{"type":"ScholarlyArticle","title":"DiSProD: Differentiable Symbolic Propagation of Distributions for Planning","authors":[{"name":"Palash Chatterjee","slug":"palash-chatterjee","given":"Palash","family":"Chatterjee","affiliation":"Department of Computer Science , Luddy School of Informatics, Computing, and Engineering , Indiana University , Bloomington , Indiana , USA"},{"name":"Ashutosh Chapagain","slug":"ashutosh-chapagain","given":"Ashutosh","family":"Chapagain","affiliation":"Department of Computer Science , Luddy School of Informatics, Computing, and Engineering , Indiana University , Bloomington , Indiana , USA"},{"name":"Weizhe Chen","slug":"weizhe-chen","given":"Weizhe","family":"Chen","affiliation":"Department of Computer Science , Luddy School of Informatics, Computing, and Engineering , Indiana University , Bloomington , Indiana , USA"},{"name":"Roni Khardon","slug":"roni-khardon","given":"Roni","family":"Khardon","affiliation":"Department of Computer Science , Luddy School of Informatics, Computing, and Engineering , Indiana University , Bloomington , Indiana , USA"}],"abstract":"The paper introduces DiSProD, an online planner developed for\r\nenvironments with probabilistic transitions in continuous state and\r\naction spaces. DiSProD builds a symbolic graph that captures the\r\ndistribution of future trajectories, conditioned on a given policy,\r\nusing independence assumptions and approximate propagation of\r\ndistributions. The symbolic graph provides a differentiable\r\nrepresentation of the policy's value, enabling efficient gradient-based\r\noptimization for long-horizon search. The propagation of approximate\r\ndistributions can be seen as an aggregation of many trajectories, making\r\nit well-suited for dealing with sparse rewards and stochastic\r\nenvironments. An extensive experimental evaluation compares DiSProD to\r\nstate-of-the-art planners in discrete-time planning and real-time\r\ncontrol of robotic systems. The proposed method improves over existing\r\nplanners in handling stochastic environments, sensitivity to search\r\ndepth, sparsity of rewards, and large action spaces. Additional\r\nreal-world experiments demonstrate that DiSProD can control ground\r\nvehicles and surface vessels to successfully navigate around obstacles.","doi":"10.24963/ijcai.2023/591","url":"https://ijcai.org/proceedings/2023/591","pdf":"https://ijcai.org/proceedings/2023/0591.pdf","markdown":"https://ijcai.org/proceedings/2023/591.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/591","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Planning and Scheduling","track":"Main Track","number":591,"pages":{"first":5324,"last":5332},"keywords":["Planning and Scheduling: PS: Planning algorithms","Planning and Scheduling: PS: Robot planning","Planning and Scheduling: PS: Planning under uncertainty"],"videos":[],"references":[{"position":1,"raw":"[Agha-Mohammadi et al., 2014] Ali-Akbar Agha- Mohammadi, Suman Chakravorty, and Nancy M Amato. Firm: Sampling-based feedback motion-planning under motion uncertainty and imperfect measurements. The In- ternational Journal of Robotics Research, 33(2):268-304, 2014.","title":"Firm: Sampling-based feedback motion-planning under motion uncertainty and imperfect measurements","authors":"Agha-Mohammadi","year":"2014","venue":"The International Journal of Robotics Research","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Barcelos et al., 2021] Lucas Barcelos, Alexander Lambert, Rafael Oliveira, Paulo Borges, Byron Boots, and Fabio Ramos. Dual Online Stein Variational Inference for Con- trol and Dynamics. In Proceedings of Robotics: Science and Systems, Virtual, July 2021.","title":"Dual Online Stein Variational Inference for Control and Dynamics","authors":"Lucas Barcelos, Alexander Lambert, Rafael Oliveira, Paulo Borges, Byron Boots, Fabio Ramos","year":"2021","venue":"Proceedings of Robotics: Science and Systems, Virtual","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Borrelli et al., 2017] Francesco Borrelli, Alberto Bempo- rad, and Manfred Morari. Predictive control for linear and hybrid systems. Cambridge University Press, 2017.","title":"Predictive control for linear and hybrid systems","authors":"Francesco Borrelli, Alberto Bemporad, Manfred Morari","year":"2017","venue":null,"doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Brockman et al., 2016] Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016.","title":"Openai