# Programming with a Differentiable Forth Interpreter

arXiv: https://arxiv.org/abs/1605.06640 · HTML: https://mlsanity.com/paper/1605.06640/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2016.

## Built on by (most cited first, 30 of 33)

- [NExT: Teaching Large Language Models to Reason about Code Execution](https://mlsanity.com/paper/2404.14662/) (2024)
- [Learning to Synthesize Programs as Interpretable and Generalizable Policies](https://mlsanity.com/paper/2108.13643/) (2021)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [NeuPSL: Neural Probabilistic Soft Logic](https://mlsanity.com/paper/2205.14268/) (2022)
- [Neuro-Symbolic Entropy Regularization](https://mlsanity.com/paper/2201.11250/) (2022)
- [How to Reuse and Compose Knowledge for a Lifetime of Tasks: A Survey on Continual Learning and Functional Composition](https://mlsanity.com/paper/2207.07730/) (2022)
- [A Pseudo-Semantic Loss for Autoregressive Models with Logical Constraints](https://mlsanity.com/paper/2312.03905/) (2023)
- [Semantic Strengthening of Neuro-Symbolic Learning](https://mlsanity.com/paper/2302.14207/) (2023)
- [Learning Differentiable Logic Programs for Abstract Visual Reasoning](https://mlsanity.com/paper/2307.00928/) (2023)
- [Static Prediction of Runtime Errors by Learning to Execute Programs with External Resource Descriptions](https://mlsanity.com/paper/2203.03771/) (2022)
- [Efficient Symbolic Policy Learning with Differentiable Symbolic Expression](https://mlsanity.com/paper/2311.02104/) (2023)
- [A Survey on Verification and Validation, Testing and Evaluations of Neurosymbolic Artificial Intelligence](https://mlsanity.com/paper/2401.03188/) (2024)
- [A Critical Review of Inductive Logic Programming Techniques for Explainable AI](https://mlsanity.com/paper/2112.15319/) (2021)
- [Hierarchical Neural Program Synthesis](https://mlsanity.com/paper/2303.06018/) (2023)
- [Smoothing Methods for Automatic Differentiation Across Conditional Branches](https://mlsanity.com/paper/2310.03585/) (2023)
- [Robot Program Parameter Inference via Differentiable Shadow Program Inversion](https://mlsanity.com/paper/2103.14452/) (2021)
- [Differentiable Tree Operations Promote Compositional Generalization](https://mlsanity.com/paper/2306.00751/) (2023)
- [Reinforcement Learning of Implicit and Explicit Control Flow in Instructions](https://mlsanity.com/paper/2102.13195/) (2021)
- [Modular Machine Learning: An Indispensable Path towards New-Generation Large Language Models](https://mlsanity.com/paper/2504.20020/) (2025)
- [Neural Machine Translation for Code Generation](https://mlsanity.com/paper/2305.13504/) (2023)
- [Semantic Loss Functions for Neuro-Symbolic Structured Prediction](https://mlsanity.com/paper/2405.07387/) (2024)
- [Compiling to linear neurons](https://mlsanity.com/paper/2511.13769/) (2025)
- [Reconciling the Discrete-Continuous Divide: Towards a Mathematical Theory of Sparse Communication](https://mlsanity.com/paper/2104.00755/) (2021)
- [RotLSTM: Rotating Memories in Recurrent Neural Networks](https://mlsanity.com/paper/2105.00357/) (2021)
- [NESTER: An Adaptive Neurosymbolic Method for Causal Effect Estimation](https://mlsanity.com/paper/2211.04370/) (2022)
- [Understanding in Artificial Intelligence](https://mlsanity.com/paper/2101.06573/) (2021)
- [Towards Neural Functional Program Evaluation](https://mlsanity.com/paper/2112.04630/) (2021)
- [Safety without alignment](https://mlsanity.com/paper/2303.00752/) (2023)
- [Toward Neurosymbolic Program Comprehension](https://mlsanity.com/paper/2502.01806/) (2025)
- [Neuro-symbolic Weak Supervision: Theory and Semantics](https://mlsanity.com/paper/2503.18509/) (2025)

## Abstract (arXiv)

Given that in practice training data is scarce for all but a small set of problems, a core question is how to incorporate prior knowledge into a model. In this paper, we consider the case of prior procedural knowledge for neural networks, such as knowing how a program should traverse a sequence, but not what local actions should be performed at each step. To this end, we present an end-to-end differentiable interpreter for the programming language Forth which enables programmers to write program sketches with slots that can be filled with behaviour trained from program input-output data. We can optimise this behaviour directly through gradient descent techniques on user-specified objectives, and also integrate the program into any larger neural computation graph. We show empirically that our interpreter is able to effectively leverage different levels of prior program structure and learn complex behaviours such as sequence sorting and addition. When connected to outputs of an LSTM and trained jointly, our interpreter achieves state-of-the-art accuracy for end-to-end reasoning about quantities expressed in natural language stories.