\rangle}}\notin I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} \\
                    \Leftrightarrow n'_{t-1} \ne n_{t-1}
                \end{array}
            \end{cases}
        \end{aligned}$$

Because $\eta$ maps nodes to nodes and $\mu$ selects options representing pairs of nodes, $\mu$ is extremely sparse and returns a value different from zero, $\eta (n_{t-1}, x_t, n_t)$, only when [$\langle n'_{t-2}, n_{t-1} \rangle$]{} and [$\langle n'_{t-1}, n_t \rangle$]{} agree on $n_{t-1}$.

Our reduction uses options with trivial policies, that execute for a single time-step, which leads to a large amount of options to compensate. In practice, we expect to be able to express policies for real-world POMDPs with much less options than the number of states an FSC would require, as shown in our simulated (Section \[sec:duplicatedinput\], 2 options) and robotic experiments (Section \[sec:khepera\], 12 options). In addition to being sufficient, the next sub-section proves that [OOIs]{}are necessary for options to be as expressive as FSCs.

Original Options are not as Expressive as FSCs {#sec:vanilla}
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![Two-nodes Finite State Controller that emits an infinite sequence ABAB... based on an uninformative observation $x_{\emptyset}$. This FSC cannot be expressed using options without [OOIs]{}.[]{data-label="fig:fsc"}](figures/hrlnext_fsc_bw)

While options with regular initiation sets are able to express some memory-based policies [@Sutton1999 page 7], the tiny but valid Finite State Controller presented 