gym","authors":"Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, Wojciech Zaremba","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bueno et al., 2019] Thiago P. Bueno, Leliane N. de Barros, Denis D. Mauá, and Scott Sanner. Deep reactive policies for planning in stochastic nonlinear domains. In Proceed- ings of the AAAI Conference on Artificial Intelligence, vol- ume 33, pages 7530-7537, 2019.","title":"Deep reactive policies for planning in stochastic nonlinear domains","authors":"P Thiago, Leliane N Bueno, Denis D De Barros, Scott Mauá, Sanner","year":"2019","venue":"Proceedings of the AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Chow et al., 2017] Yinlam Chow, Mohammad Ghavamzadeh, Lucas Janson, and Marco Pavone. Risk-constrained reinforcement learning with percentile risk criteria. Journal of Machine Learning Research, 18:167:1-167:51, 2017.","title":"Risk-constrained reinforcement learning with percentile risk criteria","authors":"Chow","year":"2017","venue":"Journal of Machine Learning Research","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Chua et al., 2018] Kurtland Chua, Roberto Calandra, Rowan McAllister, and Sergey Levine. Deep reinforce- ment learning in a handful of trials using probabilistic dynamics models. Advances in neural information processing systems, 31, 2018.","title":"Deep reinforcement learning in a handful of trials using probabilistic dynamics models","authors":"Chua","year":"2018","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Cui et al., 2018] Hao Jackson Cui, Radu Marinescu, and Roni Khardon. From stochastic planning to marginal map. Advances in Neural Information Processing Systems, 31, 2018.","title":"From stochastic planning to marginal map","authors":"Jackson Hao, Radu Cui, Roni Marinescu, Khardon","year":"2018","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Cui et al., 2019] Hao Cui, Thomas Keller, and Roni Khardon. Stochastic planning with lifted symbolic tra- jectory optimization. In Proceedings of the International Conference on Automated Planning and Scheduling, vol- ume 29, pages 119-127, 2019.","title":"Stochastic planning with lifted symbolic trajectory optimization","authors":"Thomas Hao Cui, Roni Keller, Khardon","year":"2019","venue":"Proceedings of the International Conference on Automated Planning and Scheduling","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Curi et al., 2020] Sebastian Curi, Felix Berkenkamp, and Andreas Krause. Efficient model-based reinforcement learning through optimistic policy search and planning. Advances in Neural Information Processing Systems, 33:14156-14170, 2020.","title":"Efficient model-based reinforcement learning through optimistic policy search and planning","authors":"Sebastian Curi, Felix Berkenkamp, Andreas Krause","year":"2020","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Deisenroth and Rasmussen, 2011] Marc Deisenroth and Carl E Rasmussen. Pilco: A model-based and data- efficient approach to policy search. In Proceedings of the 28th International Conference on machine learning (ICML-11), pages 465-472, 2011.","title":"Pilco: A model-based and dataefficient approach to policy search","authors":"Rasmussen Deisenroth, Marc Deisenroth, Carl E Rasmussen","year":"2011","venue":"Proceedings of the 28th International Conference on machine learning (ICML-11)","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Deisenroth et al., 2013] Marc Peter Deisenroth, Dieter Fox, and Carl Edward Rasmussen. Gaussian processes for data-efficient learning in robotics and control. IEEE transactions on pattern analysis and machine intelligence, 37(2):408-423, 2013.","title":"Gaussian processes for data-efficient learning in robotics and control","authors":"Marc Peter Deisenroth, Dieter Fox, Carl Edward Rasmussen","year":"2013","venue":"IEEE transactions on pattern analysis and machine intelligence","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Depeweg et al., 2017] Stefan Depeweg, José Miguel Hernández-Lobato, Finale Doshi-Velez, and Steffen Ud- luft. Learning and policy search in stochastic dynamical systems with bayesian neural networks. In International Conference on Learning Representations, ICLR, 2017.","title":"Learning and policy search in stochastic dynamical systems with bayesian neural networks","authors":"Stefan Depeweg, José Miguel Hernández-Lobato, Finale Doshi-Velez, Steffen Udluft","year":"2017","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Dubins, 1957] Lester E Dubins. