 $$\begin{aligned} \tilde{E}_{t}^{m}(g_{i_{0}},\cdots , g_{i_{k}},s) = \begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{0}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}  \\  \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty. } \end{cases} \label{eq:GeneralLookupTable