Formal Argumentation First International Workshop on Theory by Modgil, Sanjay; Oren, Nir; Toni, Francesca

By Modgil, Sanjay; Oren, Nir; Toni, Francesca

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Extra info for Formal Argumentation First International Workshop on Theory and Application, Tafa 2011, Barcelona, Spain, July 16-17, 2011, Revised Selected Papers

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In [8], Ext2Lab is restricted to conflict-free sets. Here it is well-defined for arbitrary sets of arguments, while it is equivalent to original definition when restricted to conflictfree sets. Now one can interpret an extension-based semantics as labeling-based semantics, using the function Ext2Lab to map each extension to a labeling. e. different extensions yield different labelings. Definition 7. Let F = (A, R) be an AF and σ an extension-based semantics. The corresponding labeling-based semantics σL is defined as follows σL (F ) = {Ext2Lab(E) | E ∈ σ(F )}.

Both Lin and Lout complete labeling. So if Lgrd (a) = in or Lgrd (a) = out we have that also J S com (F, a) = {in} or resp. J S com (F, a) = {out}. Otherwise we have that Lgrd (a) = undec and therefore undec ∈ J S com (F, a). To show (4) we use that Lundec = ∅ for every stable labeling and thus undec ∈ / J S stb (F, a). The remaining follows from the fact that for each semantic σ ∈ {prf , sem, stg, grd ∗ } it is the case that σ(F ) = ∅ and thus also that J S σ (F, a) = {}. t. some semantics. It remains to show that the remaining ones are possible.

Using this we obtain the exact complexity of JSsem , GJSsem ,JSstg and GJSstg . Proposition 10. The problems JSsem , GJSsem are DP 2 -complete. Proof. The membership follows from Theorem 1 and the fact that Versem ∈ coNP. 2 2 To show hardness we reducing the (DP 2 -hard) QBF ∀ -coQBF ∀ problem to JSsem . The 44 W. Dvoˇra´ k t¯ t c1 t¯ z b c2 y1 y¯1 y2 y¯2 y1 y¯1 y2 y¯2 t c3 z1 z¯1 y3 y¯3 y3 y¯3 b c4 z2 z¯2 Fig. 4. AF Fϕ,ψ with cϕ,1 = {y1 , y2 , z¯1 }, cϕ,2 = {¯ y1 , y¯2 , z¯1 }, cψ,3 = {y3 , y¯3 }, cψ,4 = {¯ z 1 , z2 } QBF 2∀ -coQBF 2∀ problem remains hard when the QBF s Φ, Ψ are in CNF with satisfiable clause sets φ, ψ.

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