2,714 research outputs found
Splitting Proofs for Interpolation
We study interpolant extraction from local first-order refutations. We
present a new theoretical perspective on interpolation based on clearly
separating the condition on logical strength of the formula from the
requirement on the com- mon signature. This allows us to highlight the space of
all interpolants that can be extracted from a refutation as a space of simple
choices on how to split the refuta- tion into two parts. We use this new
insight to develop an algorithm for extracting interpolants which are linear in
the size of the input refutation and can be further optimized using metrics
such as number of non-logical symbols or quantifiers. We implemented the new
algorithm in first-order theorem prover VAMPIRE and evaluated it on a large
number of examples coming from the first-order proving community. Our
experiments give practical evidence that our work improves the state-of-the-art
in first-order interpolation.Comment: 26th Conference on Automated Deduction, 201
The BEM with graded meshes for the electric field integral equation on polyhedral surfaces
We consider the variational formulation of the electric field integral
equation on a Lipschitz polyhedral surface . We study the Galerkin
boundary element discretisations based on the lowest-order Raviart-Thomas
surface elements on a sequence of anisotropic meshes algebraically graded
towards the edges of . We establish quasi-optimal convergence of
Galerkin solutions under a mild restriction on the strength of grading. The key
ingredient of our convergence analysis are new componentwise stability
properties of the Raviart-Thomas interpolant on anisotropic elements
Interpolation Properties and SAT-based Model Checking
Craig interpolation is a widespread method in verification, with important
applications such as Predicate Abstraction, CounterExample Guided Abstraction
Refinement and Lazy Abstraction With Interpolants. Most state-of-the-art model
checking techniques based on interpolation require collections of interpolants
to satisfy particular properties, to which we refer as "collectives"; they do
not hold in general for all interpolation systems and have to be established
for each particular system and verification environment. Nevertheless, no
systematic approach exists that correlates the individual interpolation systems
and compares the necessary collectives. This paper proposes a uniform
framework, which encompasses (and generalizes) the most common collectives
exploited in verification. We use it for a systematic study of the collectives
and of the constraints they pose on propositional interpolation systems used in
SAT-based model checking
Domain-Type-Guided Refinement Selection Based on Sliced Path Prefixes
Abstraction is a successful technique in software verification, and
interpolation on infeasible error paths is a successful approach to
automatically detect the right level of abstraction in counterexample-guided
abstraction refinement. Because the interpolants have a significant influence
on the quality of the abstraction, and thus, the effectiveness of the
verification, an algorithm for deriving the best possible interpolants is
desirable. We present an analysis-independent technique that makes it possible
to extract several alternative sequences of interpolants from one given
infeasible error path, if there are several reasons for infeasibility in the
error path. We take as input the given infeasible error path and apply a
slicing technique to obtain a set of error paths that are more abstract than
the original error path but still infeasible, each for a different reason. The
(more abstract) constraints of the new paths can be passed to a standard
interpolation engine, in order to obtain a set of interpolant sequences, one
for each new path. The analysis can then choose from this set of interpolant
sequences and select the most appropriate, instead of being bound to the single
interpolant sequence that the interpolation engine would normally return. For
example, we can select based on domain types of variables in the interpolants,
prefer to avoid loop counters, or compare with templates for potential loop
invariants, and thus control what kind of information occurs in the abstraction
of the program. We implemented the new algorithm in the open-source
verification framework CPAchecker and show that our proof-technique-independent
approach yields a significant improvement of the effectiveness and efficiency
of the verification process.Comment: 10 pages, 5 figures, 1 table, 4 algorithm
Bayesian interpolation
Although Bayesian analysis has been in use since Laplace, the Bayesian method of model-comparison has only recently been developed in depth. In this paper, the Bayesian approach to regularization and model-comparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other data modeling problems. Regularizing constants are set by examining their posterior probability distribution. Alternative regularizers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. “Occam's razor” is automatically embodied by this process. The way in which Bayes infers the values of regularizing constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling
Direct Numerical Simulation of decaying two-dimensional turbulence in a no-slip square box using Smoothed Particle Hydrodynamics
This paper explores the application of SPH to a Direct Numerical Simulation
(DNS) of decaying turbulence in a two-dimensional no-slip wall-bounded domain.
In this bounded domain, the inverse energy cascade, and a net torque exerted by
the boundary, result in a spontaneous spin up of the fluid, leading to a
typical end state of a large monopole vortex that fills the domain. The SPH
simulations were compared against published results using a high accuracy
pseudo-spectral code. Ensemble averages of the kinetic energy, enstrophy and
average vortex wavenumber compared well against the pseudo-spectral results, as
did the evolution of the total angular momentum of the fluid. However, while
the pseudo-spectral results emphasised the importance of the no-slip boundaries
as generators of long lived coherent vortices in the flow, no such generation
was seen in the SPH results. Vorticity filaments produced at the boundary were
always dissipated by the flow shortly after separating from the boundary layer.
The kinetic energy spectrum of the SPH results was calculated using a SPH
Fourier transform that operates directly on the disordered particles. The
ensemble kinetic energy spectrum showed the expected k-3 scaling over most of
the inertial range. However, the spectrum flattened at smaller length scales
(initially less than 7.5 particle spacings and growing in size over time),
indicating an excess of small-scale kinetic energy
Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi
We have recently presented a general method of proving the fundamental
logical properties of Craig and Lyndon Interpolation (IPs) by induction on
derivations in a wide class of internal sequent calculi, including sequents,
hypersequents, and nested sequents. Here we adapt the method to a more general
external formalism of labelled sequents and provide sufficient criteria on the
Kripke-frame characterization of a logic that guarantee the IPs. In particular,
we show that classes of frames definable by quantifier-free Horn formulas
correspond to logics with the IPs. These criteria capture the modal cube and
the infinite family of transitive Geach logics
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
It is often difficult to correctly implement a Boolean controller for a
complex system, especially when concurrency is involved. Yet, it may be easy to
formally specify a controller. For instance, for a pipelined processor it
suffices to state that the visible behavior of the pipelined system should be
identical to a non-pipelined reference system (Burch-Dill paradigm). We present
a novel procedure to efficiently synthesize multiple Boolean control signals
from a specification given as a quantified first-order formula (with a specific
quantifier structure). Our approach uses uninterpreted functions to abstract
details of the design. We construct an unsatisfiable SMT formula from the given
specification. Then, from just one proof of unsatisfiability, we use a variant
of Craig interpolation to compute multiple coordinated interpolants that
implement the Boolean control signals. Our method avoids iterative learning and
back-substitution of the control functions. We applied our approach to
synthesize a controller for a simple two-stage pipelined processor, and present
first experimental results.Comment: This paper originally appeared in FMCAD 2013,
http://www.cs.utexas.edu/users/hunt/FMCAD/FMCAD13/index.shtml. This version
includes an appendix that is missing in the conference versio
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