Markov Decision Process

A Markov Decision Process on a set of states S and a set of actions A consists of a conditional distribution, P(r,s'|a,s).

Partially Observable Markov Decision Process

A POMDP on a set of states S and a set of actions A consists of a conditional distribution, P(r, s'|a, s), and another conditional distribution P(o|s).

MDP is an assumption, but POMDP is not.