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

Treewidth-aware Reductions of Normal ASP to SAT -- Is Normal ASP Harder than SAT after All?

Answer Set Programming (ASP) is a paradigm for modeling and solving problems for knowledge representation and reasoning. There are plenty of results dedicated to studying the hardness of (fragments of) ASP. So far, these studies resulted in characterizations in terms of computational complexity as well as in fine-grained insights presented in form of dichotomy-style results, lower bounds when translating to other formalisms like propositional satisfiability (SAT), and even detailed parameterized complexity landscapes. A generic parameter in parameterized complexity originating from graph theory is the so-called treewidth, which in a sense captures structural density of a program. Recently, there was an increase in the number of treewidth-based solvers related to SAT. While there are translations from (normal) ASP to SAT, no reduction that preserves treewidth or at least keeps track of the treewidth increase is known. In this paper we propose a novel reduction from normal ASP to SAT that is aware of the treewidth, and guarantees that a slight increase of treewidth is indeed sufficient. Further, we show a new result establishing that, when considering treewidth, already the fragment of normal ASP is slightly harder than SAT (under reasonable assumptions in computational complexity). This also confirms that our reduction probably cannot be significantly improved and that the slight increase of treewidth is unavoidable. Finally, we present an empirical study of our novel reduction from normal ASP to SAT, where we compare treewidth upper bounds that are obtained via known decomposition heuristics. Overall, our reduction works better with these heuristics than existing translations

Treewidth-Aware Complexity in ASP: Not all Positive Cycles are Equally Hard

It is well-know that deciding consistency for normal answer set programs (ASP) is NP-complete, thus, as hard as the satisfaction problem for classical propositional logic (SAT). The best algorithms to solve these problems take exponential time in the worst case. The exponential time hypothesis (ETH) implies that this result is tight for SAT, that is, SAT cannot be solved in subexponential time. This immediately establishes that the result is also tight for the consistency problem for ASP. However, accounting for the treewidth of the problem, the consistency problem for ASP is slightly harder than SAT: while SAT can be solved by an algorithm that runs in exponential time in the treewidth k, it was recently shown that ASP requires exponential time in k \cdot log(k). This extra cost is due checking that there are no self-supported true atoms due to positive cycles in the program. In this paper, we refine the above result and show that the consistency problem for ASP can be solved in exponential time in k \cdot log({\lambda}) where {\lambda} is the minimum between the treewidth and the size of the largest strongly-connected component in the positive dependency graph of the program. We provide a dynamic programming algorithm that solves the problem and a treewidth-aware reduction from ASP to SAT that adhere to the above limit

Assessing composition gradients in multifilamentary superconductors by means of magnetometry methods

We present two magnetometry-based methods suitable for assessing gradients in the critical temperature and hence the composition of multifilamentary superconductors: AC magnetometry and Scanning Hall Probe Microscopy. The novelty of the former technique lies in the iterative evaluation procedure we developed, whereas the strength of the latter is the direct visualization of the temperature dependent penetration of a magnetic field into the superconductor. Using the example of a PIT Nb3Sn wire, we demonstrate the application of these techniques, and compare the respective results to each other and to EDX measurements of the Sn distribution within the sub-elements of the wire.Comment: 7 pages, 8 figures; broken hyperlinks are due to a problem with arXi

Rotating sample magnetometer for cryogenic temperatures and high magnetic fields

We report on the design and implementation of a rotating sample magnetometer (RSM) operating in the variable temperature insert of a cryostat equipped with a high-field magnet. The limited space and the cryogenic temperatures impose the most critical design parameters: the small bore size of the magnet requires a very compact pick-up coil system and the low temperatures demand a very careful design of the bearings. Despite these difficulties the RSM achieves excellent resolution at high magnetic field sweep rates, exceeding that of a typical vibrating sample magnetometer by about a factor of ten. In addition the gas-flow cryostat and the high-field superconducting magnet provide a temperature and magnetic field range unprecedented for this type of magnetometer.Comment: 10 pages, 5 figure