This paper introduces the Quantified Boolean Bayesian Network (QBBN), which provides a unified view of logical and probabilistic reasoning. The QBBN is meant to address a central problem with the Large Language Model (LLM), which has become extremely popular in Information Retrieval, which is that the LLM hallucinates. A Bayesian Network, by construction, cannot hallucinate, because it can only return answers that it can explain. We show how a Bayesian Network over an unbounded number of boolean variables can be configured to represent the logical reasoning underlying human language. We do this by creating a key-value version of the First-Order Calculus, for which we can prove consistency and completeness. We show that the model is trivially trained over fully observed data, but that inference is non-trivial. Exact inference in a Bayesian Network is intractable (i.e. $\Omega(2^N)$ for $N$ variables). For inference, we investigate the use of Loopy Belief Propagation (LBP), which is not guaranteed to converge, but which has been shown to often converge in practice. Our experiments show that LBP indeed does converge very reliably, and our analysis shows that a round of LBP takes time $O(N2^n)$, where $N$ bounds the number of variables considered, and $n$ bounds the number of incoming connections to any factor, and further improvements may be possible. Our network is specifically designed to alternate between AND and OR gates in a Boolean Algebra, which connects more closely to logical reasoning, allowing a completeness proof for an expanded version of our network, and also allows inference to follow specific but adequate pathways, that turn out to be fast.

翻译：本文提出量化布尔贝叶斯网络（QBBN），为逻辑推理与概率推理提供了统一视角。QBBN旨在解决当前信息检索领域广泛使用的大语言模型（LLM）的核心问题——幻觉现象。贝叶斯网络本质上不会产生幻觉，因为它只能返回可解释的答案。我们展示了如何通过配置包含无界布尔变量的贝叶斯网络，来表征人类语言背后的逻辑推理过程。为此，我们构建了基于键值对的一阶演算体系，并证明了其一致性和完备性。研究表明该模型可在完全观测数据上实现简单训练，但推理过程具有非平凡性。贝叶斯网络中的精确推理是难解的（即对于N个变量，时间复杂度为Ω(2^N)）。在推理方面，我们研究了环状置信传播（LBP）算法——该算法虽不保证收敛，但实践表明其通常能够收敛。实验证实LBP确实具有高度可靠的收敛性，分析表明每轮LBP的时间复杂度为O(N2^n)，其中N表示变量数量上界，n表示任意因子节点入连接数上界，且存在进一步优化空间。本网络专门设计为在布尔代数中交替使用AND门与OR门，使其更贴近逻辑推理机制，既能对扩展版网络进行完备性证明，又能使推理沿着特定且充分的路径快速执行。