476 research outputs found
Message passing for quantified Boolean formulas
We introduce two types of message passing algorithms for quantified Boolean
formulas (QBF). The first type is a message passing based heuristics that can
prove unsatisfiability of the QBF by assigning the universal variables in such
a way that the remaining formula is unsatisfiable. In the second type, we use
message passing to guide branching heuristics of a Davis-Putnam
Logemann-Loveland (DPLL) complete solver. Numerical experiments show that on
random QBFs our branching heuristics gives robust exponential efficiency gain
with respect to the state-of-art solvers. We also manage to solve some
previously unsolved benchmarks from the QBFLIB library. Apart from this our
study sheds light on using message passing in small systems and as subroutines
in complete solvers.Comment: 14 pages, 7 figure
Analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms
The computational complexity of solving random 3-Satisfiability (3-SAT)
problems is investigated. 3-SAT is a representative example of hard
computational tasks; it consists in knowing whether a set of alpha N randomly
drawn logical constraints involving N Boolean variables can be satisfied
altogether or not. Widely used solving procedures, as the
Davis-Putnam-Loveland-Logeman (DPLL) algorithm, perform a systematic search for
a solution, through a sequence of trials and errors represented by a search
tree. In the present study, we identify, using theory and numerical
experiments, easy (size of the search tree scaling polynomially with N) and
hard (exponential scaling) regimes as a function of the ratio alpha of
constraints per variable. The typical complexity is explicitly calculated in
the different regimes, in very good agreement with numerical simulations. Our
theoretical approach is based on the analysis of the growth of the branches in
the search tree under the operation of DPLL. On each branch, the initial 3-SAT
problem is dynamically turned into a more generic 2+p-SAT problem, where p and
1-p are the fractions of constraints involving three and two variables
respectively. The growth of each branch is monitored by the dynamical evolution
of alpha and p and is represented by a trajectory in the static phase diagram
of the random 2+p-SAT problem. Depending on whether or not the trajectories
cross the boundary between phases, single branches or full trees are generated
by DPLL, resulting in easy or hard resolutions.Comment: 37 RevTeX pages, 15 figures; submitted to Phys.Rev.
What makes a phase transition? Analysis of the random satisfiability problem
In the last 30 years it was found that many combinatorial systems undergo
phase transitions. One of the most important examples of these can be found
among the random k-satisfiability problems (often referred to as k-SAT), asking
whether there exists an assignment of Boolean values satisfying a Boolean
formula composed of clauses with k random variables each. The random 3-SAT
problem is reported to show various phase transitions at different critical
values of the ratio of the number of clauses to the number of variables. The
most famous of these occurs when the probability of finding a satisfiable
instance suddenly drops from 1 to 0. This transition is associated with a rise
in the hardness of the problem, but until now the correlation between any of
the proposed phase transitions and the hardness is not totally clear. In this
paper we will first show numerically that the number of solutions universally
follows a lognormal distribution, thereby explaining the puzzling question of
why the number of solutions is still exponential at the critical point.
Moreover we provide evidence that the hardness of the closely related problem
of counting the total number of solutions does not show any phase
transition-like behavior. This raises the question of whether the probability
of finding a satisfiable instance is really an order parameter of a phase
transition or whether it is more likely to just show a simple sharp threshold
phenomenon. More generally, this paper aims at starting a discussion where a
simple sharp threshold phenomenon turns into a genuine phase transition
SAT Competition 2020
The SAT Competitions constitute a well-established series of yearly open international algorithm implementation competitions, focusing on the Boolean satisfiability (or propositional satisfiability, SAT) problem. In this article, we provide a detailed account on the 2020 instantiation of the SAT Competition, including the new competition tracks and benchmark selection procedures, overview of solving strategies implemented in top-performing solvers, and a detailed analysis of the empirical data obtained from running the competition. (C) 2021 The Authors. Published by Elsevier B.V.Peer reviewe
SAT Competition 2020
The SAT Competitions constitute a well-established series of yearly open international algorithm implementation competitions, focusing on the Boolean satisfiability (or propositional satisfiability, SAT) problem. In this article, we provide a detailed account on the 2020 instantiation of the SAT Competition, including the new competition tracks and benchmark selection procedures, overview of solving strategies implemented in top-performing solvers, and a detailed analysis of the empirical data obtained from running the competition
Community structure in industrial SAT instances
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them.
In this paper, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.Peer ReviewedPostprint (published version
Structure and Problem Hardness: Goal Asymmetry and DPLL Proofs in<br> SAT-Based Planning
In Verification and in (optimal) AI Planning, a successful method is to
formulate the application as boolean satisfiability (SAT), and solve it with
state-of-the-art DPLL-based procedures. There is a lack of understanding of why
this works so well. Focussing on the Planning context, we identify a form of
problem structure concerned with the symmetrical or asymmetrical nature of the
cost of achieving the individual planning goals. We quantify this sort of
structure with a simple numeric parameter called AsymRatio, ranging between 0
and 1. We run experiments in 10 benchmark domains from the International
Planning Competitions since 2000; we show that AsymRatio is a good indicator of
SAT solver performance in 8 of these domains. We then examine carefully crafted
synthetic planning domains that allow control of the amount of structure, and
that are clean enough for a rigorous analysis of the combinatorial search
space. The domains are parameterized by size, and by the amount of structure.
The CNFs we examine are unsatisfiable, encoding one planning step less than the
length of the optimal plan. We prove upper and lower bounds on the size of the
best possible DPLL refutations, under different settings of the amount of
structure, as a function of size. We also identify the best possible sets of
branching variables (backdoors). With minimum AsymRatio, we prove exponential
lower bounds, and identify minimal backdoors of size linear in the number of
variables. With maximum AsymRatio, we identify logarithmic DPLL refutations
(and backdoors), showing a doubly exponential gap between the two structural
extreme cases. The reasons for this behavior -- the proof arguments --
illuminate the prototypical patterns of structure causing the empirical
behavior observed in the competition benchmarks
- …