Logical Neural Networks (2006.13155v1)

Abstract: We propose a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning). Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a highly intepretable disentangled representation. Inference is omnidirectional rather than focused on predefined target variables, and corresponds to logical reasoning, including classical first-order logic theorem proving as a special case. The model is end-to-end differentiable, and learning minimizes a novel loss function capturing logical contradiction, yielding resilience to inconsistent knowledge. It also enables the open-world assumption by maintaining bounds on truth values which can have probabilistic semantics, yielding resilience to incomplete knowledge.

• The paper introduces Logical Neural Networks to integrate neural learning with symbolic logic, enhancing interpretability and robust reasoning.
• It maps each neuron to logical expressions and employs bidirectional inference to ensure consistency and manage computational contradictions.
• The framework features a novel loss function that penalizes logical conflicts, supporting reliable performance under incomplete or uncertain data.

Logical Neural Networks: Integrating Neural and Logical Paradigms

The paper Logical Neural Networks introduces a novel neuro-symbolic framework designed to integrate the powerful learning capabilities of neural networks (NNs) with the rigorous reasoning and interpretability features of symbolic logic. This innovative approach aims at achieving a seamless combination that retains the advantages of both paradigms while mitigating their individual limitations.

Key Contributions

The paper makes several significant contributions to the field of neuro-symbolic AI:

1. Neuron-Formula Interoperability: Every neuron in the proposed Logical Neural Networks (LNNs) directly corresponds to elements of logical formulae. This neuron-formula mapping ensures high interpretability and composability by allowing networks to explicitly represent and manipulate logical expressions.
2. Omnidirectional Inference: LNNs support omnidirectional inference due to the bidirectional relationships between neurons and their operational neighbors. This characteristic enables inference in any direction and facilitates theorem proving, demonstrating a generality beyond typical neural networks focused on predefined target variables.
3. Novel Learning Mechanism: The LNN framework introduces a unique loss function that captures logical contradictions, thereby inherently providing resilience to inconsistent knowledge. Additionally, the model supports the open-world assumption by maintaining bounds on truth values, thus accommodating incomplete knowledge.
4. Probabilistic Semantics: LNNs maintain upper and lower bounds on truth values, which can represent probabilistic semantics. This capacity permits the network to handle uncertain or imprecise information effectively.

Detailed Overview

The LNN model is designed as a recurrent neural network, with its architecture aligned to logical operations. Neurons implement logical functions such as $\land$, $\lor$, $\neg$, and $\rightarrow$, and when necessary, are extended to include first-order logic operations ($\forall, \exists$).

LNNs permit flexible yet precise `truth value bounds' on neurons, distinguishing them between known, unknown, approximately known, and contradictory states. The inference process is achieved through alternating upward and downward passes over represented formulae's syntax trees, ensuring the bounds tightening process converges reliably in finite steps.

Inference and Learning

LNN inference operates through multilayered bidirectional propagation, leveraging logical rules to tighten truth value bounds. For each formula, the system traverses the graph:

1. Upward Pass: Computes bounds at each node based on its subformulae.
2. Downward Pass: Updates bounds at subformulae based on the formula's bounds and neighboring inferences.

This recursive evaluation algorithm is not constrained by closed-world assumptions, thus supporting real-world applications where the complete state of knowledge cannot be assumed.

The learning mechanism extends traditional gradient-based optimization, introducing constraints to ensure logical operations are respected even after training. Constraints ensure:

• Conjunctions and disjunctions uphold classical truth functions when presented with classical input values.
• Logical coherence by minimizing contradictions, facilitated by a unique loss function that penalizes contradictory outcomes.

Empirical Evaluation

The authors validate the LNN framework through experiments including:

• Smokers and Friends Data: Demonstrating how LNNs can learn neuron weights and adjust bounds to resolve inconsistencies while aligning with known axioms and facts.
• LUBM (Lehigh University Benchmark): Confirming the soundness and completeness of LNN-driven reasoning on benchmark OWL data, which involves complex ontological interpretations.
• TPTP (Thousands of Problems for Theorem Provers): Successfully solving first-order logic problems without relying on functions and equality, showcasing the model's robust theorem-proving capabilities.

Implications and Future Work

LNNs represent a significant step towards explainable AI by offering:

• Transparent and interpretable neuron-level logical correspondences.
• A framework capable of handling inconsistent and incomplete knowledge realms.
• Facilitation of complex, multi-task general-purpose inference within a unified model.

The potential applications of the proposed framework span numerous AI domains, from automated theorem proving to knowledge-based systems in structured and unstructured environments. Future research can explore extending the model to support functions and equality, potentially broadening its applicability and further enhancing its robustness.

The synthesis of neural learning and symbolic reasoning encapsulated in Logical Neural Networks lays the groundwork for advancements in neuro-symbolic AI, presenting a versatile and interpretable approach towards tackling complex reasoning tasks while leveraging the merits of both paradigms.