The Quantified Boolean Bayesian Network: Theory and Experiments with a Logical Graphical Model

TL;DR

The paper introduces the Quantified Boolean Bayesian Network (QBBN), a unified framework that integrates logical deduction with probabilistic reasoning to address hallucinations and planning deficiencies in large language models. It achieves this by mapping a First-Order Calculus to a key-value semantic-role language and implementing a bipartite graph of AND/OR junctions whose inference uses loopy belief propagation with a theoretical time complexity of $O(N2^n)$ per full pass. The authors prove consistency and completeness within this graphical calculus, provide a running example with an implication graph, and demonstrate empirical convergence of belief propagation on synthetic logical structures. They also describe an efficient, on-demand construction of the proposition graph at inference time and discuss potential speedups and future research directions, including learning from unlabeled text and improving inference scalability. The work situates QBBN as a principled bridge between symbolic reasoning and probabilistic inference with practical implications for reliable, non-hallucinatory language understanding and reasoning systems.

Abstract

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. $Ω(2^N)$ 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.

Figures (9)

• Figure 1: A labeled dependency parse. Without labels, we could not do semantics, so this is the most simple structure that can support semantics.
• Figure 2: A boolean network that alternates between and and or gates.
• Figure 3: The prior state of the network, with no observations.
• Figure 4: Assume that $like({\bf x}_{jill}, {\bf x}_{jack})$ is true. Forward inferences only.
• Figure 5: Assume that $like({\bf x}_{jack}, {\bf x}_{jill})$ is true. Forward and backwards inference.