3,186 research outputs found
Shingle 2.0: generalising self-consistent and automated domain discretisation for multi-scale geophysical models
The approaches taken to describe and develop spatial discretisations of the
domains required for geophysical simulation models are commonly ad hoc, model
or application specific and under-documented. This is particularly acute for
simulation models that are flexible in their use of multi-scale, anisotropic,
fully unstructured meshes where a relatively large number of heterogeneous
parameters are required to constrain their full description. As a consequence,
it can be difficult to reproduce simulations, ensure a provenance in model data
handling and initialisation, and a challenge to conduct model intercomparisons
rigorously. This paper takes a novel approach to spatial discretisation,
considering it much like a numerical simulation model problem of its own. It
introduces a generalised, extensible, self-documenting approach to carefully
describe, and necessarily fully, the constraints over the heterogeneous
parameter space that determine how a domain is spatially discretised. This
additionally provides a method to accurately record these constraints, using
high-level natural language based abstractions, that enables full accounts of
provenance, sharing and distribution. Together with this description, a
generalised consistent approach to unstructured mesh generation for geophysical
models is developed, that is automated, robust and repeatable, quick-to-draft,
rigorously verified and consistent to the source data throughout. This
interprets the description above to execute a self-consistent spatial
discretisation process, which is automatically validated to expected discrete
characteristics and metrics.Comment: 18 pages, 10 figures, 1 table. Submitted for publication and under
revie
The complexity of conservative finite-valued CSPs
We study the complexity of valued constraint satisfaction problems (VCSP). A
problem from VCSP is characterised by a \emph{constraint language}, a fixed set
of cost functions over a finite domain. An instance of the problem is specified
by a sum of cost functions from the language and the goal is to minimise the
sum. We consider the case of so-called \emph{conservative} languages; that is,
languages containing all unary cost functions, thus allowing arbitrary
restrictions on the domains of the variables. This problem has been studied by
Bulatov [LICS'03] for -valued languages (i.e. CSP), by
Cohen~\etal\ (AIJ'06) for Boolean domains, by Deineko et al. (JACM'08) for
-valued cost functions (i.e. Max-CSP), and by Takhanov (STACS'10) for
-valued languages containing all finite-valued unary cost
functions (i.e. Min-Cost-Hom).
We give an elementary proof of a complete complexity classification of
conservative finite-valued languages: we show that every conservative
finite-valued language is either tractable or NP-hard. This is the \emph{first}
dichotomy result for finite-valued VCSPs over non-Boolean domains.Comment: 15 page
The complexity of approximating conservative counting CSPs
We study the complexity of approximately solving the weighted counting
constraint satisfaction problem #CSP(F). In the conservative case, where F
contains all unary functions, there is a classification known for the case in
which the domain of functions in F is Boolean. In this paper, we give a
classification for the more general problem where functions in F have an
arbitrary finite domain. We define the notions of weak log-modularity and weak
log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in
FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem
of counting independent sets in bipartite graphs. #BIS is complete with respect
to approximation-preserving reductions for a logically-defined complexity class
#RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard
case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as
a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is
NP-hard to approximate. Finally, we give a full trichotomy for the arity-2
case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in
difficulty to #SAT, the problem of approximately counting the satisfying
assignments of a Boolean formula in conjunctive normal form. We also discuss
the algorithmic aspects of our classification.Comment: Minor revisio
Hybrid tractability of soft constraint problems
The constraint satisfaction problem (CSP) is a central generic problem in
computer science and artificial intelligence: it provides a common framework
for many theoretical problems as well as for many real-life applications. Soft
constraint problems are a generalisation of the CSP which allow the user to
model optimisation problems. Considerable effort has been made in identifying
properties which ensure tractability in such problems. In this work, we
initiate the study of hybrid tractability of soft constraint problems; that is,
properties which guarantee tractability of the given soft constraint problem,
but which do not depend only on the underlying structure of the instance (such
as being tree-structured) or only on the types of soft constraints in the
instance (such as submodularity). We present several novel hybrid classes of
soft constraint problems, which include a machine scheduling problem,