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. American Journal of mathematics, 79(3):497-516, 1957.","title":"On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents","authors":"E Lester, Dubins","year":"1957","venue":"American Journal of mathematics","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Gal et al., 2016] Yarin Gal, Rowan McAllister, and Carl Ed- ward Rasmussen. Improving pilco with bayesian neu- ral network dynamics models. In Data-Efficient Machine Learning workshop, ICML, volume 4, page 25, 2016.","title":"Improving pilco with bayesian neural network dynamics models","authors":"Yarin Gal, Rowan Mcallister, Carl Edward Rasmussen","year":"2016","venue":"Data-Efficient Machine Learning workshop","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Garrett et al., 2021] Caelan Reed Garrett, Rohan Chitnis, Rachel Holladay, Beomjoon Kim, Tom Silver, Leslie Pack Kaelbling, and Tomás Lozano-Pérez. Integrated task and motion planning. Annual review of control, robotics, and autonomous systems, 4:265-293, 2021.","title":"Integrated task and motion planning","authors":"Caelan Reed Garrett, Rohan Chitnis, Rachel Holladay, Beomjoon Kim, Tom Silver, Leslie Pack Kaelbling, Tomás Lozano-Pérez","year":"2021","venue":"Annual review of control, robotics, and autonomous systems","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Hafner et al., 2020] Danijar Hafner, Timothy Lillicrap, Jimmy Ba, and Mohammad Norouzi. Dream to control: Learning behaviors by latent imagination. In International Conference on Learning Representations, 2020.","title":"Dream to control: Learning behaviors by latent imagination","authors":"Hafner","year":"2020","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Heess et al., 2015] Nicolas Heess, Gregory Wayne, David Silver, Timothy Lillicrap, Tom Erez, and Yuval Tassa. Learning continuous control policies by stochastic value gradients. Advances in neural information processing sys- tems, 28, 2015.","title":"Learning continuous control policies by stochastic value gradients","authors":"Nicolas Heess, Gregory Wayne, David Silver, Timothy Lillicrap, Tom Erez, Yuval Tassa","year":"2015","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Kaelbling and Lozano-Pérez, 2013] Leslie Pack Kaelbling and Tomás Lozano-Pérez. Integrated task and motion planning in belief space. The International Journal of Robotics Research, 32(9-10):1194-1227, 2013.","title":"Integrated task and motion planning in belief space","authors":"Lozano-Pérez ; Kaelbling, Tomás Lozano-Pérez","year":"2013","venue":"The International Journal of Robotics Research","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kamthe and Deisenroth, 2018] Sanket Kamthe and Marc Deisenroth. Data-efficient reinforcement learning with probabilistic model predictive control. In International conference on artificial intelligence and statistics, pages 1701-1710. PMLR, 2018.","title":"Data-efficient reinforcement learning with probabilistic model predictive control","authors":"Sanket Kamthe, Marc Deisenroth","year":"2018","venue":"International conference on artificial intelligence and statistics","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Keller and Helmert, 2013] Thomas Keller and Malte Helmert. Trial-based heuristic tree search for finite hori- zon mdps. In Proceedings of the International Conference on Automated Planning and Scheduling, volume 23, pages 135-143, 2013.","title":"Trial-based heuristic tree search for finite horizon mdps","authors":"Helmert Keller, Thomas Keller, Malte Helmert","year":"2013","venue":"Proceedings of the International Conference on Automated Planning and Scheduling","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Kingma and Ba, 2015] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Inter- national Conference on Learning Representations, ICLR, 2015.","title":"Adam: A method for stochastic optimization","authors":"P Diederik, Jimmy Kingma, Ba","year":"2015","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Kobilarov, 2012] Marin Kobilarov. Cross-entropy motion planning. The International Journal of Robotics Research, 31(7):855-871, 2012.","title":"Marin Kobilarov. Cross-entropy motion planning","authors":"Kobilarov","year":"2012","venue":"The International Journal of Robotics Research","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Kolobov et al., 2012] Andrey Kolobov, Peng Dai, Mausam Mausam, and Daniel Weld. Reverse