Language.Dung.AF

This module implements Dung's argumentation frameworks.

Basic definitions

data DungAF arg

An abstract argumentation framework is a set of arguments (represented as a list) and an attack relation on these arguments.

Constructors

AF [arg] [(arg, arg)]

Instances

Eq arg => Eq (DungAF arg)

Show arg => Show (DungAF arg)

setAttacks :: Eq arg => DungAF arg -> [arg] -> arg -> Bool

Given an argumentation framework, determines whether args (subset of the arguments in the AF), attacks an argument arg (in the AF).

aplus :: Eq arg => DungAF arg -> arg -> [arg]

Given an argumentation framework, determines the set of arguments that are attacked by an argument (in the AF).

amin :: Eq arg => DungAF arg -> arg -> [arg]

Given an argumentation framework, determines the set of arguments attacking an argument (in the AF).

argplus :: Eq arg => DungAF arg -> [arg] -> [arg]

Given an argumentation framework, determines the set of arguments that are attacked by the given subset of arguments (in the AF).

argmin :: Eq arg => DungAF arg -> [arg] -> [arg]

Given an argumentation framework, determines the set of arguments that attack a given subset of arguments (in the AF).

conflictFree :: Eq arg => DungAF arg -> [arg] -> Bool

Given an argumentation framework, determines whether args (subset of the arguments in the AF) is conflict-free.

acceptable :: Eq arg => DungAF arg -> 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).

f :: Eq arg => DungAF arg -> [arg] -> [arg]

Given an argumentation framework, returns the set of arguments that are acceptable with respect to args (subset of the arguments in the AF).

admissible :: Eq arg => DungAF arg -> [arg] -> Bool

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

Grounded, preferred, semi-stable and stable semantics through fixpoints

groundedF :: Eq arg => ([arg] -> [arg]) -> [arg]

Given a characteristic function f, computes the grounded extension by iterating on the empty set (list) until it reaches a fixpoint.

Basic labelling definitions

The following functions are implementations of the definitions in "An algorithm for Computing Semi-Stable Semantics" in "Symbolic and Quantitative Approaches to Reasoning with Uncertainty", pages 222--234, Springer, 2007.

data Status

Labelling status of arguments.

Constructors

In

Out

Undecided

Instances

Eq Status

Ord Status

Show Status

type Labelling arg = [(arg, Status)]

Labelling of arguments.

inLab :: Labelling arg -> [arg]

Given a labelling of arguments, give back the arguments labelled In.

outLab :: Labelling arg -> [arg]

Given a labelling of arguments, give back the arguments labelled Out.

undecLab :: Labelling arg -> [arg]

Given a labelling of arguments, give back the arguments labelled Undecided.

allIn :: [arg] -> Labelling arg

The allIn labelling is a Labelling that labels every argument In.

allOut :: [arg] -> Labelling arg

The allOut labelling is a Labelling that labels every argument Out.

allUndec :: [arg] -> Labelling arg

The allUndec labelling is a Labelling that labels every argument Undecided.

unattacked :: Eq arg => [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.

attacked :: 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.

labAttackers :: Eq arg => DungAF arg -> arg -> Labelling arg -> Labelling arg

Given an argumentation framework, determines the list of attackers of an argument, from a given labelling, returning the labelled attackers.

illegallyIn :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is illegally In iff a is labelled In, but not all its attackers are labelled Out.

illegallyOut :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is illegally Out iff a is labelled Out but does not have an attacker labelled In.

illegallyUndec :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is illegally Undecided iff a is labelled Undecided but either all its attackers are labelled Out or it has an attacker that is labelled In.

legallyIn :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is legally In iff a is labelled In and it's not illegallyIn.

legallyOut :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is legally Out iff a is labelled Out and it's not illegallyOut.

legallyUndec :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is legally Undecided iff a is labelled Undecided and it's not illegallyUndec.

isAdmissible :: Eq arg => DungAF arg -> Labelling arg -> Bool

Given an AF, an admissible labelling is a Labelling without arguments that are illegallyIn and without arguments that are illegallyOut.

