Labellings for assumption-based and abstract argumentation

Abstract

The semantics of Assumption-Based Argumentation (ABA) frameworks are traditionally characterised as assumption extensions, i.e. sets of accepted assumptions. Assumption labellings are an alternative way to express the semantics of flat ABA frameworks, where one of the labels in, out, or undec is assigned to each assumption. They are beneficial for applications where it is important to distinguish not only between accepted and non-accepted assumptions, but further divide the non-accepted assumptions into those which are clearly rejected and those which are neither accepted nor rejected and thus undecided. We prove one-to-one correspondences between assumption labellings and extensions for the admissible, grounded, complete, preferred, ideal, semi-stable and stable semantics. We also show how the definition of assumption labellings for flat ABA frameworks can be extended to assumption labellings for any (flat and non-flat) ABA framework, enabling reasoning with a wider range of scenarios. Since flat ABA frameworks are structured instances of Abstract Argumentation (AA) frameworks, we furthermore investigate the relation between assumption labellings for flat ABA frameworks and argument labellings for AA frameworks. Building upon prior work on complete assumption and argument labellings, we prove one-to-one correspondences between grounded, preferred, ideal, and stable assumption and argument labellings, and a one-to-many correspondence between admissible assumption and argument labellings. Inspired by the notion of admissible assumption labellings we introduce committed admissible argument labellings for AA frameworks, which correspond more closely to admissible assumption labellings of ABA frameworks than admissible argument labellings do

On the Equivalence between Assumption-Based Argumentation and Logic Programming

“…As pointed out in the introduction, it is not surprising that there exists a semantic correspondence for some translation from an ABA framework into a logic program due to the semantic correspondence of ABA and AA (Dung et al, 2007;Caminada et al, 2015b;Schulz & Toni, 2017) and AA and LP (Osorio et al, 2005;Wu et al, 2009;Strass, 2013;Caminada et al, 2015a). However, the translation obtained from concatenating the ABA to AA and the AA to LP translations is different from our translation, as illustrated in the following.…”

“…Translating F into an AA framework and then into a logic program using existing translations (Dung et al, 2007;Caminada et al, 2015b;Schulz & Toni, 2017;Osorio et al, 2005;Wu et al, 2009;Strass, 2013;Caminada et al, 2015a) yields the logic program P = {x ← ; y ← ; z ← not y ; v ← ; w ← not y} (where the names of the atoms could be anything) 10 . In comparison, according to our translation we obtain the logic program P F = {a ← ; b ← a, not d ; c ← a, not b}, which is much closer to the underlying ABA framework.…”

“…The notion of assumption labellings has been extended from the complete semantics as introduced by Schulz and Toni (2014) to other well-known ABA semantics (Schulz & Toni, 2017), which were previously defined in terms of extensions rather than labellings (Bondarenko et al, 1997;Toni, 2014). Note that there exists a one-to-one correspondence between the assumption labellings and the assumption extensions of an ABA framework (Schulz & Toni, 2014.…”

“…sets of "acceptable" assumptions (Bondarenko et al, 1997), rather than as assumption labellings. However, the two notions of ABA semantics coincide (Schulz & Toni, 2017), so 3-valued stable (respectively well-founded, regular, (2-valued) stable, ideal) models also correspond to complete (respectively grounded, preferred, stable, ideal) assumption extensions.…”

“…Correspondence under certain semantics between an ABA framework and some translation into a logic program is not surprising since existing translations from ABA to AA (Dung et al, 2007;Caminada et al, 2015b;Schulz & Toni, 2017) and from AA to LP (Osorio, Zepeda, Nieves, & Cortés, 2005;Wu, Caminada, & Gabbay, 2009;Strass, 2013;Caminada, Sá, Alcântara, & Dvořák, 2015a) can be used to show correspondence under certain semantics between an ABA framework and the logic program resulting from concatenation. However, the concatenation may lead to an exponential blow-up as it requires the construction of all arguments from an ABA framework and is therefore not suitable for practical matters such as the computation of ABA semantics using LP tools.…”