18,192 research outputs found
Proof Automation in the Theory of Finite Sets and Finite Set Relation Algebra
{log} ('setlog') is a satisfiability solver for formulas of the theory of
finite sets and finite set relation algebra (FSTRA). As such, it can be used as
an automated theorem prover (ATP) for this theory. {log} is able to
automatically prove a number of FSTRA theorems, but not all of them.
Nevertheless, we have observed that many theorems that {log} cannot
automatically prove can be divided into a few subgoals automatically
dischargeable by {log}. The purpose of this work is to present a prototype
interactive theorem prover (ITP), called {log}-ITP, providing evidence that a
proper integration of {log} into world-class ITP's can deliver a great deal of
proof automation concerning FSTRA. An empirical evaluation based on 210
theorems from the TPTP and Coq's SSReflect libraries shows a noticeable
reduction in the size and complexity of the proofs with respect to Coq
Propagators and Solvers for the Algebra of Modular Systems
To appear in the proceedings of LPAR 21.
Solving complex problems can involve non-trivial combinations of distinct
knowledge bases and problem solvers. The Algebra of Modular Systems is a
knowledge representation framework that provides a method for formally
specifying such systems in purely semantic terms. Formally, an expression of
the algebra defines a class of structures. Many expressive formalism used in
practice solve the model expansion task, where a structure is given on the
input and an expansion of this structure in the defined class of structures is
searched (this practice overcomes the common undecidability problem for
expressive logics). In this paper, we construct a solver for the model
expansion task for a complex modular systems from an expression in the algebra
and black-box propagators or solvers for the primitive modules. To this end, we
define a general notion of propagators equipped with an explanation mechanism,
an extension of the alge- bra to propagators, and a lazy conflict-driven
learning algorithm. The result is a framework for seamlessly combining solving
technology from different domains to produce a solver for a combined system.Comment: To appear in the proceedings of LPAR 2
Adapting Real Quantifier Elimination Methods for Conflict Set Computation
The satisfiability problem in real closed fields is decidable. In the context
of satisfiability modulo theories, the problem restricted to conjunctive sets
of literals, that is, sets of polynomial constraints, is of particular
importance. One of the central problems is the computation of good explanations
of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the
input constraints whose conjunction is already unsatisfiable. We adapt two
commonly used real quantifier elimination methods, cylindrical algebraic
decomposition and virtual substitution, to provide such conflict sets and
demonstrate the performance of our method in practice
Multiparameter spectral analysis for aeroelastic instability problems
This paper presents a novel application of multiparameter spectral theory to
the study of structural stability, with particular emphasis on aeroelastic
flutter. Methods of multiparameter analysis allow the development of new
solution algorithms for aeroelastic flutter problems; most significantly, a
direct solver for polynomial problems of arbitrary order and size, something
which has not before been achieved. Two major variants of this direct solver
are presented, and their computational characteristics are compared. Both are
effective for smaller problems arising in reduced-order modelling and
preliminary design optimization. Extensions and improvements to this new
conceptual framework and solution method are then discussed.Comment: 20 pages, 8 figure
Resolution of Linear Algebra for the Discrete Logarithm Problem Using GPU and Multi-core Architectures
In cryptanalysis, solving the discrete logarithm problem (DLP) is key to
assessing the security of many public-key cryptosystems. The index-calculus
methods, that attack the DLP in multiplicative subgroups of finite fields,
require solving large sparse systems of linear equations modulo large primes.
This article deals with how we can run this computation on GPU- and
multi-core-based clusters, featuring InfiniBand networking. More specifically,
we present the sparse linear algebra algorithms that are proposed in the
literature, in particular the block Wiedemann algorithm. We discuss the
parallelization of the central matrix--vector product operation from both
algorithmic and practical points of view, and illustrate how our approach has
contributed to the recent record-sized DLP computation in GF().Comment: Euro-Par 2014 Parallel Processing, Aug 2014, Porto, Portugal.
\<http://europar2014.dcc.fc.up.pt/\>
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
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