Abstract
AbstractPowerful formalisms for abstract argumentation have been proposed, among them abstract dialectical frameworks (ADFs) that allow for a succinct and flexible specification of the relationship between arguments and the GRAPPA framework which allows argumentation scenarios to be represented as arbitrary edge-labeled graphs. The complexity of ADFs and GRAPPA is located beyond NP and ranges up to the third level of the polynomial hierarchy. The combined complexity of Answer Set Programming (ASP) exactly matches this complexity when programs are restricted to predicates of bounded arity. In this paper, we exploit this coincidence and present novel efficient translations from ADFs and GRAPPA to ASP. More specifically, we provide reductions for the five main ADF semantics of admissible, complete, preferred, grounded, and stable interpretations, and exemplify how these reductions need to be adapted for GRAPPA for the admissible, complete, and preferred semantics.
Highlights
Since the DIAMOND systems rely on static encodings, i.e. the encoding does not change for different framework instances, this approach is limited by the data complexity of Answer Set Programming (ASP) (which only reaches the second level of the polynomial hierarchy (Eiter and Gottlob 1995; Eiter et al 1997))
We introduce a new method for implementing reasoning tasks related to both abstract dialectical frameworks (ADFs) and GRAPPA such that even the hardest among the problems are treated with a single call to an ASP solver
In the encoding of the preferred semantics for an ADF D we extend πadm(D) by making use of the saturation technique to verify that all interpretations of D that are greater w.r.t. ≤i than the interpretation determined by the assignments guessed in the program fragment πguess are either identical to the interpretation in question or not admissible
Summary
Argumentation is an active area of research with applications in legal reasoning (Bench-Capon and Dunne 2005), decision making (Amgoud and Prade 2009), e-governance (Cartwright and Atkinson 2009) and multi-agent systems (McBurney et al 2012). Since the DIAMOND systems rely on static encodings, i.e. the encoding does not change for different framework instances, this approach is limited by the data complexity of ASP (which only reaches the second level of the polynomial hierarchy (Eiter and Gottlob 1995; Eiter et al 1997)). We introduce a new method for implementing reasoning tasks related to both ADFs and GRAPPA such that even the hardest among the problems are treated with a single call to an ASP solver (and avoiding any exponential blow-up in data or program size). Our approach makes use of the fact that the combined complexity of ASP for programs with predicates of bounded arity (Eiter et al 2007) exactly matches the complexity of ADFs and GRAPPA This approach is called dynamic, because the encodings are generated individually for every instance. The paper is based on (Section 3.2 of) the second author’s thesis (Diller 2019)
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