31 research outputs found
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
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 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
Parallel Model Counting with CUDA: Algorithm Engineering for Efficient Hardware Utilization
Propositional model counting (MC) and its extensions as well as applications in the area of probabilistic reasoning have received renewed attention in recent years. As a result, also the need for quickly solving counting-based problems with automated solvers is critical for certain areas. In this paper, we present experiments evaluating various techniques in order to improve the performance of parallel model counting on general purpose graphics processing units (GPGPUs). Thereby, we mainly consider engineering efficient algorithms for model counting on GPGPUs that utilize the treewidth of a propositional formula by means of dynamic programming. The combination of our techniques results in the solver GPUSAT3, which is based on the programming framework Cuda that -compared to other frameworks- shows superior extensibility and driver support. When combining all findings of this work, we show that GPUSAT3 not only solves more instances of the recent Model Counting Competition 2020 (MCC 2020) than existing GPGPU-based systems, but also solves those significantly faster. A portfolio with one of the best solvers of MCC 2020 and GPUSAT3 solves 19% more instances than the former alone in less than half of the runtime
A Time Leap Challenge for SAT Solving
We compare the impact of hardware advancement and algorithm advancement for
SAT solving over the last two decades. In particular, we compare 20-year-old
SAT-solvers on new computer hardware with modern SAT-solvers on 20-year-old
hardware. Our findings show that the progress on the algorithmic side has at
least as much impact as the progress on the hardware side.Comment: Authors' version of a paper which is to appear in the proceedings of
CP'202