Structured knowledge fact checking aims to determine the truthfulness of natural language claims by reasoning over structured evidence. Recent program-generation approaches leverage large language models (LLMs) to generate executable graph reasoning programs, achieving strong performance on structured knowledge fact checking benchmarks. However, these methods remain limited by invalid relation generation, single-path reasoning that lacks self-correction, and biased evidence assessment that tends to overestimate supporting signals. We propose Dialectical Agentic REasoning (DARE), a multi-agent framework that formulates structured knowledge fact checking as an iterative retrieve-reason-reflect process. DARE integrates relation-grounded evidence retrieval to constrain reasoning to valid structures, dialectical bidirectional verification to evaluate evidence from both supporting and refuting perspectives, and confidence-driven meta-reflection to dynamically determine whether additional evidence exploration is necessary. Extensive experiments demonstrate the effectiveness of DARE in structured knowledge fact checking, with an 8B backbone achieving 88.12% accuracy and matching or surpassing GPT-4o-based program-generation baselines, which attests to the efficacy of dialectical agentic reasoning in eliciting the latent reasoning capabilities of LLMs. Our code is available at https://github.com/inflee/DARE.

Query embedding (QE)—which aims to embed entities and first-order logical (FOL) queries in a vector space, has shown great power in answering FOL queries on knowledge graphs (KGs). Existing QE methods divide a complex query into a sequence of mini-queries according to its computation graph and perform logical operations on the answer sets of mini-queries to get answers. However, most of them assume that answer sets satisfy an individual distribution (e.g., Uniform, Beta, or Gaussian), which is often violated in real applications and limit their performance. In this paper, we propose a Neural-based Mixture Probabilistic Query Embedding Model (NMP-QEM) that encodes the answer set of each mini-query as a mixed Gaussian distribution with multiple means and covariance parameters, which can approximate any random distribution arbitrarily well in real KGs. Additionally, to overcome the difficulty in defining the closed solution of negation operation, we introduce neural-based logical operators of projection, intersection and negation for a mixed Gaussian distribution to answer all the FOL queries. Extensive experiments demonstrate that NMP-QEM significantly outperforms existing state-of-the-art methods on benchmark datasets. In NELL995, NMP-QEM achieves a 31% relative improvement over the state-of-the-art.