Paul E. Dunne

Weighted argument systems: Basic definitions, algorithms, and complexity results

We introduce and investigate a natural extension of Dungs well-known model of argument systems in which attacks are associated with a weight, indicating the relative strength of the attack. A key concept in our framework is the notion of an inconsistency budget, which characterises how much inconsistency we are prepared to tolerate: given an inconsistency budget @b, we would be prepared to disregard attacks up to a total weight of @b. The key advantage of this approach is that it permits a much finer grained level of analysis of argument systems than unweighted systems, and gives useful solutions when conventional (unweighted) argument systems have none. We begin by reviewing Dungs abstract argument systems, and motivating weights on attacks (as opposed to the alternative possibility, which is to attach weights to arguments). We then present the framework of weighted argument systems. We investigate solutions for weighted argument systems and the complexity of computing such solutions, focussing in particular on weighted variations of grounded extensions. Finally, we relate our work to the most relevant examples of argumentation frameworks that incorporate strengths.

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.

On the computational complexity of qualitative coalitional games

Abstract We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices , with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games ( qcg s) are a natural tool for modelling goal-oriented multiagent systems. After introducing and formally defining qcg s, we systematically formulate fourteen natural decision problems associated with them, and determine the computational complexity of these problems. For example, we formulate a notion of coalitional stability inspired by that of the core from conventional coalitional games, and prove that the problem of showing that the core of a qcg is non-empty is D p 1 -complete. (As an aside, we present what we believe is the first “natural” problem that is proven to be complete for D p 2 .) We conclude by discussing the relationship of our work to other research on coalitional reasoning in multiagent systems, and present some avenues for future research.

On the computational complexity of coalitional resource games

We study Coalitional Resource Games (CRGs), a variation of Qualitative Coalitional Games (QCGs) in which each agent is endowed with a set of resources, and the ability of a coalition to bring about a set of goals depends on whether they are collectively endowed with the necessary resources. We investigate and classify the computational complexity of a number of natural decision problems for CRGs, over and above those previously investigated for QCGs in general. For example, we show that the complexity of determining whether conflict is inevitable between two coalitions with respect to some stated resource bound (i.e., a limit value for every resource) is co-NP-complete. We then investigate the relationship between CRGs and QCGs, and in particular the extent to which it is possible to translate between the two models. We first characterise the complexity of determining equivalence between CRGS and QCGs. We then show that it is always possible to translate any given CRG into a succinct equivalent QCG, and that it is not always possible to translate a QCG into an equivalent CRG; we establish some necessary and some sufficient conditions for a translation from QCGs to CRGs to be possible, and show that even where an equivalent CRG exists, it may have size exponential in the number of goals and agents of its source QCG.

Algorithms for decision problems in argument systems under preferred semantics

For Dung@?s model of abstract argumentation under preferred semantics, argumentation frameworks may have several distinct preferred extensions: i.e., in informal terms, sets of acceptable arguments. Thus the acceptance problem (for a specific argument) can consider deciding whether an argument is in at least one such extensions (credulously accepted) or in all such extensions (skeptically accepted). We start by presenting a new algorithm that enumerates all preferred extensions. Following this we build algorithms that decide the acceptance problem without requiring explicit enumeration of all extensions. We analyze the performance of our algorithms by comparing these to existing ones, and present experimental evidence that the new algorithms are more efficient with respect to the expected running time. Moreover, we extend our techniques to solve decision problems in a widely studied development of Dung@?s model: namely value-based argumentation frameworks (vafs). In this regard, we examine analogous notions to the problem of enumerating preferred extensions and present algorithms that decide subjective, respectively objective, acceptance.

Computing Preferred Extensions in Abstract Argumentation: A SAT-Based Approach

This paper presents a novel SAT-based approach for the computation of extensions in abstract argumentation, with focus on preferred semantics, and an empirical evaluation of its performances. The approach is based on the idea of reducing the problem of computing complete extensions to a SAT problem and then using a depth-first search method to derive preferred extensions. The proposed approach has been tested using two distinct SAT solvers and compared with three state-of-the-art systems for preferred extension computation. It turns out that the proposed approach delivers significantly better performances in the large majority of the considered cases.

Complexity in Value-Based Argument Systems

We consider a number of decision problems formulated in value-based argumentation frameworks (VAFs), a development of Dung’s argument systems in which arguments have associated abstract values which are considered relative to the orderings induced by the opinions of specific audiences. In the context of a single fixed audience, it is known that those decision questions which are typically computationally hard in the standard setting admit efficient solution methods in the value-based setting. In this paper we show that, in spite of this positive property, there still remain a number of natural questions that arise solely in value-based schemes for which there are unlikely to be efficient decision processes.

Solving coalitional resource games

Coalitional Resource Games (crgs) are a form of Non-Transferable Utility (ntu) game, which provide a natural formal framework for modelling scenarios in which agents must pool scarce resources in order to achieve mutually satisfying sets of goals. Although a number of computational questions surrounding crgs have been studied, there has to date been no attempt to develop solution concepts for crgs, or techniques for constructing solutions. In this paper, we rectify this omission. Following a review of the crg framework and a discussion of related work, we formalise notions of coalition structures and the core for crgs, and investigate the complexity of questions such as determining nonemptiness of the core. We show that, while such questions are in general computationally hard, it is possible to check the stability of a coalition structure in time exponential in the number of goals in the system, but polynomial in the number of agents and resources. As a consequence, checking stability is feasible for systems with small or bounded numbers of goals. We then consider constructive approaches to generating coalition structures. We present a negotiation protocol for crgs, give an associated negotiation strategy, and prove that this strategy forms a subgame perfect equilibrium. We then show that coalition structures produced by the protocol satisfy several desirable properties: Pareto optimality, dummy player, and pseudo-symmetry.

Representation and Complexity in Boolean Games

Boolean games are a class of two-player games which may be defined via a Boolean form over a set of atomic actions. A particular game on some form is instantiated by partitioning these actions between the players – player 0 and player 1 – each of whom has the object of employing its available actions in such a way that the game’s outcome is that sought by the player concerned, i.e. player i tries to bring about the outcome i. In this paper our aim is to consider a number of issues concerning how such forms are represented within an algorithmic setting. We introduce a concept of concise form representation and compare its properties in relation to the more frequently used “extensive form” descriptions. Among other results we present a “normal form” theorem that gives a characterisation of winning strategies for each player. Our main interest, however, lies in classifying the computational complexity of various decision problems when the game instance is presented as a concise form. Among the problems we consider are: deciding existence of a winning strategy given control of a particular set of actions; determining whether two games are “equivalent”.

The Computational Complexity of Agent Verification

In this paper, we investigate the computational complexity of the agent verification problem. Informally, this problem can be understood as follows. Given representations of an agent, an environment, and a task we wish the agent to carry out in this environment, can the agent be guaranteed to carry out the task successfully? Following a formal definition of agents, environments, and tasks, we establish two results, which relate the computational complexity of the agent verification problem to the complexity of the task specification (how hard it is to decide whether or not an agent has succeeded). We first show that for tasks with specifications that are in ?up, the corresponding agent verification problem is ?u+1p-complete; we then show that for pspace-complete task specifications, the corresponding verification problem is no worse -- it is also pspace-complete. Some variations of these problems are investigated. We then use these results to analyse the computational complexity of various common kinds of tasks, including achievement and maintenance tasks, and tasks that are specified as arbitrary Boolean combinations of achievement and maintenance tasks.