-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | An implementation of the Dung argumentation frameworks.
--
-- An implementation of Dung's argumentation frameworks, an abstract
-- argumentation model used to either directly represent conflicting
-- information, or used as a translation target for more complex
-- (structured) argumentation models. For an introduction to Dung's
-- frameworks see
-- http://en.wikipedia.org/wiki/Argumentation_framework and Dung's
-- paper from 1995: "On the acceptability of arguments and its
-- fundamental role in nonmonotonic reasoning, logic programming, and
-- n-person games", Artificial Intelligence 77: 321-357. For the paper
-- accompanying this library see "Towards a framework for the
-- implementation and verification of translations between argumentation
-- models" available at http://www.cs.nott.ac.uk/~bmv/Dung/.
@package Dung
@version 0.9
-- | This module implements Dung's argumentation frameworks.
module Language.Dung.AF
-- | An abstract argumentation framework is a set of arguments (represented
-- as a list) and an attack relation on these arguments.
data DungAF arg
AF :: [arg] -> [(arg, arg)] -> DungAF arg
-- | Given an argumentation framework, determines whether args (subset of
-- the arguments in the AF), attacks an argument arg (in the AF).
setAttacks :: Eq arg => DungAF arg -> [arg] -> arg -> Bool
-- | Given an argumentation framework, determines whether args (subset of
-- the arguments in the AF) is conflict-free.
conflictFree :: Eq arg => DungAF arg -> [arg] -> Bool
-- | Given an argumentation framework, determines whether an argument is
-- acceptable with respect to a list of args (subset of the
-- arguments in the AF).
acceptable :: Eq arg => DungAF arg -> arg -> [arg] -> Bool
-- | Given an argumentation framework, determines whether an argument is
-- acceptable with respect to args (subset of the arguments in
-- the AF).
f :: Eq arg => DungAF arg -> [arg] -> [arg]
-- | Given an argumentation framework, determines whether the set of
-- arguments args (subset of the arguments in the AF) is
-- admissible, i.e. if args is conflictFree and args is a
-- subset of f af args
admissible :: Eq arg => DungAF arg -> [arg] -> Bool
-- | Given a list of arguments that are Out in an argumentation
-- framework af, an argument arg is unattacked if the list of
-- its attackers, ignoring the outs, is empty.
unattacked :: Eq arg => [arg] -> DungAF arg -> arg -> Bool
-- | Given a list of arguments that are In in an argumentation
-- framework af, an argument arg is attacked if there exists an
-- attacker that is In.
attacked :: Eq arg => [arg] -> DungAF arg -> arg -> Bool
-- | Given a characteristic function f, computes the grounded extension by
-- iterating on the empty set (list) until it reaches a fixpoint.
groundedF :: Eq arg => ([arg] -> [arg]) -> [arg]
-- | Labelling of arguments.
data Status
In :: Status
Out :: Status
Undecided :: Status
-- | Computes the grounded labelling for a Dung argumentation framework,
-- returning a list of arguments with statuses.
--
-- Based on section 4.1 of Proof Theories and Algorithms for Abstract
-- Argumentation Frameworks by Modgil and Caminada
grounded :: Eq arg => DungAF arg -> [(arg, Status)]
-- | The grounded extension of an argumentation framework is just the
-- grounded labelling, keeping only those arguments that were labelled
-- In.
groundedExt :: Eq arg => DungAF arg -> [arg]
instance Eq arg => Eq (DungAF arg)
instance Show arg => Show (DungAF arg)
instance Eq Status
instance Show Status
-- | This is the examples module accompanying the implementation of Dung's
-- argumentation frameworks.
--
-- This module contains a collection of examples, showing how to define
-- arguments, argumentation frameworks and how to use the standard
-- definitions.
--
-- To run these examples, or your own: start GHCi and do the following:
--
--
-- :l Language.Dung.Examples
--
module Language.Dung.Examples
-- | The simplest abstract argument is an argument identifiable by its name
type AbsArg = String
-- | Example AF: A -> B -> C
exampleAF :: DungAF AbsArg
-- | Example AF: A <-> B
--
-- Now follow a few example outputs using the above argumentation
-- frameworks.
--
--
--
-- [a,b] setAttacks c in the argumentation
-- framework exampleAF:
--
--
-- >>> setAttacks exampleAF [a,b] c
-- True
--
--
--
-- >>> setAttacks exampleAF [b,c] a
-- False
--
--
--
-- >>> setAttacks exampleAF2 [] b
-- False
--
--
--
--
-- [a,c] is conflictFree in the argumentation framework
-- exampleAF:
--
--
-- >>> conflictFree exampleAF [a,c]
-- True
--
--
--
-- >>> conflictFree exampleAF [a,b,c]
-- False
--
--
--
-- >>> conflictFree exampleAF2 [a,b]
-- False
--
--
--
--
-- c is acceptable w.r.t. [a,b] in the argumentation
-- framework exampleAF:
--
--
-- >>> acceptable exampleAF c [a,b]
-- True
--
--
--
-- >>> acceptable exampleAF c []
-- False
--
--
--
-- >>> acceptable exampleAF b [a,b,c]
-- False
--
--
--
--
-- [a,b,c] is admissible in the argumentation framework
-- exampleAF:
--
--
-- >>> admissible exampleAF [a,b,c]
-- False
--
--
--
-- >>> admissible exampleAF [a,c]
-- True
--
--
--
-- >>> admissible exampleAF [a]
-- True
--
--
--
--
-- The grounded labelling of the argumentation framework
-- exampleAF:
--
--
-- >>> grounded exampleAF
-- [("A",In),("C",In),("B",Out)]
--
--
--
-- >>> grounded exampleAF2
-- [("A",Undecided),("B",Undecided)]
--
--
--
--
-- The grounded extension of the argumentation framework
-- exampleAF:
--
--
-- >>> groundedExt exampleAF
-- ["A", "C"]
--
-- >>> groundedExt exampleAF2
-- []
--
exampleAF2 :: DungAF AbsArg
-- | fixed point function for a specific argumentation framework, faf =
-- f exampleAF.
--
--
--
-- The grounded extension of the argumentation framework exampleAF
-- using the fixpoint definition:
--
--
-- >>> groundedF faf
-- ["A","C"]
--
--
--
-- >>> groundedF (f exampleAF2)
-- []
--
faf :: [AbsArg] -> [AbsArg]