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A POMDP (Partially Observable Markov Decision Process) is a mathematical model used in the field of artificial intelligence and reinforcement learning to represent decision-making problems in which an agent interacts with an environment that is not fully observable:

i) Markov Decision Process (MDP): An MDP is a model for decision-making in situations where an agent makes decisions in an environment. The key assumption is the Markov property, which states that the future state depends only on the current state and action, and not on the sequence of states and actions that preceded it. 

ii) Partially Observable: In a POMDP, the agent does not have complete information about the current state of the environment. Instead, it receives observations that are partially informative about the underlying state. 

iii) Decision Process: The agent makes decisions by selecting actions to maximize some notion of cumulative reward over time. 

iv) Observations: At each time step, the agent receives observations based on the underlying state of the environment. These observations are often noisy or incomplete, adding an element of uncertainty to the decision-making process. 

The primary challenge in a POMDP is to develop policies that map from current observations to actions, taking into account the uncertainty introduced by partial observability. Solving a POMDP involves finding an optimal policy that maximizes the expected cumulative reward over time. POMDPs have applications in various fields, including robotics, autonomous systems, and natural language processing, where the agent needs to make decisions in environments with uncertainty and partial observability.

POMDP can have an optimal policy π^*. Formally, the optimal policy π^* is defined as: 

              ---------------------------- [3657a]

where,

         π is a policy (a mapping from observations to actions). 

         γ is the discount factor that determines the importance of future rewards. 

         R_t is the reward obtained at time step t. 

The challenge in POMDPs is that the agent does not have direct access to the true underlying state of the environment but receives observations that are only partially informative. Consequently, finding the optimal policy involves reasoning about uncertainty and making decisions based on the available information. Solving POMDPs and finding the optimal policy often requires sophisticated algorithms, such as POMDP solvers, which take into account both the current observation and the history of observations to make decisions. Algorithms like the POMDP value iteration or POMCP (Partially Observable Monte Carlo Planning) are commonly used for solving POMDPs and finding the optimal policy. 

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