= Abstract Dialectical Frameworks Well-founded Model =

== Problem Description ==

Abstract Dialectical Frameworks (ADFs for short) are a 
formalism to model argumentation. They consist of statements and one 
acceptance condition (ac) for each statement. The acceptance conditions are 
boolean formulas, to denote relations between statements. Each formula has 
statements as its variables. 

The well-founded model now computes all statements, which are necessarily 
accepted or rejected. The well-founded model is defined via the following 
"algorithm":

1.) Accept := {}, Reject := {}
2a.) Add each statement s with a valid formula to Accept
2b.) Add each statement s with an unsatisfiable formula to Reject
3.) Replace the accepted variables in all acceptance conditions
(i.e. variables in Accept) with the constant true and all rejected variables 
with false
4.) If new statements have been accepted/rejected in this iteration go to 
step 2a, else stop.

Complexity-wise, the verification if a given solution is indeed a solution is DP-complete and the problem of computing a solution is NP-easy.

Reference Paper:
G. Brewka, S. Ellmauthaler, H. Strass, J. P. Wallner, and S. Woltran, Abstract Dialectical Frameworks Revisited, in IJCAI'13.


== Predicates ==

 * '''Input''': {{{statement/1, ac/2}}}

 * '''Output''': {{{accept/1, reject/1}}}

== Input format ==

We encode a statement X by {{{statement(X)}}}.

The acceptance condition of a statement X is given by
{{{ac(X,F)}}} where F is a formula, specified as follows in BNF, with s being a statement:

<F> ::= s | c(v) | c(f) | neg(<F>) | and(<F>,<F>) | or(<F>,<F>)
by c(v) and c(f) we denote the truth constants true and false respectively.


== Output format ==


== Example(s) ==

A possible input is:
{{{ 
statement(1). statement(2). statement(3). statement(4). 
ac(1,c(v)). ac(2,1). ac(3,or(neg(3),4). ac(4,c(f)). 
}}}

Then a correct result would be:
{{{
accept(1). accept(2). reject(4).
}}}

Additional sample instances: [[https://www.mat.unical.it/aspcomp2013/files/samples/abstract_dialectical_frameworks-sample.zip|download]]

== Problem Peculiarities ==

'''Type''': Optimization
'''Competition''': System Track


== Notes and Updates ==



== Author(s) ==
 * Authors: Stefan Ellmauthaler, Johannes Wallner	
  * Affiliation: DBAI, Vienna University of Technology, Austria