Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics. When asking for projected counts we are interested in counting the number of extensions of a given argumentation framework while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming (DP). Our algorithms run in time double or triple exponential in the treewidth depending on the considered semantics. Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension.Comment: Extended version of a paper published at AAAI-1

The counterSharp Model Counting Benchmark

We present the counterSharp benchmark consisting of 123 projected model counting instances. The instances originate from work on the reliability quantification of programs written in C. We briefly introduce the application field and describe the benchmark selection process

Is the largest aqueous gold cluster a superatom complex? Electronic structure & optical response of the structurally determined Au146(pMBA)57

The new water-soluble gold cluster Au146(pMBA)57, the structure of which has been recently determined at sub-atomic resolution by Vergara et al. [1], is the largest aqueous gold cluster ever structurally determined and likewise the smallest cluster with a stacking fault. The core presents a twinned truncated octahedron, while additional peripheral gold atoms follow a C2 rotational symmetry. According to the usual counting rules of the superatom complex (SAC) model, the compound attains a number of 92 SAC electrons if the overall net charge is 3- (three additional electrons). As this is the number of electrons required for a major shell closing, the question arises if Au146(pMBA)57 should be regarded as a superatom complex. Starting from the experimental coordinates we have analyzed the structure using density-functional theory. The optimized (relaxed) structure retains all the connectivity of the experimental coordinates, while removing much of its irregularities in interatomic distances, thereby enhancing the C2-symmetry feature. Analyzing the angular-momentum projected states, we show that, despite a small gap, the electronic structure does not exhibit SAC model character. In addition, optical absorption spectra are found to be relatively smooth compared to the example of the Au144(SR)60 cluster. The Au146(SR)57 cluster does not derive its stability from SAC character; it cannot be considered a superatom complex