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Logical Neural Networks

Ryan Riegel, Alexander Gray, Francois Luus, Naweed Khan, Ndivhuwo Makondo, Ismail Yunus Akhalwaya, Haifeng Qian, Ronald Fagin, Francisco Barahona, Udit Sharma, Shajith Ikbal, Hima Karanam, Sumit Neelam, Ankita Likhyani, Santosh Srivastava

TL;DR

Logical Neural Networks (LNNs) constitute a neuro-symbolic framework that unifies learning with symbolic logic by mapping neurons to elements of logical formulae and operating within weighted real-valued logics. The core innovations include bidirectional inference via an Upward–Downward algorithm, truth-value bounds that support open-world semantics, and differentiable learning using a contradiction-focused loss while preserving interpretability through tailored activations. The approach enables end-to-end differentiable deduction, supports first-order logic representations with grounding, and demonstrates competitive reasoning on benchmarks such as LUBM and TPTP, while locating and down-weighting inconsistent axioms. These capabilities advance explainable, domain-knowledge-rich AI with robust, provable reasoning integrated into learning.

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.

Logical Neural Networks

TL;DR

Logical Neural Networks (LNNs) constitute a neuro-symbolic framework that unifies learning with symbolic logic by mapping neurons to elements of logical formulae and operating within weighted real-valued logics. The core innovations include bidirectional inference via an Upward–Downward algorithm, truth-value bounds that support open-world semantics, and differentiable learning using a contradiction-focused loss while preserving interpretability through tailored activations. The approach enables end-to-end differentiable deduction, supports first-order logic representations with grounding, and demonstrates competitive reasoning on benchmarks such as LUBM and TPTP, while locating and down-weighting inconsistent axioms. These capabilities advance explainable, domain-knowledge-rich AI with robust, provable reasoning integrated into learning.

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.

Paper Structure

This paper contains 66 sections, 3 theorems, 51 equations, 7 figures, 9 tables.

Table of Contents

  1. Introduction and related work
  2. Overview
  3. Model structure
  4. Activation functions for connectives
  5. Weighted nonlinear logic
  6. Activation functions for atoms
  7. Inference
  8. Bidirectional inference
  9. Inference rules in weighted nonlinear logic
  10. The Upward--Downward algorithm
  11. LNN bounds as probability bounds
  12. Learning
  13. Tailored activation functions
  14. Empirical evaluation
  15. Conclusions
  16. ...and 51 more sections

Key Result

Theorem 1

Given monotonic $\neg$, $\oplus$, and $f$, Algorithm algo-inference converges to within $\epsilon$ in finite time.

Figures (7)

  • Figure 1: Neurons (\ref{['fig:structure']}) with alternative activation functions (\ref{['fig:logistic']}, \ref{['fig:tailored']}) configured to match the truth functions of their corresponding operations, with established regions of unambiguously True, unambiguously False, and intermediate truth.
  • Figure 2: Regions of classicality as alpha varies; left - minimum alpha, right - maximum alpha
  • Figure 3: Piecewise linear activation function; left: alpha = 0.7, right: alpha = 0.95
  • Figure 4: Nonlinear activation function; left: alpha = 0.7, right: maximum alpha
  • Figure 5: Activation function for a binary conjunction $(\alpha=1)$, with one weight = 0
  • ...and 2 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:1']}
  • proof : Proof of Theorem 2