  The interaction between an artificial agent and its environment is
bi-directional. The agent extracts relevant information from the environment,
and affects the environment by its actions in return to accumulate high
expected reward. Standard reinforcement learning (RL) deals with the expected
reward maximization. However, there are always information-theoretic
limitations that restrict the expected reward, which are not properly
considered by the standard RL. In this work we consider RL objectives with
information-theoretic limitations. For the first time we derive a Bellman-type
recursive equa- tion for the causal information between the environment and the
agent, which is combined plausibly with the Bellman recursion for the value
function. The unified equitation serves to explore the typical behavior of
artificial agents in an infinite time horizon.
