In this section we discuss work which is related to the content of this thesis. As our work is dedicated to an analysis of thecf2 argumentation semantics, we start with related work on the systematic evaluation of semantics. Here one can mainly mention the work done by Baroni and Giacomin [7, 8, 10, 11]. They introduced several general evaluation criteria a semantics should fulfill. As none of the previously existing semantics satisfied all those criteria, they defined the resolution-based grounded semantics [9] which closed this gap. Also thecf2 semantics has been defined by them, to overcome the problems which arise on AFs with odd-length cycles.
12seehttp://homepage.univie.ac.at/wolfgang.dvorak/files/cf2-challenge.pdf
Caminada and Amgoud defined rationality postulatesfor argumentation systems [29]. In particular these postulates are defined for the ASPIC system [89] which is based on strict and defeasible rules. In contrast to the evaluation criteria proposed by Baroni and Giacomin, these postulates refer to the whole argumentation system and not to the individual semantics. We did not consider these postulates in more detail in this work because we concentrated on abstract argumentation only. Furthermore, the instantiation process in the ASPIC system is not adequate for naive-based semantics likecf2, because if one instantiates within this framework and then applies a semantics which is not admissible-based, the outcome turns out to be inconsistent.
Next, we consider different approaches which deal with the problematic ofcyclesin AFs, and in particular with odd-length cycles. Of course thecf2 semantics is not the only attempt to solve this problem. Bodanza and Tohmé introduced thetolerantsemantics [23], which has the intuition that the defense of a set of arguments should not be defined in absolute terms but relative to other possible challenging sets of arguments. They propose an application of this semantics in the field of strategic argumentation games, where each player has to choose a set of arguments to confront with and defend against the possible choices of the other agent.
Gabbay introduced severalloop-bustingsemantics in [64]. One of them, the LB2 semantics has shown to be equivalent tocf2. All these semantics are involved in theequational approachto argumentation networks [65]. The author also defines an equational approach tostage2 seman-tics in [64], namely LB2 −stage. Furthermore in [1] the authors propose theShkopsemantics which has been shown to be equivalent to LB4 from the loop-busting semantics in [64].
Roos proposed in [91] thepreferential model semantics which also handles odd loops in a special way. The motivation for this semantics comes from the preferential model semantics for non-monotonic reasoning systems [73]. There, the attack relation is used to define preferences over states. So, not one argument is preferred over another one, but one prefers a state where the attacking argument is valid, over a state where the attacked argument is valid. This semantics results in different extensions thancf2, for example consider the AFF = (A, R) with A = {a, b, c}andR ={(a, b),(b, c),(c, a),(c, c)}. Then,{a}and{b}arecf2 extensions, but only {a}is apmextension, because the stateais preferred over the stateb, as the only attacker ofa is the argumentcwhich is self-attacking.
Next, we consider work related to the investigation of equivalence as we did in Chapter 6.
We start with Amgoud and Vesic who studied equivalence of logic-based argumentation [3]
with respect to stable semantics. In particular the authors refined and extended the criteria from Oikarinen and Woltran for logic-based argumentation systems by taking the internal structure of the arguments into account.
Baumann characterized two new notions of equivalence, namely normal and strong expan-sion equivalence which lie in-between standard and strong equivalence [18]. There new ments and attacks can be added with the condition that the attacks between the original argu-ments remain unchanged.
Cayrol et al. studied therevisionof an argument system in [33] oriented on the field of belief revision [2]. There, the authors study the impact of adding a single new argument to an AF.
Finally we want to mention that the idea of considering strong equivalence of argumentation framework arose from the work on strong equivalence of logic programs [75, 100]. Therefore also the notion of strong equivalence for AFs is very similar to the one of logic programs.
Last, we say some words about related work on implementations. The only mentionable reference about a system supporting the cf2 semantics is the work done by Osorio et al. [86].
They presented ASP encodings for thecf2 semantics at COMMA 2010. Thus, at the same time as we presented the alternative characterization ofcf2 and the respective ASP encodings [67].
However, these encodings are not implemented in any system and as they are based on the original definition ofcf2they are very hard to follow. Moreover, one can observe that disjunction has been used in some rule heads, thus they are not even adequate from a complexity point of view. As disjunctive logic programs have a data complexity ofΣP2 (resp.ΠP2), but the complexity ofcf2 isNP-complete (resp.coNP-complete).
Regarding other reductions from argumentation to logic programing one can mention the work of Nieves et al. [81, 82]. One aspect in their work is to use a fixed encodingschemato represent AFs as logic programs, and then show how different semantics for logic programs can be used to compute different forms of extensions using this particular schema. Most notably, they showed that in their setting the stable semantics (for logic programs) captures stable exten-sions of AFs, the well-founded semantics captures the grounded extension of AFs, and a novel stratification semantics [82] captures thecf2 semantics. Osorio et al. [85] present an algorithm for computing preferred extensions (based on abductive logic programming) using a fixed logic program to characterize the admissible sets in the same manner as it is done in the ASPAR-TIX approach. In [81], a different approach to compute preferred extensions by means of logic programs has been proposed. However, this work requires a recompilation of the encoding for each particular AF. Similarly, Wakaki and Nitta [99] also provide ASP encodings for different semantics. In contrast to the ASPARTIX approach, their encodings for complete and stable se-mantics are based on labelings, whereas for grounded, preferred and semi-stable sese-mantics they use a meta-programming technique applying additional translations for each AF into normal logic programs.
8.4 Future Work
In the following we will list some possible future directions. Regarding the alternative charac-terization, we note that it can also be seen as a general schema, where one can exchange the parts. For example, sem(F) = {S | σ(F)∩τ([[F −∆F,S]])}, where for naive-based se-mantics σ = naive and for admissible-based semantics σ = adm. One special case of this instantiation isstable2(F) ={S |S ∈naive(F)∩stable([[F −∆F,S]])}and it clearly holds thatstable2(F) = stable(F). The investigation of other such combinations might reveal new options.
As it turned out that strong equivalence is indeed very strong for many semantics, it can be beneficial to relax the notion of equivalence and for example consider a relativized notion, where source and target of attacks are restricted. This can be interesting in the course of two agents, where one can only point attacks from and to a specific set of arguments. Also a more fine grained classification of the semantics with respect to different notions of succinctness can be identified as a future direction. The information obtained there can help to improve instantiation methods.
Regarding implementations we would like to investigate if an optimization of the ASP en-codings by using for example aggregates or symmetry breaking can improve the performance.
As far es we know there does not exist yet an appropriate instantiation method for naive-based semantics. It has been shown that both stage and cf2 semantics produce inconsistent solutions when instantiated within the ASPIC+ system. Thus, the identification of possible application scenarios forcf2 andstage2 semantics and the respective instantiation methods is still open.
With the use of the concept of Modular Logic Programming (MLP) [62] one can implement the whole argumentation process, from the instantiation of the arguments and attacks to the computation of the semantics. Due to the modularity of this approach, we plan to instantiate the frameworks from an input database and embed the existing ASP encodings in one program with several modules.
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