On the Computational Complexity of Naive-based Semantics for Abstract Dialectical Frameworks

On the Computational Complexity of Naive-based Semantics for Abstract Dialectical Frameworks

Sarah Gaggl¹ Sebastian Rudolph¹ Hannes Strass²

1Technische Universität Dresden, Germany

2Universität Leipzig, Germany

Buenos Aires, 31st July 2015

Sarah Gaggl (TU-Dresden) Complexity of Naive-based Semantics for ADFs 1 / 4

Motivation

Abstract dialectical frameworks (ADFs): a powerfulgeneralisationof Dung’s abstract argumentation frameworks

ADFs allow to model argumentation scenarios s.t. that ADFsemantics provide interpretations of scenarios

Naive-based semanticsare built upon the fundamental concept of conflict-freeness

Exhaustive analysis of computational complexity of naive-based semantics

Interesting results: some involve little-known classes of so-calledBoolean hierarchy(another hierarchy in between classes of polynomial hierarchy) Credulous and sceptical entailment:different complexity depending on whether we check fortruthorfalsityof statement

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ADFs and Semantics

Example

a

¬a

b

¬a∧d

c b

d c

cfi(D) ={{a7→u,b7→u,c7→u,d7→u}, {a7→u,b7→t,c7→u,d7→u}, {a7→u,b7→u,c7→t,d7→u}, {a7→u,b7→u,c7→u,d7→t}, {a7→u,b7→t,c7→t,d7→u}, {a7→u,b7→t,c7→u,d7→t}, {a7→u,b7→u,c7→t,d7→t}, {a7→u,b7→t,c7→t,d7→t}, {a7→u,b7→f,c7→f,d7→f}}

nai(D) ={{a7→u,b7→t,c7→t,d7→t}, {a7→u,b7→f,c7→f,d7→f}}

Definition

LetD= (S,L,C)be an ADF. A three-valued interpretationv :S→ {t,f,u}is conflict-free, i.e.v ∈cfi(D), iff for alls∈Swe have:

I v(s) =timplies thatϕ^v_s is satisfiable,

I v(s) =fimplies thatϕ^vs is unsatisfiable;

naive, i.e.v ∈nai(D), iffv is≤_i-maximal conflict-free;

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Main Results

cfi nai stg nai₂ Verσ DP-c Π^P₂-c Π^P₂-c Π^P₂-c Existsσ coDP-c coDP-c coDP-c coDP-c

Cred^t_σ NP-c NP-c Σ^P₃-c Σ^P₃-c Cred^f_σ Σ^P₂-c Σ^P₂-c Σ^P₃-c Σ^P₃-c Scep^t_σ trivial Π^P₂-c Π^P₃-c Π^P₃-c Scep^f_σ trivial Π^P₃-c Π^P₃-c Π^P₃-c

Decision Problems

LetD= (S,L,C)be an ADF,σa semantics.

Verσ: givenv :S→ {t,f,u}, isv ∈σ(D)?

Existsσ: does thereexistanon-trivialinterpretationv ∈σ(D)?

Cred^t_σ/ Cred^f_σ:s∈S, does thereexistan interpretationv ∈σ(D)with v(s) =t/v(s) =f?

Scep^t_σ/ Scep^f_σ:s ∈S, isv(s) =t/v(s) =ffor allv ∈σ(D)?

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