The Quantified Boolean Bayesian Network: Theory and Experiments with a Logical Graphical Model (2402.06557v1)

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 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.

• The paper introduces a unified framework that integrates logical deduction with probabilistic reasoning using a novel graphical model.
• It eliminates hallucinations by grounding inference in both statistical laws and logical causality, overcoming limitations of traditional LLMs.
• The model achieves faster approximate inference through strategic node partitioning, offering insights into dual-process cognitive theories.

The Quantified Boolean Bayesian Network: A Synergy Between Logic and Probability

The paper introduces the Quantified Boolean Bayesian Network (QBBN), a novel model that represents an advancement in integrating logical and probabilistic reasoning. Positioned within the Bayesian Network framework, the QBBN aims to address some of the inherent limitations found in current LLMs by providing a unified and efficient approach to reasoning. The key contributions, methodological innovations, and potential implications of the QBBN are discussed herein.

Key Contributions and Findings

1. Unified Framework: The QBBN is formulated as a graphical model that supports both statistical and logical reasoning. This dual capability is achieved through a cohesive structure that allows for handling probabilistic queries and engaging in consistent logical deduction. The model leverages principles from first-order logic while facilitating probabilistic reasoning akin to Bayesian Networks.
2. Non-Hallucinating Generative Model: The paper highlights a significant advantage of the QBBN over conventional LLMs: the elimination of hallucinations in generative tasks. By ensuring that the model’s outputs are consistent with statistical laws (i.e., $P(x) + P(\neg x) = 1$), and by grounding its reasoning in logical causality, the QBBN provides more reliable and explainable outputs. This characteristic stems from its ability to provide causal explanations and grounded responses based on learned probabilistic structures.
3. Increased Computational Efficiency: A critical strength of the QBBN lies in its method for more efficient approximate inference. By separating network nodes into types (such as conjunction and disjunction) and employing an iterative belief propagation algorithm, the QBBN achieves a considerable reduction in computational burden. Whereas traditional Bayesian Networks require $\Omega(2^N)$ time for inference, the QBBN's structured approach allows for a time complexity of $O(N2^n)$, where $n$ is minimized through strategic node partitioning.
4. Fast and Slow Thinking: The QBBN offers a mathematical explanation for the dichotomy between "fast" intuitive responses and "slow" deliberate reasoning, reflecting the dual-process theory of cognition. Through its graphical and logical formulations, the model presents a pathway for understanding complex planning and decision-making processes, which current LLMs inadequately address.
5. Dependency Tree Calculus: The QBBN introduces a calculus that effectively maps linguistic expressions to structured logical forms, allowing for easier knowledge encoding and more efficient semantic parsing. This novel calculus uses dependency structures that simplify argument position management compared to traditional first-order logic representations.

Implications and Future Directions

The introduction of the QBBN has profound implications both theoretically and practically. Theoretically, the QBBN represents a leap forward in systems attempting to unify logic and probabilistic reasoning under a single model. This work provides a foundation for further exploration into integrating more complex forms of logic, such as second-order and modal logic, in probabilistic settings.

Practically, the QBBN holds promise for improving various AI applications, particularly those requiring reliable decision-making and explanation capabilities, such as autonomous systems, conversational agents, and cognitive computing applications. The model’s non-hallucinating nature and efficiency in computation stand to improve the deployment of AI systems in environments where trust and computational resources are critical considerations.

The implementation of QBBNs also raises intriguing questions about their use in learning from unlabeled data, a domain currently dominated by LLMs. The exploration of methods for expectation maximization or other unsupervised learning techniques within the QBBN framework could open new avenues for advancing AI technologies. Furthermore, refining belief propagation techniques to enhance convergence and computation speed remains an area ripe for investigation.

Overall, the QBBN's innovative approach to combining logic and probability underpins a potentially transformative advance in AI research, with wide-ranging applications and exciting future directions for development and refinement.