Artificial Intelligence 171 (2007) 642–674
www.elsevier.com/locate/artint
Computing ideal sceptical argumentation
P.M. Dung a, P. Mancarella b, F. Toni c,∗
a Division of Computer Science, Asian Institute of Technology, PO Box 2754, Bangkok 10501, Thailand
b Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy
c Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Received 8 November 2006; received in revised form 24 April 2007; accepted 2 May 2007
Available online 10 May 2007
Abstract
We present two dialectic procedures for the sceptical ideal semantics for argumentation. The first procedure is defined in terms
of dispute trees, for abstract argumentation frameworks. The second procedure is defined in dialectical terms, for assumption-
based argumentation frameworks. The procedures are adapted from (variants of) corresponding procedures for computing the
credulous admissible semantics for assumption-based argumentation, proposed in [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic
proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114–159]. We prove that the
first procedure is sound and complete, and the second procedure is sound in general and complete for a special but natural class of
assumption-based argumentation frameworks, that we refer to as p-acyclic. We also prove that in the case of p-acyclic assumption-
based argumentation frameworks (a variant of) the procedure of [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for
assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114–159] for the admissible semantics is complete.
Finally, we present a variant of the procedure of [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-
based, admissible argumentation, Artificial Intelligence 170 (2006) 114–159] that is sound for the sceptical grounded semantics.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Abstract argumentation; Assumption-based argumentation; Ideal semantics; Proof procedure; Dispute
1. Introduction
Argumentation has proven to be a useful abstraction mechanism for understanding several AI problems, for exam-
ple non-monotonic reasoning (e.g. see [3,7]), defeasible logic (e.g. see [13]) and several forms of reasoning needed to
be performed by agents (e.g. see [15]).
Several formulations of argumentation have been proposed, including the frameworks of abstract argumentation
[7] and assumption-based argumentation [3,6,8]. For these two frameworks, several semantics have been proposed
defining what it means for a set of arguments to be deemed “acceptable” to a rational reasoner. All these semantics
rely upon the semantics of admissible arguments [3,7]. This semantics is credulous, in that it sanctions a set as
“acceptable” if it can successfully dispute every argument against it, without disputing itself. However, there might be
conflicting admissible sets. In some applications, it is more appropriate to adopt a sceptical semantics, whereby, for
* Corresponding author.
E-mail addresses: dung@cs.ait.ac.th (P.M. Dung), paolo.mancarella@unipi.it (P. Mancarella), ft@doc.ic.ac.uk (F. Toni).0004-3702/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.artint.2007.05.003

P.M. Dung et al. / Artificial Intelligence 171 (2007) 642–674 643example, only beliefs sanctioned by all (maximally) admissible sets of assumptions are held. For example, in the legal
domain, different members of a jury could hold different admissible sets of assumptions but a guilty verdict must be
the result of sceptical reasoning. Also, in a multi-agent setting, agents may have competing plans for achieving goals
(where a plan can be interpreted as an argument for the goal it allows to achieve), and, when negotiating resources,
they may decide to give away a resource only if that resource is not needed to support any of their plans. Furthermore,
in the same setting, agents may decide to request an “expensive” resource from another agent only if that resource is
useful to render all plans for achieving its goals executable.
Several sceptical semantics have been proposed for argumentation frameworks, notably the grounded semantics
[7] and the semantics whereby beliefs held within all maximally admissible sets of arguments are drawn, referred to
as the sceptically preferred semantics. The grounded semantics can be easily computed but is often overly sceptical.
Procedures for the computation of the sceptically preferred semantic exist, e.g. the TPI procedure [21] for coherent
argumentation frameworks [10], namely frameworks where all preferred sets of arguments are guaranteed to be stable,
and the procedure of [5], for any argumentation framework, defined in non-dialectical terms. To the best of our
knowledge, no dialectical procedure exists for checking whether a given belief can be deemed to hold under the
sceptically preferred semantics for non-coherent cases.
In this paper we present two novel procedures for computing sceptical argumentation under the ideal semantics,
originally proposed for extended logic programming in [1]. We adapt this semantics for abstract [7] and assumption-
based [3] argumentation frameworks. The ideal semantics is sceptical, and has the advantage of being easily com-
putable by a modification of the machinery presented in [8], but without being overly sceptical.
We define a procedure for the ideal semantics in abstract argumentation frameworks in terms of a form of dispute
trees adapted from corresponding trees for computing the admissibility semantics in [8]. We prove that this procedure
is sound and complete for all abstract argumentation frameworks. We define a procedure for the ideal semantics in
assumption-based argumentation frameworks in terms of a form of dispute derivations adapted from corresponding
derivations for computing the credulous admissibility semantics in [8]. These derivations use arguments which can
be computed effectively by backward deductions in assumption-based frameworks. We prove that this procedure is
sound for all assumption-based frameworks, and complete for a special class of assumption-based frameworks we
define, with the property of being p-acyclic.
The paper is organised as follows. In Section 2 we review abstract and assumption-based argumentation frame-
works, define the ideal semantics for both types of frameworks and relate it to other well-known semantics. We also
provide a formal link between the two types of frameworks. In Section 3 we define dispute trees for abstract argu-
mentation frameworks under the ideal semantics, by extending corresponding trees from the admissibility semantics
proposed in [8]. In Section 4 we define dispute derivations for assumption-based argumentation under the ideal se-
mantics, by extending corresponding derivations from the admissibility semantics proposed in [8]. We also provide
two new variants of the derivations of [8], and prove that the first is sound for the sceptical grounded semantics, and
the second is sound in general and complete for p-acyclic frameworks. In Section 5 we discuss some related work.
Finally, in Section 6 we conclude.
This paper is an extended and improved version of the paper [9]: with respect to its predecessor, it presents a formal
analysis of the ideal semantics also for abstract argumentation, a procedure (in terms of dispute trees) for the ideal
semantics for abstract argumentation, a procedure in terms of dispute derivations for the grounded semantics, and
contains proofs of all results.
2. Argumentation frameworks and semantics
In this section we briefly review the notions of abstract argumentation [7] and assumption-based argumentation
[3,4,8,14,16], and present the ideal semantics for argumentation, adapted from [1].
2.1. Abstract argumentation
Definition 2.1. An abstract argumentation framework is a pair (Arg,attacks) where Arg is a finite set, whose ele-
ments are referred to as arguments, and attacks ⊆ Arg × Arg is a binary relation over Arg. Given sets X,Y ⊆ Arg of
arguments, X attacks Y iff there exists x ∈ X and y ∈ Y such that (x, y) ∈ attacks.