set of **acceptance conditions**, one condition per each argument. An acceptance condition is a total function $C_s : 2^{par(s)} \rightarrow \{in, out\}$, where $par(s) = \{ p \in S \mid (p,s) \in L\}$ is the set of parents of an argument $s$.

One can also represent the acceptance conditions by propositional formulas [@thesis:stefan] rather than functions. By this we mean that given an argument $s \in S$, $C_s = \varphi_s$, where $\varphi_s$ is a propositional formula over arguments $S$. As we will be making use of both extension and labeling–based semantics, we need to provide necessary information on interpretations first (more details can be found in [@tofix:newadf; @inproc:adm]). Please note that the links in ADFs only represent connections between arguments, while the burden of deciding the nature of these connections falls to the acceptance conditions. Moreover, parents of an argument can be easily extracted from the conditions. Thus, we will use of shortened notation $D = (S,C)$ through the rest of this paper.

Interpretations and decisiveness {#sec:premdec}
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A two (or three–valued) interpretation is simply a mapping that assigns the truth values $\{{\mathbf{t}}, {\mathbf{f}}\}$ (respectively $\{{\mathbf{t}}, {\mathbf{f}}, {\mathbf{u}}\}$) to arguments. We will be making use both of partial (i.e. defined 