---
title: "On the Practical Computational Power of Finite — Weiss 2018"
description: "Classic proofs that RNNs are Turing complete depend on unlimited state precision and unbounded computation time, which do not match how RNNs are used in…"
authors: ["Gail Weiss", "Yoav Goldberg", "Eran Yahav"]
year: 2018
venue: "Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)"
source: "https://aclanthology.org/P18-2117/"
doi: "10.18653/v1/p18-2117"
arxiv: "1805.04908"
licence: "CC BY 4.0 (ACL Anthology)"
licence_url: "https://creativecommons.org/licenses/by/4.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/weiss-2018-practical-computational-power-finite-precision/"
---
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# On the Practical Computational Power of Finite Precision RNNs for Language Recognition

**Authors:** Gail Weiss, [Yoav Goldberg](https://md.research.elamur.ai/free/en/authors/yoav-goldberg/), Eran Yahav · **Institutions:** [Bar-Ilan University](https://md.research.elamur.ai/free/en/universities/bar-ilan-university/) · [Technion – Israel Institute of Technology](https://md.research.elamur.ai/free/en/universities/technion-israel-institute-of-technology/) · **Year:** 2018 · **Venue:** Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers) · **Licence:** [CC BY 4.0 (ACL Anthology)](https://creativecommons.org/licenses/by/4.0/)

**Links:** [Publisher's page](https://aclanthology.org/P18-2117/) · [PDF](https://aclanthology.org/P18-2117.pdf) · [DOI 10.18653/v1/p18-2117](https://doi.org/10.18653/v1/p18-2117) · [arXiv 1805.04908](https://arxiv.org/abs/1805.04908) · [OpenAlex](https://openalex.org/W2798579706)

**Citations:** 217 citations (OpenAlex, 2026-10-14) · [93](https://md.research.elamur.ai/free/en/papers/weiss-2018-practical-computational-power-finite-precision/cited-by/) in our library, of the 140884 papers whose bibliographies we read

## Summary
Classic proofs that RNNs are Turing complete depend on unlimited state precision and unbounded computation time, which do not match how RNNs are used in NLP. This paper asks what different recurrent architectures can compute when states have finite precision and the network performs one update per input symbol (input-bound, finite-precision RNNs, or IBFP-RNNs). The authors compare four architectures: the Elman SRNN with squashing activation, the ReLU-based IRNN, the GRU and the LSTM. They relate each to a simplified k-counter machine (SKCM).

Their analysis finds that the LSTM and IRNN can maintain unbounded counters in dedicated dimensions. The SRNN and GRU cannot, because their state values are confined to a bounded range, so they recognize only regular languages. Experiments on aⁿbⁿ and aⁿbⁿcⁿ support this. Trained LSTMs develop interpretable counting dimensions and generalize well past the training lengths, while GRUs do neither.

The work shows that the LSTM and GRU, often treated as interchangeable, differ in practical expressive power. This may explain why LSTMs are preferred for tasks requiring bracket counting, such as processing linearized parse trees.

## Key points
- **Theoretical claim:** under finite precision and real-time computation, the LSTM and IRNN can do everything SKCMs can, and their expressive power strictly exceeds that of both the SRNN and the GRU. The GRU's interpolation between the previous state and a tanh-bounded proposal prevents unbounded counting.
- **Setup:** 10-dimensional, single-layer LSTM and GRU networks were implemented in DyNet and trained with SGD. Training lengths went up to n = 100 for aⁿbⁿ and n = 50 for aⁿbⁿcⁿ.
- **Generalization:** the LSTM handled aⁿbⁿ correctly up to n = 256 and aⁿbⁿcⁿ up to n = 100. The GRU erred even inside its training range, for example by accepting aⁿbⁿ⁺¹ for n > 38.
- **Test accuracy (1000-sample random test sets):** the LSTM scored 100% on aⁿbⁿ and 98.6% on aⁿbⁿcⁿ, against 87.0% and 86.9% for the GRU.
- **Counter imprecision:** the LSTM's counters were slightly inexact, for instance stepping up by about 0.99 but down by about 0.98, because its gates did not fully saturate. This drift explains its eventual failures at very large n.

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

Fields: [Natural language processing](https://md.research.elamur.ai/free/en/shelves/nlp/)

## Cite

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

```bibtex
@inproceedings{weiss2018practical,
  title = {{On the Practical Computational Power of Finite Precision RNNs for Language Recognition}},
  author = {Weiss, Gail and Goldberg, Yoav and Yahav, Eran},
  booktitle = {Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)},
  year = {2018},
  doi = {10.18653/v1/p18-2117},
  eprint = {1805.04908},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.18653/v1/p18-2117}
}
```

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