10,349 research outputs found
Guaranteed Proofs Using Interval Arithmetic
International audienceThis paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic. The tools implement two techniques for reducing decorrelation: interval splitting and Taylor's series expansions. Although the tools are designed for the proof assistant system PVS, expertise on PVS is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal human-theorem prover interaction
Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics
In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr-Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr-Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire's problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators
Certifying floating-point implementations using Gappa
High confidence in floating-point programs requires proving numerical
properties of final and intermediate values. One may need to guarantee that a
value stays within some range, or that the error relative to some ideal value
is well bounded. Such work may require several lines of proof for each line of
code, and will usually be broken by the smallest change to the code (e.g. for
maintenance or optimization purpose). Certifying these programs by hand is
therefore very tedious and error-prone. This article discusses the use of the
Gappa proof assistant in this context. Gappa has two main advantages over
previous approaches: Its input format is very close to the actual C code to
validate, and it automates error evaluation and propagation using interval
arithmetic. Besides, it can be used to incrementally prove complex mathematical
properties pertaining to the C code. Yet it does not require any specific
knowledge about automatic theorem proving, and thus is accessible to a wide
community. Moreover, Gappa may generate a formal proof of the results that can
be checked independently by a lower-level proof assistant like Coq, hence
providing an even higher confidence in the certification of the numerical code.
The article demonstrates the use of this tool on a real-size example, an
elementary function with correctly rounded output
Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling
An L-space is a rational homology 3-sphere with minimal Heegaard Floer
homology. We give the first examples of hyperbolic L-spaces with no symmetries.
In particular, unlike all previously known L-spaces, these manifolds are not
double branched covers of links in S^3. We prove the existence of infinitely
many such examples (in several distinct families) using a mix of hyperbolic
geometry, Floer theory, and verified computer calculations. Of independent
interest is our technique for using interval arithmetic to certify symmetry
groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the
process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus
3 with two distinct lens space fillings. These are the first examples where
multiple Dehn fillings drop the Heegaard genus by more than one, which answers
a question of Gordon.Comment: 19 pages, 2 figures. v2: minor changes to intro. v3: accepted
version, to appear in Math. Res. Letter
A library of Taylor models for PVS automatic proof checker
We present in this paper a library to compute with Taylor models, a technique
extending interval arithmetic to reduce decorrelation and to solve differential
equations. Numerical software usually produces only numerical results. Our
library can be used to produce both results and proofs. As seen during the
development of Fermat's last theorem reported by Aczel 1996, providing a proof
is not sufficient. Our library provides a proof that has been thoroughly
scrutinized by a trustworthy and tireless assistant. PVS is an automatic proof
assistant that has been fairly developed and used and that has no internal
connection with interval arithmetic or Taylor models. We built our library so
that PVS validates each result as it is produced. As producing and validating a
proof, is and will certainly remain a bigger task than just producing a
numerical result our library will never be a replacement to imperative
implementations of Taylor models such as Cosy Infinity. Our library should
mainly be used to validate small to medium size results that are involved in
safety or life critical applications
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