2,716 research outputs found
Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
We present a formal tool for verification of multivariate nonlinear
inequalities. Our verification method is based on interval arithmetic with
Taylor approximations. Our tool is implemented in the HOL Light proof assistant
and it is capable to verify multivariate nonlinear polynomial and
non-polynomial inequalities on rectangular domains. One of the main features of
our work is an efficient implementation of the verification procedure which can
prove non-trivial high-dimensional inequalities in several seconds. We
developed the verification tool as a part of the Flyspeck project (a formal
proof of the Kepler conjecture). The Flyspeck project includes about 1000
nonlinear inequalities. We successfully tested our method on more than 100
Flyspeck inequalities and estimated that the formal verification procedure is
about 3000 times slower than an informal verification method implemented in
C++. We also describe future work and prospective optimizations for our method.Comment: 15 page
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Certified Roundoff Error Bounds Using Semidefinite Programming.
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementation. This problem becomes challenging when the program does not employ solely linear operations as non-linearities are inherent to many interesting computational problems in real-world applications. Existing solutions to reasoning are limited in the presence of nonlinear correlations between variables, leading to either imprecise bounds or high analysis time. Furthermore, while it is easy to implement a straightforward method such as interval arithmetic, sophisticated techniques are less straightforward to implement in a formal setting. Thus there is a need for methods which output certificates that can be formally validated inside a proof assistant. We present a framework to provide upper bounds on absolute roundoff errors. This framework is based on optimization techniques employing semidefinite programming and sums of squares certificates, which can be formally checked inside the Coq theorem prover. Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization and space control. Our tool produces more precise error bounds for 37 percent of all programs and yields better performance in 73 percent of all programs
Certification of Bounds of Non-linear Functions: the Templates Method
The aim of this work is to certify lower bounds for real-valued multivariate
functions, defined by semialgebraic or transcendental expressions. The
certificate must be, eventually, formally provable in a proof system such as
Coq. The application range for such a tool is widespread; for instance Hales'
proof of Kepler's conjecture yields thousands of inequalities. We introduce an
approximation algorithm, which combines ideas of the max-plus basis method (in
optimal control) and of the linear templates method developed by Manna et al.
(in static analysis). This algorithm consists in bounding some of the
constituents of the function by suprema of quadratic forms with a well chosen
curvature. This leads to semialgebraic optimization problems, solved by
sum-of-squares relaxations. Templates limit the blow up of these relaxations at
the price of coarsening the approximation. We illustrate the efficiency of our
framework with various examples from the literature and discuss the interfacing
with Coq.Comment: 16 pages, 3 figures, 2 table
Algorithmic Verification of Continuous and Hybrid Systems
We provide a tutorial introduction to reachability computation, a class of
computational techniques that exports verification technology toward continuous
and hybrid systems. For open under-determined systems, this technique can
sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661
Certification of Real Inequalities -- Templates and Sums of Squares
We consider the problem of certifying lower bounds for real-valued
multivariate transcendental functions. The functions we are dealing with are
nonlinear and involve semialgebraic operations as well as some transcendental
functions like , , , etc. Our general framework is to use
different approximation methods to relax the original problem into polynomial
optimization problems, which we solve by sparse sums of squares relaxations. In
particular, we combine the ideas of the maxplus estimators (originally
introduced in optimal control) and of the linear templates (originally
introduced in static analysis by abstract interpretation). The nonlinear
templates control the complexity of the semialgebraic relaxations at the price
of coarsening the maxplus approximations. In that way, we arrive at a new -
template based - certified global optimization method, which exploits both the
precision of sums of squares relaxations and the scalability of abstraction
methods. We analyze the performance of the method on problems from the global
optimization literature, as well as medium-size inequalities issued from the
Flyspeck project.Comment: 27 pages, 3 figures, 4 table
Satisfiability Modulo Transcendental Functions via Incremental Linearization
In this paper we present an abstraction-refinement approach to Satisfiability
Modulo the theory of transcendental functions, such as exponentiation and
trigonometric functions. The transcendental functions are represented as
uninterpreted in the abstract space, which is described in terms of the
combined theory of linear arithmetic on the rationals with uninterpreted
functions, and are incrementally axiomatized by means of upper- and
lower-bounding piecewise-linear functions. Suitable numerical techniques are
used to ensure that the abstractions of the transcendental functions are sound
even in presence of irrationals. Our experimental evaluation on benchmarks from
verification and mathematics demonstrates the potential of our approach,
showing that it compares favorably with delta-satisfiability /interval
propagation and methods based on theorem proving
A formal proof of the Kepler conjecture
This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project
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