and one option [$\langle \emptyset, n_1 \rangle$]{} per node in the FSC. Assuming that $n_0 = \emptyset$ and $\eta (\emptyset, x_1, \cdot) = \eta^0 (x_1, \cdot)$, the options are defined by:

$$\begin{aligned}
            \label{eq:fsc_beta}
            \beta_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} (x_t)     &= 1 \\
            \label{eq:fsc_pio}
            \pi_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} (x_t, a_t)  &= \psi(n_t, a_t) \\
            \label{eq:fsc_pi}
            \mu (x_t, {\ensuremath{\langle n'_{t-1}, n_t \rangle}})        &= \eta (n'_{t-1}, x_t, n_t) \\
            \nonumber
            I_{{\ensuremath{\langle \emptyset, n_1 \rangle}}}           &= \Omega \times \{ \emptyset \} \\
            \nonumber
            I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}}               &= \Omega \times \{{\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}: n'_{t-1} = n_{t-1} \}
        \end{aligned}$$

Each option corresponds to an edge of the FSC. Equation \[eq:fsc\_beta\] ensures that every option stops after having emitted a single action, as the FSC takes one transition every time-step. Equation \[eq:fsc\_pio\] maps the current option to the action emitted by the destination node of its corresponding FSC edge. We show that $\mu$ and $I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}}$ implement $\eta (n_{t-1}, x_t, n_t)$, with $\omega_{t-1} = {\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}$, by:

$$\begin{aligned}
            \mu   &(x_t, {\ensuremath{\langle n'_{t-1}, n_t \rangle}}) = \\ &\begin{cases}
                \eta (n_{t-1}, x_t, n_t) &\begin{array}{l}
                    {\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}\in I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} \\
                    \Leftrightarrow n'_{t-1} = n_{t-1}
                \end{array} \\
                0 &\begin{array}{l}
                    {\ensuremath{\langle n'_{t-2}, n_{t-1} 