A partially observable Markov decision process (POMDP) is a generalization of a Markov decision process (MDP). A POMDP models an agent decision process in which it is assumed that the system dynamics are determined by an MDP, but the agent cannot directly observe the underlying state. Instead, it must maintain a sensor model (the probability distribution of different observations given the underlying state) and the underlying MDP. Unlike the policy function in MDP which maps the underlying states to the actions, POMDP's policy is a mapping from the history of observations (or belief states) to the actions. The POMDP framework is general enough to model a variety of real-world sequential decision processes. Applications include robot navigation problems, machine maintenance, and planning under uncertainty in general. The general framework of Markov decision processes with imperfect information was described by Karl Johan Åström in 1965 in the case of a discrete state space, and it was further studied in the operations research community where the acronym POMDP was coined. It was later adapted for problems in artificial intelligence and automated planning by Leslie P. Kaelbling and Michael L. Littman. An exact solution to a POMDP yields the optimal action for each possible belief over the world states. The optimal action maximizes the expected reward (or minimizes the cost) of the agent over a possibly infinite horizon. The sequence of optimal actions is known as the optimal policy of the agent for interacting with its environment. Definition Formal definition A discrete-time POMDP models the relationship between an agent and its environment. Formally, a POMDP is a 7-tuple , where is a set of states, is a set of actions, is a set of conditional transition probabilities between states, is the reward function. is a set of observations, is a set of conditional observation probabilities, and is the discount factor. At each time period, the environment is in some state . The agent takes an action , which causes the environment to transition to state with probability . At the same time, the agent receives an observation which depends on the new state of the environment, , and on the just taken action, , with probability (or sometimes depending on the sensor model). Finally, the agent receives a reward equal to . Then the process repeats. The goal is for the agent to choose actions at each time step that maximize its expected future