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. 2021 Aug 16;12:4942. doi: 10.1038/s41467-021-25123-3

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 2 — a The DR corresponding to a three-level decision tree task is shown. Each row of the DR represents weighted future expectancies starting from the corresponding state and following the default policy. Therefore, the DR is independent of the goals and optimized policy. b, c The optimized path for planning from home (H) to the food (F) state is computed based on the DR. The linear RL model is efficient because the same DR is sufficient for planning towards a new goal, such as the water (W) state. d The path between every two states in a 10 × 10 maze environment (d) computed by linear RL exactly matches the optimal (shortest) path computed by exhaustive search. The DR has been computed once and reused (in combination with techniques from matrix identities) to compute each optimal path.