Complexity of Abstract Argumentation

argumentation may be further advanced. We stress that our aim is to focus on general areas rather than particular open questions as such: the reader who has followed the earlier exposition will have noted that a number of specific open issues have already been raised in the text. 6.1 Average case properties As discussed in Section 5.2, the lower bounds on problem complexity are worstcase, so leaving open the possibility that feasible algorithms may be available in suitable contexts. In addition to the use of restrictions on the form of instances one other approach that has been widely considered in the theory of algorithms is the study of average-case complexity. Underpinning this approach one considers a probability distribution, μ , on instances of a decision problem – often, but not invariably so, μ is the uniform distribution whereby each instance is equally likely, proceeding to define the average-case run time of an algorithm P on instances of size n of L as ∑x∈I(n) μ(x)ρ(P,x) where ρ(P,x) is the run-time of P on instance x. Formal definitions of average-case complexity classes may be found in [36]. To date surprisingly little work has been carried out concerning the application of average-case methods to decision problems in AFs either in terms of algorithmic development or in considering the limitations of such approaches. It remains open to what extent techniques such as those applied to other intractable problems, e.g., [1] for the NP–complete Hamiltonian cycle problem, or [46] for CNF satisfiability could be replicated in AF settings. Of some relevance to such approaches are so-called “phase-transition” effects, which received much attention in the mid-late 1990s as potential indicators of factors separating tractable and intractable classes of problem instances, e.g., the studies of random CNF-SAT from [37, 40]. Analytic studies of such effects appears to indicate connections between suitable witnessing structures, e.g., satisfying assignment, being present “almost certainly” and the performance of algorithms to identify such structures. Of some interest in the context of AF semantics are the results of [41, 17] which give conditions ensuring that a random AF “almost certainly” has a stable extension. There has as yet, however, been no detailed study of the implications of these results for fast on average methods for identifying or enumerating stable extensions. In the same way that the analyses of [41, 17] relate 102 Paul E. Dunne and Michael Wooldridge to the existence of stable extensions in AFs, it would be of some interest to examine to consider existence properties of other solution structures in random AFs and algorithmic consequences. 6.2 Approaches to dynamic updates An important feature of the argumentation forms discussed so far is that, in practice, these are not static systems: typically an AF, 〈A,R〉, represents only a “snapshot” of the environment, and, as further facts, information and opinions emerge the form of the initial view may change significantly in order to accommodate these. For example, additional arguments may have to be considered so changing A; existing attacks may cease to apply and new attacks (arising from changes to A) come into force. It is clear that accounting for such dynamic aspects raises a number of issues in terms of assessing the acceptability status of individual arguments. As with the study of average-case properties, the treatment of algorithms and complexity issues relating to determining argument status in dynamically changing environments has been somewhat neglected. Thus, given 〈A,R〉 and S ⊆A for which S ∈ Es(〈A,R〉) according to some semantics s, natural decision questions are: does x ∈ S continue to be credulously accepted (w.r.t. to semantics s) in the AF 〈B,S〉 where B results by removing some arguments from A and replacing these; similarly T modifies the attack relation R.