Neural Scalable Symbolic Search Framework for Complex Logical Queries with Multiple Free Variables

Complex Query Answering (CQA) is a fundamental knowledge representation and reasoning task over incomplete knowledge graphs (KGs). Answering existential first-order queries with 𝑘 free variables (i.e., EFO 𝑘 queries) is a crucial yet challenging problem, as it requires ranking answer tuples in E 𝑘 , where E denotes the entity set of a KG. This quickly becomes intractable as 𝑘 grows. Consequently, existing benchmarks and methods rely on marginal rankings over individual variables; however, marginal rankings are a poor proxy for the true joint ranking of tuples. Building on neural symbolic search for EFO 1 queries, we propose Neural Scalable Symbolic Search (NS3), a budgeted framework that approximates joint ranking without enumerating E 𝑘 . NS3 (i) answers marginalized sub-queries to obtain necessary candidate sets, (ii) merges multiple free variables into hypernodes whose domains are pruned and controlled by a dynamic budget 𝐵, and (iii) progressively reduces an EFO 𝑘 query to an EFO 𝑘 -1 query over a budgeted reduced domain. Across three standard KG datasets, NS3 substantially improves joint ranking performance while retaining strong marginal accuracy. We further release a joint-ranking benchmark that extends existing EFO 1 datasets to 𝑘 = 3, enabling systematic evaluation of multi-variable queries. Our code is provided in https://github.com/HKUST-KnowComp/NS3_KDD2026.