---
title: "Programming with a Differentiable Forth Interpreter (2016)"
description: "Bošnjak et al. present ∂4, a continuous, differentiable version of the abstract machine behind the… 36 references, cited by 27 papers here."
authors: ["Matko Bošnjak", "Tim Rocktäschel", "Jason Naradowsky", "Sebastian Riedel"]
year: 2016
venue: "arXiv:1605.06640"
source: "https://arxiv.org/abs/1605.06640"
doi: "10.48550/arxiv.1605.06640"
arxiv: "1605.06640"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/bosnjak-2016-programming-differentiable-forth-interpreter/"
---
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# Programming with a Differentiable Forth Interpreter

**Authors:** [Matko Bošnjak](https://md.research.elamur.ai/free/en/authors/matko-bosnjak/), [Tim Rocktäschel](https://md.research.elamur.ai/free/en/authors/tim-rocktaschel/), [Jason Naradowsky](https://md.research.elamur.ai/free/en/authors/jason-naradowsky/) and [Sebastian Riedel](https://md.research.elamur.ai/free/en/authors/sebastian-riedel/) · **Year:** 2016 · **Venue:** arXiv:1605.06640 · **Licence:** [arXiv — non-exclusive licence to distribute](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/1605.06640) · [PDF](https://arxiv.org/pdf/1605.06640) · [DOI 10.48550/arxiv.1605.06640](https://doi.org/10.48550/arxiv.1605.06640) · [arXiv 1605.06640](https://arxiv.org/abs/1605.06640) · [OpenAlex](https://openalex.org/W2963055053)

**Citations:** 89 citations (OpenAlex, 2026-10-14) · [27](https://md.research.elamur.ai/free/en/papers/bosnjak-2016-programming-differentiable-forth-interpreter/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
Neural networks that learn algorithms usually depend on large amounts of input-output data. A programmer, however, often knows the rough shape of a solution, such as how to walk through a sequence, without knowing the exact operation needed at each step. Bošnjak et al. present ∂4, a continuous, differentiable version of the abstract machine behind the stack-based language Forth. Users write program "sketches": ordinary Forth code with gaps (slots) that neural networks fill in. These slots are trained by backpropagation on input-output pairs. Execution runs as an RNN in TensorFlow over differentiable stacks, a heap and a program counter.

The authors tested sketches with different amounts of prior structure on sorting and addition tasks from Reed & de Freitas. With only the recursive structure provided, the learned programs generalised to input sizes not seen in training. Coupled with a BiLSTM and trained jointly on word algebra problems, ∂4 achieved state-of-the-art end-to-end accuracy without hand-built intermediate representations. This shows that partial code knowledge can act as a strong prior when data is scarce.

## Key points
- The machine state has a data stack D, a return stack R, a heap H and a program counter c. Each is implemented as a differentiable memory buffer with read/write operations similar to the Neural Turing Machine.
- Forth words such as DUP, DROP and SWAP are turned into differentiable state transitions. Control flow reduces to jumps via BRANCH and BRANCH0.
- Two sketch styles are demonstrated for Bubble sort:
  - **PERMUTE** learns how to reorder the top stack elements.
  - **COMPARE** learns whether to apply NOP or SWAP.
- For word algebra problems, BiLSTM outputs (a representation of the whole problem, context vectors of the numbers, and the numbers themselves) are written into the heap H. This makes the whole pipeline end-to-end differentiable.
- The paper also introduces code optimisations based on symbolic execution that speed up neural execution. It was published at ICML 2017 (PMLR 70), with authors from UCL, Oxford and Cambridge.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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- [University of Cambridge](https://md.research.elamur.ai/free/en/universities/university-of-cambridge/) — University, Cambridge, United Kingdom · 287 works · 125,174 citations
- [University of Oxford](https://md.research.elamur.ai/free/en/universities/university-of-oxford/) — University, Oxford, United Kingdom · 138 works · 161,448 citations
- [University College London](https://md.research.elamur.ai/free/en/universities/university-college-london/) — University, London, United Kingdom · 119 works · 23,545 citations

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## Cite

Cite as: Matko Bošnjak, Tim Rocktäschel, Jason Naradowsky and Sebastian Riedel. 2016. Programming with a Differentiable Forth Interpreter. arXiv:1605.06640.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{bosnjak2016programming,
  title = {{Programming with a Differentiable Forth Interpreter}},
  author = {Matko Bošnjak, Tim Rocktäschel, Jason Naradowsky and Sebastian Riedel},
  year = {2016},
  doi = {10.48550/arxiv.1605.06640},
  eprint = {1605.06640},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.1605.06640}
}
```

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*Updated 3 November 2026*