dataflow, allowing efficient query containment tests and supplying the combinatorial backbone for algebraic completeness results in database theory (Bonchi et al., 2018).
Justification and Explanation: In logic program interpretation, dependency graphs facilitate extraction of justification trees for computed models, aiding both automated reasoning and human interpretability (Li et al., 2021).

6. Limitations and Open Problems

Domain-Specific Graph Construction: Generalizing the construction of logic dependency graphs from LLM-prompted decomposition or classical rule analysis to arbitrary propositional, first-order, or non-grounded settings remains an open question (Fandinno et al., 2022).
Component Selection for Reasoning Tasks: The minimal dependency graph sufficient for tightness/completion (e.g., $G^{sp}$ ) may be too weak for loop formulas or splitting results, which demand richer annotations (e.g., $G^{pnn}$ ) (Fandinno et al., 2022).
Scalability and Overhead: Graph-based interpreters introduce 2–5× overhead in stable model computation compared to direct solvers, but supply unified semantics and automatic explanations (Li et al., 2021).
Semantic Ambiguities: The empirical benefit of global disambiguation (rerewriting sub-queries in light of dependency graph answers) is demonstrated in RAG, but optimal reranking equations and integration strategies remain a subject of ongoing research (Li et al., 7 Jun 2025).

A plausible implication is that continued refinement and application-specific tailoring of query logic dependency graph extraction and annotation will underpin advances in scalable, explainable, and consistent reasoning—across symbolic logic, LLM frameworks, and retrieval-augmented systems.