iterative deepen- ing for finite-horizon mdps with large branching factors. In Proceedings of the International Conference on Auto- mated Planning and Scheduling, volume 22, pages 146- 154, 2012.","title":"Reverse iterative deepening for finite-horizon mdps with large branching factors","authors":"Andrey Kolobov, Peng Dai, Mausam Mausam, Daniel Weld","year":"2012","venue":"Proceedings of the International Conference on Automated Planning and Scheduling","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Kurniawati et al., 2011] Hanna Kurniawati, Yanzhu Du, David Hsu, and Wee Sun Lee. Motion planning under un- certainty for robotic tasks with long time horizons. The In- ternational Journal of Robotics Research, 30(3):308-323, 2011.","title":"Motion planning under uncertainty for robotic tasks with long time horizons","authors":"Hanna Kurniawati, Yanzhu Du, David Hsu, Wee Sun, Lee","year":"2011","venue":"The International Journal of Robotics Research","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Lambert et al., 2021] Alexander Lambert, Fabio Ramos, Byron Boots, Dieter Fox, and Adam Fishman. Stein vari- ational model predictive control. In Conference on Robot Learning, pages 1278-1297. PMLR, 2021.","title":"Stein variational model predictive control","authors":"Alexander Lambert, Fabio Ramos, Byron Boots, Dieter Fox, Adam Fishman","year":"2021","venue":"Conference on Robot Learning","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[LaValle, 2006] Steven M LaValle. Planning algorithms. Cambridge university press, 2006.","title":"Planning algorithms","authors":"Steven M Lavalle","year":"2006","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Lenz et al., 2015] Ian Lenz, Ross A Knepper, and Ashutosh Saxena. Deepmpc: Learning deep latent features for model predictive control. In Robotics: Science and Sys- tems, volume 10. Rome, Italy, 2015.","title":"Deepmpc: Learning deep latent features for model predictive control","authors":"Ian Lenz, Ross A Knepper, Ashutosh Saxena","year":"2015","venue":"Robotics: Science and Systems","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Li and Littman, 2005] Lihong Li and Michael L. Littman. Lazy approximation for solving continuous finite-horizon mdps. In Proceedings of the 20th National Conference on Artificial Intelligence -Volume 3, AAAI'05, page 1175-1180. AAAI Press, 2005.","title":"Lazy approximation for solving continuous finite-horizon mdps","authors":"Lihong Li, Michael L Littman","year":"2005","venue":"Proceedings of the 20th National Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Mania et al., 2018] Horia Mania, Aurelia Guy, and Ben- jamin Recht. Simple random search of static linear policies is competitive for reinforcement learning. In Advances in Neural Information Processing Systems, volume 31, 2018.","title":"Simple random search of static linear policies is competitive for reinforcement learning","authors":"Horia Mania, Aurelia Guy, Benjamin Recht","year":"2018","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Mohamed et al., 2022] Ihab S Mohamed, Kai Yin, and Lan- tao Liu. Autonomous navigation of agvs in unknown cluttered environments: log-mppi control strategy. IEEE Robotics and Automation Letters, 2022.","title":"Autonomous navigation of agvs in unknown cluttered environments: log-mppi control strategy","authors":"Kai Ihab S Mohamed, Lantao Yin, Liu","year":"2022","venue":"IEEE Robotics and Automation Letters","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Parmas et al., 2018] Paavo Parmas, Carl Edward Ras- mussen, Jan Peters, and Kenji Doya. Pipps: Flexible model-based policy search robust to the curse of chaos. In International Conference on Machine Learning, pages 4065-4074. PMLR, 2018.","title":"Pipps: Flexible model-based policy search robust to the curse of chaos","authors":"Paavo Parmas, Carl Edward Rasmussen, Jan Peters, Kenji Doya","year":"2018","venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Raghavan et al., 2017] Aswin Raghavan, Scott Sanner, Roni Khardon, Prasad Tadepalli, and Alan Fern. Hind- sight optimization for hybrid state and action mdps. In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 31, 2017.","title":"Hindsight optimization for hybrid state and action mdps","authors":"Aswin Raghavan, Scott Sanner, Roni Khardon, Prasad Tadepalli, Alan Fern","year":"2017","venue":"Proceedings of the AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Tamar et al., 2016] Aviv Tamar, Yi Wu, Garrett Thomas, Sergey Levine, and Pieter Abbeel. Value iteration net- works. Advances in neural information processing sys- tems, 29, 2016. Tassa et al., 2012] Yuval Tassa, Tom Erez, and Emanuel Todorov. Synthesis and stabilization of complex behaviors through online trajectory optimization. