74 research outputs found
Theorem of three circles in Coq
The theorem of three circles in real algebraic geometry guarantees the
termination and correctness of an algorithm of isolating real roots of a
univariate polynomial. The main idea of its proof is to consider polynomials
whose roots belong to a certain area of the complex plane delimited by straight
lines. After applying a transformation involving inversion this area is mapped
to an area delimited by circles. We provide a formalisation of this rather
geometric proof in Ssreflect, an extension of the proof assistant Coq,
providing versatile algebraic tools. They allow us to formalise the proof from
an algebraic point of view.Comment: 27 pages, 5 figure
05021 Abstracts Collection -- Mathematics, Algorithms, Proofs
From 09.01.05 to 14.01.05, the Dagstuhl Seminar 05021 ``Mathematics, Algorithms, Proofs\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
LinkstFo extended abstracts or full papers are provided, if available
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Towards justifying computer algebra algorithms in Isabelle/HOL
As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results.
This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis.
In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains.
The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework
Affine functions and series with co-inductive real numbers
We extend the work of A. Ciaffaglione and P. Di Gianantonio on mechanical
verification of algorithms for exact computation on real numbers, using
infinite streams of digits implemented as co-inductive types. Four aspects are
studied: the first aspect concerns the proof that digit streams can be related
to the axiomatized real numbers that are already axiomatized in the proof
system (axiomatized, but with no fixed representation). The second aspect
re-visits the definition of an addition function, looking at techniques to let
the proof search mechanism perform the effective construction of an algorithm
that is correct by construction. The third aspect concerns the definition of a
function to compute affine formulas with positive rational coefficients. This
should be understood as a testbed to describe a technique to combine
co-recursion and recursion to obtain a model for an algorithm that appears at
first sight to be outside the expressive power allowed by the proof system. The
fourth aspect concerns the definition of a function to compute series, with an
application on the series that is used to compute Euler's number e. All these
experiments should be reproducible in any proof system that supports
co-inductive types, co-recursion and general forms of terminating recursion,
but we performed with the Coq system [12, 3, 14]
PolyARBerNN: A Neural Network Guided Solver and Optimizer for Bounded Polynomial Inequalities
Constraints solvers play a significant role in the analysis, synthesis, and
formal verification of complex embedded and cyber-physical systems. In this
paper, we study the problem of designing a scalable constraints solver for an
important class of constraints named polynomial constraint inequalities (also
known as non-linear real arithmetic theory). In this paper, we introduce a
solver named PolyARBerNN that uses convex polynomials as abstractions for
highly nonlinear polynomials. Such abstractions were previously shown to be
powerful to prune the search space and restrict the usage of sound and complete
solvers to small search spaces. Compared with the previous efforts on using
convex abstractions, PolyARBerNN provides three main contributions namely (i) a
neural network guided abstraction refinement procedure that helps selecting the
right abstraction out of a set of pre-defined abstractions, (ii) a Bernstein
polynomial-based search space pruning mechanism that can be used to compute
tight estimates of the polynomial maximum and minimum values which can be used
as an additional abstraction of the polynomials, and (iii) an optimizer that
transforms polynomial objective functions into polynomial constraints (on the
gradient of the objective function) whose solutions are guaranteed to be close
to the global optima. These enhancements together allowed the PolyARBerNN
solver to solve complex instances and scales more favorably compared to the
state-of-art non-linear real arithmetic solvers while maintaining the soundness
and completeness of the resulting solver. In particular, our test benches show
that PolyARBerNN achieved 100X speedup compared with Z3 8.9, Yices 2.6, and
NASALib (a solver that uses Bernstein expansion to solve multivariate
polynomial constraints) on a variety of standard test benches
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
Proving formally the implementation of an efficient gcd algorithm for polynomials
Last version published in the proceedings of IJCAR 06, part of FLOC 06.International audienceWe describe here a formal proof in the Coq system of the structure theorem for subresultants, which allows to prove formally the correction of our implementation of the subresultant algorithm. Up to our knowledge, it is the first mechanized proof of this result
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Deciding Univariate Polynomial Problems Using Untrusted Certificates in Isabelle/HOL
We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates, separating solving from verifying: efficient external tools perform expensive real algebraic computations, producing evidence that is formally checked within Isabelle’s logic. This allows us to exploit highly-tuned computer algebra systems like Mathematica to guide our procedure without impacting the correctness of its results. We present experiments demonstrating the efficacy of this approach, in many cases yielding orders of magnitude improvements over previous methods.The first author was funded by the China Scholarship Council, via the CSC Cambridge Scholarship programme. The development of MetiTarski was supported by the Engineering and Physical Sciences Research Council [Grant Numbers EP/I011005/1, EP/I010335/1]
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