isComplete :: Eq arg => DungAF arg -> Labelling arg -> Bool

Given an AF, a complete labelling is a labelling without arguments that are illegallyIn, without arguments that are illegallyOut and without arguments that are illegallyUndec.

isPreferred :: Eq arg => DungAF arg -> [Labelling arg] -> Labelling arg -> Bool

Let labs be a complete labelling, i.e. isComplete af labs, we say that labs is a preferred labelling iff inLab labs is maximal (w.r.t. set inclusion).

isStable :: Eq arg => DungAF arg -> [Labelling arg] -> Labelling arg -> Bool

Let labs be a complete labelling, i.e. 'isComplete af labs', we say that labs is a preferred labelling iff undecLab(labs) == []

isSemiStable :: Eq arg => DungAF arg -> [Labelling arg] -> Labelling arg -> Bool

Let labs be a complete labelling, i.e. isComplete af labs, we say that labs is a semi-stable labelling iff undecLab labs is minimal (w.r.t. set inclusion).

transitionStep :: Eq arg => DungAF arg -> Labelling arg -> arg -> Labelling arg

Given an AF, a labelling labs and an illegally in argument a in the af, (i.e. illegallyIn af a labs => True), a transition step on a in labs consists of the following: 1. the label of a is changed from In to Out 2. for every b in {a} cup a+, if b is illegally out, then change the label from b from Out to Undecided

terminatedTransition :: Eq arg => DungAF arg -> Labelling arg -> Bool

Given an AF, a labelling, labs, is terminated iff labs does not contain any argument that is illegally in, i.e. not (illegallyIn af lab arg) for all arg in labs.

superIllegallyIn :: Eq arg => DungAF arg -> Labelling arg -> (arg, Status) -> Bool

Given an AF and Labelling, an argument a (in the AF) is superillegally In iff a is labelled In, and it is attacked by an argument that is legally In or legally Undecided.

Grounded, preferred, semi-stable and stable labellings

The following functions are implementations of the definitions in "An algorithm for Computing Semi-Stable Semantics" in "Symbolic and Quantitative Approaches to Reasoning with Uncertainty", pages 222--234, Springer, 2007 and Section 4.1 of Proof Theories and Algorithms for Abstract Argumentation Frameworks by Modgil and Caminada.

grounded :: Eq arg => DungAF arg -> Labelling arg

Computes the grounded labelling for a Dung argumentation framework, returning a (unique) list of arguments with statuses.

Based on section 4.1 of Proof Theories and Algorithms for Abstract Argumentation Frameworks by Modgil and Caminada.

groundedExt :: Eq arg => DungAF arg -> [arg]

The grounded extension of an argumentation framework is just the grounded labelling, keeping only those arguments that were labelled In.

complete :: Ord arg => DungAF arg -> [Labelling arg]

Computes all complete labellings for a Dung argumentation framework. This is based on Caminada's algorithm for computing semi-stable labellings, with all checks removed.

preferred :: Ord arg => DungAF arg -> [Labelling arg]

Computes all preferred labellings for a Dung argumentation framework, by taking the maximally in complete labellings.

stable :: Ord arg => DungAF arg -> [Labelling arg]

Computes all stable labellings for a Dung argumentation framework, by keeping only those labellings with no Undecided labels.

semiStable :: Ord arg => DungAF arg -> [Labelling arg]

Computes all semi-stable labellings for a Dung argumentation framework, by taking the minimally undecided complete labellings.

completeExt :: Ord arg => DungAF arg -> [[arg]]

The complete extension of an argumentation framework is just the complete labelling, keeping only those arguments that were labelled In.

preferredExt :: Ord arg => DungAF arg -> [[arg]]

The preferred extension of an argumentation framework is just the preferred labelling, keeping only those arguments that were labelled In.

stableExt :: Ord arg => DungAF arg -> [[arg]]

The stable extension of an argumentation framework is just the stable labelling, keeping only those arguments that were labelled In.

semiStableExt :: Ord arg => DungAF arg -> [[arg]]

The semi-stable extension of an argumentation framework is just the semi-stable labelling, keeping only those arguments that were labelled In.