constraint problems of arbitrary arities with no overlapping nogoods, and the
SoftAllDiff constraint with arbitrary unary soft constraints. An important tool
in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page
Transfer Function Synthesis without Quantifier Elimination
Traditionally, transfer functions have been designed manually for each
operation in a program, instruction by instruction. In such a setting, a
transfer function describes the semantics of a single instruction, detailing
how a given abstract input state is mapped to an abstract output state. The net
effect of a sequence of instructions, a basic block, can then be calculated by
composing the transfer functions of the constituent instructions. However,
precision can be improved by applying a single transfer function that captures
the semantics of the block as a whole. Since blocks are program-dependent, this
approach necessitates automation. There has thus been growing interest in
computing transfer functions automatically, most notably using techniques based
on quantifier elimination. Although conceptually elegant, quantifier
elimination inevitably induces a computational bottleneck, which limits the
applicability of these methods to small blocks. This paper contributes a method
for calculating transfer functions that finesses quantifier elimination
altogether, and can thus be seen as a response to this problem. The
practicality of the method is demonstrated by generating transfer functions for
input and output states that are described by linear template constraints,
which include intervals and octagons.Comment: 37 pages, extended version of ESOP 2011 pape
The power of Sherali-Adams relaxations for general-valued CSPs
We give a precise algebraic characterisation of the power of Sherali-Adams
relaxations for solvability of valued constraint satisfaction problems to
optimality. The condition is that of bounded width which has already been shown
to capture the power of local consistency methods for decision CSPs and the
power of semidefinite programming for robust approximation of CSPs.
Our characterisation has several algorithmic and complexity consequences. On
the algorithmic side, we show that several novel and many known valued
constraint languages are tractable via the third level of the Sherali-Adams
relaxation. For the known languages, this is a significantly simpler algorithm
than the previously obtained ones. On the complexity side, we obtain a
dichotomy theorem for valued constraint languages that can express an injective
unary function. This implies a simple proof of the dichotomy theorem for
conservative valued constraint languages established by Kolmogorov and Zivny
[JACM'13], and also a dichotomy theorem for the exact solvability of
Minimum-Solution problems. These are generalisations of Minimum-Ones problems
to arbitrary finite domains. Our result improves on several previous
classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and
Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301
The Power of Linear Programming for Valued CSPs
A class of valued constraint satisfaction problems (VCSPs) is characterised
by a valued constraint language, a fixed set of cost functions on a finite
domain. An instance of the problem is specified by a sum of cost functions from
the language with the goal to minimise the sum. This framework includes and
generalises well-studied constraint satisfaction problems (CSPs) and maximum
constraint satisfaction problems (Max-CSPs).
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation. Using this result, we obtain tractability of several novel and
previously widely-open classes of VCSPs, including problems over valued
constraint languages that are: (1) submodular on arbitrary lattices; (2)
bisubmodular (also known as k-submodular) on arbitrary finite domains; (3)
weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo
Using Program Synthesis for Program Analysis
In this paper, we identify a fragment of second-order logic with restricted
quantification that is expressive enough to capture numerous static analysis
problems (e.g. safety proving, bug finding, termination and non-termination
proving, superoptimisation). We call this fragment the {\it synthesis
fragment}. Satisfiability of a formula in the synthesis fragment is decidable
over finite domains; specifically the decision problem is NEXPTIME-complete. If
a formula in this fragment is satisfiable, a solution consists of a satisfying
assignment from the second order variables to \emph{functions over finite
domains}. To concretely find these solutions, we synthesise \emph{programs}
that compute the functions. Our program synthesis algorithm is complete for
finite state programs, i.e. every \emph{function} over finite domains is
computed by some \emph{program} that we can synthesise. We can therefore use
our synthesiser as a decision procedure for the synthesis fragment of
second-order logic, which in turn allows us to use it as a powerful backend for
many program analysis tasks. To show the tractability of our approach, we
evaluate the program synthesiser on several static analysis problems.Comment: 19 pages, to appear in LPAR 2015. arXiv admin note: text overlap with
arXiv:1409.492
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