Exploring the Paradigm Shift from Grounding to Skolemization for Complex Query Answering on Knowledge Graphs

TL;DR

This work addresses Complex Query Answering (CQA) over incomplete Knowledge Graphs by formalizing a Grounding–Skolemization dichotomy and advocating a Skolemization-based paradigm. It introduces LVSA, a neuro-symbolic framework built on Vector Symbolic Architecture that combines a differentiable Skolemization module and a neural negator within a logic-driven optimization protocol, and proves universal expressivity for all $EFO_1$ queries with favorable $O(k d^2 + |oldsymbol{V}| d)$ complexity. Empirically, LVSA outperforms state-of-the-art Skolemization-based methods and demonstrates substantial efficiency gains over Grounding-based baselines, with strong generalization to challenging, partially observable and fully unobservable benchmarks. The results highlight the practical potential of Skolemization-based CQA for scalable, transparent reasoning in real-world KGs and point to future directions for enhancing Skolem function modeling and logical constraint enforcement.

Abstract

Complex Query Answering (CQA) over incomplete Knowledge Graphs (KGs), typically formalized as reasoning with Existential First-Order predicate logic with one free variable (EFO\textsubscript{1}), faces a fundamental tradeoff between logic fidelity and computational efficiency. This work establishes a Grounding-Skolemization dichotomy to systematically analyze this challenge and motivate a paradigm shift in CQA. While Grounding-based methods inherently suffer from combinatorial explosion, most Skolemization-based methods neglect to explicitly model Skolem functions and compromise logical consistency. To address these limitations, we propose the Logic-constrained Vector Symbolic Architecture (LVSA), a neuro-symbolic framework that unifies a differentiable Skolemization module and a neural negator, as well as a logical constraint-driven optimization protocol to harmonize geometric and logical requirements. Theoretically, LVSA guarantees universality for all EFO\textsubscript{1} queries with low computational complexity. Empirically, it outperforms state-of-the-art Skolemization-based methods and reduces inference costs by orders of magnitude compared to Grounding-based baselines.

Figures (10)

• Figure 1: (i) An example KG and a complex query in EFO1 form. $-1$ denotes the inverse relation. (ii) Grounding-based reasoning workflow. (iii) Skolemization-based reasoning workflow.
• Figure 2: (i) Taxonomy of CQA methods along the Grounding-Skolemization and Neuro-Symbolic dimensions. (ii) Comparison of Skolemization-based and Grounding-based CQA methods in efficiency and logic fidelity.
• Figure 3: Diagram of the query graph and topological sorting workflow for an example query.
• Figure 4: LVSA's inference process for the sample complex query $Q[V_?]=V_?: \exists V: r_A(\text{A}, V) \wedge r_A^{-1}(V, V_?) \wedge r_W^{-1}(\text{N}, V_?)$.
• Figure 5: (i) Query graphs of 14 standard query types. Specifically, query types labeled with "p" (except the atomic query 1p) contain existential quantifiers. Besides, "i", "u", and "n" indicate the presence of logical conjunction, disjunction, and negation operations, respectively. (ii) Query graphs of novel query types from new benchmarks. (iii) Samples of partially observable queries in the standard benchmarks and fully unobservable queries in the new benchmarks.

Theorems & Definitions (6)

• Definition 1: Grounding
• Definition 2: Skolemization
• Theorem 1: Universal Expressivity for EFO1
• proof
• Proposition 1: Computational Complexity of LVSA for Existential Quantification
• proof