A Novel Deontic Argumentation Semantics

This repository contains an Answer Set Programming implementation of the novel WP-semantics for Deontic Argumentation.

The main feature of the WP-sematnics is its ability to handle weak permission (in paticuler in the presence of deontic conflicts).

The wp-semantics is a bipolar sceptical semantics and assumes two types of arguments: natural arguments and imaginary arguments. The semantics adopts two relationships among arguments: attack and sub-argument. Thus, given two argumetns $A$ and $B$, we can have that $A$ attacks $B$ (written as $A > B$), and we can have that argument $C$ is a sub-argument of $A$ ( $C\in Sub(A)$ ).

Definition (Support) An set of argument $S$ supports an argument $A$ if every proper subargument of $A$ is in $S$.

Definition (Undercut) A set of arguments $S$ undercuts an argument $B$ if $S$ supports an argument $A$ that attacks a proper natural subargument of $B$.

Definition (wp-Acceptable) An argument $A$ is wp-acceptable by the sets of arguments $R$ and $S$ iff

1. $A$ is an imaginary argument and $\forall B\in Args$, $B>A$, $B$ is wp-rejected by $R$ and $S$; or
2. $A$ is a natural argument, $S$ supports $A$, and $\forall C\in Args$, $C>A$, $S$ undercuts $C$.

Definition (wp-Rejected) An argument $A$ is \emph{wp-rejected} by the sets of arguments $R$ and $S$ iff

1. $A$ is an imaginary argument and $\exists B\in Args$, $B>A$, $B\in S$; or
2. $A$ is a natural argument and
   1. $\exists B\in Sub(A)$, $B\neq A$, and $B\in R$; or
   2. $\exists B\in Args$, $B>A$, and $S$ supports $B$.

Definition (wp-extension) The \emph{wp-extension} of a Deontic Argumentation Theory $T$ is the pair

$$(JArgs,RArgs)$$

such that

$$ JArgs=\bigcup_{i=1}^{\infty}J^T_i \qquad\qquad RArgs=\bigcup_{i=1}^{\infty}R^T_i $$

• $J^T_0=\emptyset$; $R^T_0=\emptyset$;
• $J^T_{n+1}={ A\in Args: A \text{ is wp-acceptable by } R^T_n \text{ and } J^T_n}$;
• $R^T_{n+1}= {A\in Args: A \text{ is wp-rejected by } R^T_n \text{ and } J^T_n }$.