Profiles
Properties
Flat assumption-based argumentation (Bondarenko et al. 1997; Dung, Kowalski and Toni 2009), as the instance of ASPIC+ it is (Prakken 2010): strict rules, assumptions (assumption/1) as ordinary premises, contraries (contrary(Assumption, Contrary)) one way only, no preference, no other conflict (not even negation). A theory with defeasible rules, or an assumption a rule concludes (not flat), is refused: the semantics of non-flat ABA are those of sets of assumptions, which the arguments do not capture.
ABA+ (Čyras and Toni 2016): flat ABA with preferences between assumptions, sup(Stronger, Weaker), an attack by an argument resting on a weaker assumption reversed (see the extension abaPref). The theory satisfies Weak Contraposition (Axiom 4 of Čyras and Toni), and is refused otherwise: without it, the semantics of ABA+ are those of sets of assumptions, which the arguments do not capture.
Dung frameworks: the arguments, a. (or arg(a)), and the attacks, a -> b. (or att(a, b)), and a <-> b. for a mutual one, stated or derived by Prolog clauses (X -> Y :- beats(X, Y).); an argument attacking or attacked is an argument as well. A theory with the rules, premises, preferences or conflicts of the structured profiles is refused.
The defeasible part of ASPIC+: defeasible rules alone, facts as rules with no premise (r : [] => a.), their preferences weighed by the last link, normal comparator; a theory with strict rules, axioms or premises is refused. Strict closure and the consistency postulates hold with nothing to close.
ASPIC+ (Modgil and Prakken 2013, 2014): strict and defeasible rules, axioms and ordinary premises, contraries, preferences weighed as orderingPrinciple and orderingComparator state, restricted rebut (only on a conclusion of a defeasible rule, rebutRestriction); the strict rules closed under transposition, which the rationality postulates need (Caminada and Amgoud 2007)