Semi-Qualified Weak Local Function

The semi-qualified weak local function $g_{\mathsf{sq}\text{-}\sigma}$ of an SCC-decomposable semantics $\sigma$ computes the labellings of an SCC like the local function $f_\sigma$ of $\sigma$, after relabelling an $\mathsf{undec}$ attacker from outside the SCC as $\mathsf{out}$ exactly when no other given labelling of these attackers labels it $\mathsf{in}$.

Formal definition

Let $S_{\mathrm{in}}$ be the set of possible labellings of the outparents of the current SCC. Then $g_{\mathsf{sq}\text{-}\sigma}((A,R), A_{\mathrm{in}}, R_{\mathrm{in}}, L_{\mathrm{in}}, S_{\mathrm{in}}) = f_\sigma((A,R), A_{\mathrm{in}}, R_{\mathrm{in}}, L'_{\mathrm{in}})$, where $L'_{\mathrm{in}}(x) = \mathsf{out}$ if $L_{\mathrm{in}}(x) = \mathsf{undec}$ and no $L \in S_{\mathrm{in}}$ labels $x$ as $\mathsf{in}$, and $L'_{\mathrm{in}}(x) = L_{\mathrm{in}}(x)$ otherwise (Dauphin et al., 2020).