and previously-executed option $\omega_{t-1}$:

$$\begin{aligned}
        I_\omega &\subseteq \Omega \times (O \cup \{ \emptyset \}) \\
        {\ensuremath{\mathcal{O}_t}}&\equiv \{ \omega \in O : (x_t, \omega_{t-1}) \in I_\omega \}
    \end{aligned}$$

with $\omega_0 = \emptyset$, $\Omega$ the set of observations and $O$ the set of options. [$\mathcal{O}_t$]{} allows the agent to condition the option selected at time $t$ on the one that has just terminated, even if the top-level policy does not observe $\omega_{t-1}$. The top-level and option policies remain memoryless. Not having to observe $\omega_{t-1}$ keeps the observation space of the top-level policy small, instead of extending it to $\Omega \times O$, without impairing the representational power of [OOIs]{}, as shown in the next sub-section.

[OOIs]{}Make Options as Expressive as FSCs {#sec:fsc}
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Finite State Controllers are state-of-the-art in policies applicable to POMDPs [@Meuleau1999]. By proving that options with [OOIs]{}are as expressive as FSCs, we provide a lower bound on the expressiveness of [OOIs]{}and ensure that they are applicable to a wide range of POMDPs.

\[thm:fsc\] [OOIs]{}allow options to represent any policy that can be expressed using a Finite State Controller.

The reduction from any FSC to options requires one option [$\langle n'_{t-1}, n_t \rangle$]{} per ordered pair of nodes in the FSC, 