# From Neural Networks to Logical Theories: The Correspondence between Fibring Modal Logics and Fibring Neural Networks

arXiv: https://arxiv.org/abs/2509.23912 · HTML: https://mlsanity.com/paper/2509.23912/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 23 references that are in the map.
- First posted on arXiv 28 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [Graph Neural Networks Meet Neural-Symbolic Computing: A Survey and Perspective](https://mlsanity.com/paper/2003.00330/) (2020)
- [Formal Language Recognition by Hard Attention Transformers: Perspectives from Circuit Complexity](https://mlsanity.com/paper/2204.06618/) (2022)
- [Logical Languages Accepted by Transformer Encoders with Hard Attention](https://mlsanity.com/paper/2310.03817/) (2023)
- [The Descriptive Complexity of Graph Neural Networks](https://mlsanity.com/paper/2303.04613/) (2023)
- [Decidability of Graph Neural Networks via Logical Characterizations](https://mlsanity.com/paper/2404.18151/) (2024)
- [A Logic for Reasoning About Aggregate-Combine Graph Neural Networks](https://mlsanity.com/paper/2405.00205/) (2024)
- [The Correspondence Between Bounded Graph Neural Networks and Fragments of First-Order Logic](https://mlsanity.com/paper/2505.08021/) (2025)

## Built on by (most cited first, 1 of 1)

- [Neurosymbolic Deep Learning Semantics](https://mlsanity.com/paper/2511.02825/) (2025)

## Abstract (arXiv)

Fibring of modal logics is a well-established formalism for combining countable families of modal logics into a single fibred language with common semantics, characterized by fibred models. Inspired by this formalism, fibring of neural networks was introduced as a neurosymbolic framework for combining learning and reasoning in neural networks. Fibring of neural networks uses the (pre-)activations of a trained network to evaluate a fibring function computing the weights of another network whose outputs are injected back into the original network. However, the exact correspondence between fibring of neural networks and fibring of modal logics was never formally established. In this paper, we close this gap by formalizing the idea of fibred models compatible with fibred neural networks. Using this correspondence, we then derive non-uniform logical expressiveness results for Graph Neural Networks (GNNs), Graph Attention Networks (GATs) and Transformer encoders. Longer-term, the goal of this paper is to open the way for the use of fibring as a formalism for interpreting the logical theories learnt by neural networks with the tools of computational logic.