A CNN-based approach to the reverse game of life problem
Ángel Fernández Caravaca 
David Guijo-Rubio 
Víctor Manuel Vargas-Yun Bioinspired intelligent systems: From robotics and computer vision to trustworthy applications
Abstract
The Game of Life (GoL) is a cellular automaton characterised by non-linear evolution and emergent complexity. Its global state transition function is non-injective and irreversible, leading to information loss. Consequently, the Reverse GoL, i.e., finding a predecessor that evolves into a given target after a given number of generations, is an NP-complete task. In this paper, we introduce a differentiable GoL transition function within a convolutional neural network-based model to reconstruct the probability distribution, i.e. a heatmap, of a possible initial state associated with the given final board. In this study, the models are validated on $$15\times 15$$15×15boards after one generation by analysing structure-based metrics on the heatmaps of the predicted initial states. In particular, we computed the fuzziness index to measure the degree of binarisation, the Earth Mover's Distance, to evaluate the accuracy of the spatial mass distribution, and the percentage of high uncertainty cells within a range, to quantify prediction confidence. Our results demonstrate that integrating the differentiable layer reduces the fuzziness index by approximately 40% compared to the baseline approach. Furthermore, the analysis indicate that pixel-wise metrics, such as Mean Squared Error, can be misleading in this context, as they ignore the spatial context of cells. In contrast, the use of structural metrics reveals that the proposed architecture effectively captures the underlying physics and spatial organisation of the automaton.
BibTex Citation
@inproceedings{Fernandez2026CNN,
author = {Fern{\' a}ndez Caravaca, {\' A}ngel and Guijo-Rubio, David and Vargas-Yun, V{\' i}ctor Manuel},
booktitle = {Bioinspired intelligent systems: From robotics and computer vision to trustworthy applications},
doi = {10.1007/978-3-032-27317-8_25},
year = {2026},
pages = {258--267},
organization = {Springer Nature Switzerland},
title = {A {CNN}-based approach to the reverse game of life problem},
}
BibTex Unicode Citation
@inproceedings{Fernandez2026CNN,
author = {Fernández Caravaca, {\' A}ngel and Guijo-Rubio, David and Vargas-Yun, Víctor Manuel},
booktitle = {Bioinspired intelligent systems: From robotics and computer vision to trustworthy applications},
doi = {10.1007/978-3-032-27317-8_25},
year = {2026},
pages = {258--267},
organization = {Springer Nature Switzerland},
title = {A {CNN}-based approach to the reverse game of life problem},
}
APA Citation
Fernández Caravaca, Á., Guijo-Rubio, D., & Vargas-Yun, V. M. (2026). A CNN-based approach to the reverse game of life problem. Bioinspired Intelligent Systems: From Robotics and Computer Vision to Trustworthy Applications, 258–267. https://doi.org/10.1007/978-3-032-27317-8_25
RIS Citation
TY - CONF
AU - Fernández Caravaca, Ángel
AU - Guijo-Rubio, David
AU - Vargas-Yun, Víctor Manuel
C3 - Bioinspired intelligent systems: From robotics and computer vision to
trustworthy applications
DA - 2026///
C2 - 2026
DO - 10.1007/978-3-032-27317-8_25
ID - temp_id_555974444083
PB - Springer Nature Switzerland
SP - 258-267
TI - A CNN-based approach to the reverse game of life problem
ER -
CV Citation
Á. Fernández Caravaca (CA), D. Guijo-Rubio, V.M. Vargas-Yun (3/3). "A CNN-based approach to the reverse game of life problem". Bioinspired intelligent systems: From robotics and computer vision to trustworthy applications, pp. 258–267, 2026.