Argument Systems provide a rich abstraction within which divers concepts of reasoning, acceptability and defeasibility of arguments, etc., may be studied using a unified framework. Two important concepts of the acceptability of an argument p in such systems are credulous acceptance to capture the notion that p can be `believed'; and sceptical acceptance capturing the idea that if anything is believed, then p must be. One important aspect affecting the computational complexity of these problems concerns whether the admissibility of an argument is defined with respect to `preferred' or `stable' semantics. One benefit of so-called `coherent' argument systems being that the preferred extensions coincide with stable extensions. In this note we consider complexity-theoretic issues regarding deciding if finitely presented argument systems modelled as directed graphs are coherent. Our main result shows that the related decision problem is (p) 2 --complete and is obtained solely via the graph-theoretic representation of an argument system, thus independent of the specific logic underpinning the reasoning theory.

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 (QCGs) are a natural tool for modelling goal-oriented multiagent systems. After introducing and formally defining QCGs, 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 1 -complete. (As an aside, we present what we believe is the first "natural" problem that is proven to be complete for D 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.

A great number of complexity classes between P and PSPACE can be defined via leaf languages for computation trees of nondeterministic polynomial time machines. Jenner, McKenzie, and Th'erien (Proceedings of the 9th Conference on Structure in Complexity Theory, 1994) raised the issue of whether considering balanced or unbalanced trees makes any difference. For a number of leaf language classes, coincidence of both models was shown, but for the very prominent example of leaf language classes from the alternating logarithmic time hierarchy the question was left open. It was only proved that in the balanced case these classes exactly characterize the classes from the polynomial time hierarchy. Here, we show that balanced trees apparently make a difference: In the unbalanced case, a class from the logarithmic time hierarchy characterizes the corresponding class from the polynomial time hierarchy with a PP-oracle. Along the way, we get an interesting normal form for PP computations.

It is a well-known result of Fagin that the complexity class NP coincides with the class of

In the spring of 1989, Seinosuke Toda of the University of Electro-Communications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To-89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest in computer science. In particular, we will introduce the reader to The Counting Hierarchy: a hierarchy of complexity classes contained in PSPACE and containing the Polynomial Hierarchy. Threshold Circuits: circuits constructed of MAJORITY gates; this notion of circuit is being studied not only by complexity theoreticians, but also by researchers in an active subfield of AI studying "neural networks". Along the way, we'll review the important notion of an operator on a complexity class. 1. The Counting Hierarchy, and Operators on Complexity Classes The counting hierarchy was defined in [Wa-86] and independently by Parberry and Schnitger in [PS-88]. (The motivation for [Wa-86] was the desir...

We present a Garey/Johnson-style list of problems known to be complete for the second and higher levels of the polynomial-time Hierarchy (polynomial hierarchy, or PH for short). We also include the best-known hardness of approximation results. The list will be updated as necessary. Updates The compendium currently lists more than 80 problems. Latest changes include: • added [GT26] SUCCINCT k-KING, • added [GT25] SUCCINCT k-DIAMETER, • added [GT4] SUCCINCT k-RADIUS at third level, • added [GT24] MINIMUM VERTEX COLORING DEFINING SET, • added [GT23] GRAPH SANDWICH PROBLEM FOR Π, • added [L24] MINIMUM 3SAT DEFINING SET,

Abstract. We introduce a simple and practically efficient method for repairing inconsistent databases. The idea is to properly represent the underlying problem, and then use off-the-shelf applications for efficiently computing the corresponding solutions. Given a possibly inconsistent database, we represent the possible ways to restore its consistency in terms of signed formulae. Then we show how the ‘signed theory ’ that is obtained can be used by a variety of computational models for processing quantified Boolean formulae, or by constraint logic program solvers, in order to rapidly and efficiently compute desired solutions, i.e., consistent repairs of the database. 1

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.

We study query order within the polynomial hierarchy. P C:D denotes the class of languages computable by a polynomial-time machine that is allowed one query to C followed by one query to D [HHW]. We prove that the levels of the polynomial hierarchy are order-oblivious: P \Sigma p j :\Sigma p k = P \Sigma p k :\Sigma p j . Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes---and thus essentially all standard complexity classes---inherit all order-obliviousness results that hold for P. 1 Introduction Query order was introduced by Hemaspaandra, Hempel, and Wechsung [HHW] in order to study whether the order in which information sources are accessed has any effect on the class of problems that can be solved. In the everyday world, the order in which we access information is crucial, and the work of Hemaspaandra, Hempel, and Wechsung [HHW] shows that this...

We consider a special kind of non-deterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the machine-independend concept of cluster sets which is the starting point to establish cluster operators. Cluster operators map complexity classes of sets into complexity classes of functions where for the domain classes only cluster sets are allowed. For the counting operator c#\Delta and the optimization operators cmax\Delta and cmin\Delta the structural relationships between images resulting from these operators on the polynomial-time hierarchy are investigated. Furthermore, we compare cluster operators with the corresponding common operators #\Delta, max\Delta and min\Delta [Tod90b, HW97].