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Sys- tems, pages 4906-4913. IEEE, 2012.","title":"Synthesis and stabilization of complex behaviors through online trajectory optimization","authors":"Tamar","year":"2012","venue":"IEEE/RSJ International Conference on Intelligent Robots and Systems","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Tassa et al., 2014] Yuval Tassa, Nicolas Mansard, and Emo Todorov. Control-limited differential dynamic program- ming. In 2014 IEEE International Conference on Robotics and Automation (ICRA), pages 1168-1175. IEEE, 2014.","title":"Control-limited differential dynamic programming","authors":"Yuval Tassa, Nicolas Mansard, Emo Todorov","year":"2014","venue":"IEEE International Conference on Robotics and Automation (ICRA)","doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Van Den Berg et al., 2012] Jur Van Den Berg, Sachin Patil, and Ron Alterovitz. Motion planning under uncertainty using iterative local optimization in belief space. The In- ternational Journal of Robotics Research, 31(11):1263- 1278, 2012.","title":"Motion planning under uncertainty using iterative local optimization in belief space","authors":"Van Den, Berg","year":"2012","venue":"The International Journal of Robotics Research","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Vega-Brown and Roy, 2018] William Vega-Brown and Nicholas Roy. Admissible abstractions for near-optimal task and motion planning. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, pages 4852-4859, 2018.","title":"William Vega-Brown and Nicholas Roy. Admissible abstractions for near-optimal task and motion planning","authors":"Roy Vega-Brown","year":"2018","venue":"Proceedings of the 27th International Joint Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Wagener et al., 2019] Nolan Wagener, Ching an Cheng, Ja- cob Sacks, and Byron Boots. An online learning approach to model predictive control. In Proceedings of Robotics: Science and Systems, pages 1-10, 2019.","title":"An online learning approach to model predictive control","authors":"Nolan Wagener, Jacob Ching An Cheng, Byron Sacks, Boots","year":"2019","venue":"Proceedings of Robotics: Science and Systems","doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Williams et al., 2017] Grady Williams, Nolan Wagener, Brian Goldfain, Paul Drews, James M Rehg, Byron Boots, and Evangelos A Theodorou. Information theoretic mpc for model-based reinforcement learning. In 2017 IEEE International Conference on Robotics and Automation (ICRA), pages 1714-1721. IEEE, 2017.","title":"Information theoretic mpc for model-based reinforcement learning","authors":"Grady Williams, Nolan Wagener, Brian Goldfain, Paul Drews, James M Rehg, Byron Boots, Evangelos A Theodorou","year":"2017","venue":"IEEE International Conference on Robotics and Automation (ICRA)","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Wu et al., 2020a] Ga Wu, Buser Say, and Scott Sanner. Scalable planning with deep neural network learned tran- sition models. Journal of Artificial Intelligence Research, 68:571-606, 2020.","title":"Scalable planning with deep neural network learned transition models","authors":"Wu","year":"2020","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"[Wu et al., 2020b] Qiaoyun Wu, Xiaoxi Gong, Kai Xu, Di- nesh Manocha, Jingxuan Dong, and Jun Wang. Towards target-driven visual navigation in indoor scenes via gener- ative imitation learning. IEEE Robotics and Automation Letters, 6(1):175-182, 2020.","title":"Towards target-driven visual navigation in indoor scenes via generative imitation learning","authors":"Wu","year":"2020","venue":"IEEE Robotics and Automation Letters","doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Zamani et al., 2012] Zahra Zamani, Scott Sanner, and Cheng Fang. Symbolic dynamic programming for contin- uous state and action mdps. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 26, pages 1839-1845, 2012.","title":"Symbolic dynamic programming for continuous state and action mdps","authors":"Zahra Zamani, Scott Sanner, Cheng Fang","year":"2012","venue":"Proceedings of the AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}