extension. The complete extensions form a complete semilattice w.r.t. set inclusion. [^3]

Argumentation Frameworks with Support {#sec:bip}
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Currently the most recognized frameworks with support are the Bipolar Argumentation Framework BAF [@article:newbaf], Argumentation Framework with Necessities AFN [@incoll:newafn] and Evidential Argumentation System EAS [@inproc:eas]. We will now briefly recall them in order to further motivate the directions of the semantics we have taken in ADFs.

The original bipolar argumentation framework BAF [@incoll:bipolar] studied a relation we will refer to as abstract support:

A **bipolar argumentation framework** is a tuple $(A, R, S)$, where $A$ is a set of **arguments**, $R \subseteq A \times A$ represents the **attack** relation and $S \subseteq A \times A$ the **support**.

The biggest difference between this abstract relation and any other interpretation of support is the fact that it did not affect the acceptability of an argument, i.e. even a supported argument could be accepted “alone”. The positive interaction was used to derive additional indirect forms of attack and based on them, stronger versions of conflict–freeness were developed.

We say that an argument $a$ **support attacks** argument $b$, if there exists some argument $c$ s.t. there is a sequence of supports from $a$ to $c$ 