Qualified Local Function

The qualified local function $f_{\mathsf{q}\text{-}\sigma}$ of an SCC-decomposable semantics $\sigma$ computes the labellings of an SCC like the local function $f_\sigma$ of $\sigma$, after relabelling every $\mathsf{undec}$ attacker from outside the SCC as $\mathsf{out}$.

Formal definition

$f_{\mathsf{q}\text{-}\sigma}((A,R), A_{\mathrm{in}}, R_{\mathrm{in}}, L_{\mathrm{in}}) = f_\sigma((A,R), A_{\mathrm{in}}, R_{\mathrm{in}}, L'_{\mathrm{in}})$, where $L'_{\mathrm{in}}(x) = \mathsf{out}$ if $L_{\mathrm{in}}(x) = \mathsf{undec}$, and $L'_{\mathrm{in}}(x) = L_{\mathrm{in}}(x)$ otherwise (Dauphin et al., 2020).