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Generating certified code from formal proofs: a case study in homological algebra

Jesús Aransay, Clemens Ballarin, Julio Rubio
2009 Formal Aspects of Computing  
We apply current theorem proving technology to certified code in the domain of abstract algebra.  ...  In the formal proof, algebraic structures occurring in the Basic Perturbation Lemma are represented in a way, which is not directly amenable to code generation with the available tools.  ...  Acknowledgments We would like to thank the anonymous referees for their suggestions, which have lead to substantial improvement of the original manuscript. The first author thanks also Prof.  ... 
doi:10.1007/s00165-009-0120-0 fatcat:ehqm47d3zfc65hwr62cwcza3fi

A new algorithm for computing certified numerical approximations of the roots of a zero-dimensional system

Stef Graillat, Philippe Trébuchet
2009 Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09  
The paper will only focus on the second step. More precisely, the purpose of this paper is to compute a certified numerical approximation of the solutions.  ...  As a symbolic representation, its computation involves computations with big coefficients, and furthermore, it involves the computation of the full multiplication table in the quotient algebra, which is  ...  The algebraic polynomial system solving process mainly relies on two time-consuming steps : • The first one is to compute a representation of the quotient algebra A = K[x]/(f1, . . . , fs). • The second  ... 
doi:10.1145/1576702.1576727 dblp:conf/issac/GraillatT09 fatcat:5oyqjzxcbnfm7a7euaadmtlkea

On the Complexity of Computing the Topology of Real Algebraic Space Curves [article]

Kai Jin, Jin-San Cheng
2019 arXiv   pre-print
We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm.  ...  In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve.  ...  Acknowledgement The work is partially supported by NKBRPC (2011CB302400), NSFC Grants (11001258, 60821002), SRF for ROCS, SEM.  ... 
arXiv:1901.10317v1 fatcat:rfmkowmcy5gwvaiojky2wo2gja

Numerical Algebraic Geometry for Macaulay2 [article]

Anton Leykin
2011 arXiv   pre-print
It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic geometry.  ...  Numerical Algebraic Geometry uses numerical data to describe algebraic varieties.  ...  The author would also like to thank the referees and editors for their careful remarks and suggestions. This work is partially supported by NSF grant DMS-0914802.  ... 
arXiv:0911.1783v2 fatcat:zfgu7z5uafgvpasz6sy7nt37d4

Towards the Certification of Covert Channel Freeness in Cloud-based Systems

Jason Jaskolka, Ridha Khedri
2015 Procedia Computer Science  
certify it to be free from covert channels on the basis of the non-existence of the potential for communication amongst its agents.  ...  We also discuss how the proposed schema can be used for identifying ways in which an analyst may amend, modify, or redesign a system in order to make it more resilient to covert channels, and to potentially  ...  Acknowledgements This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the grant RGPIN 2014-06115.  ... 
doi:10.1016/j.procs.2015.05.100 fatcat:rd4erjyusne4rgt3tiyt6erway

Certifying zeros of polynomial systems using interval arithmetic [article]

Paul Breiding, Kemal Rose, Sascha Timme
2021 arXiv   pre-print
Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated solution to a square system of polynomial equations.  ...  The implementation rests on Krawczyk's method. We demonstrate that it dramatically outperforms earlier approaches to certification.  ...  The field of computational algebraic geometry is often associated with symbolic computations based on Gröbner bases.  ... 
arXiv:2011.05000v3 fatcat:hvj5kk5m6vgzrfulu43s4iynam

A Certified Module to Study Digital Images with the Kenzo System [chapter]

Jónathan Heras, Vico Pascual, Julio Rubio
2012 Lecture Notes in Computer Science  
Kenzo is a Computer Algebra system devoted to Algebraic Topology, written in the Common Lisp programming language.  ...  In this paper, programs which allow us to analyze monochromatic digital images with the Kenzo system are presented. Besides a complete automated proof of the correctness of our programs is provided.  ...  Introduction In the field of Intelligent Information Processing, mechanized reasoning systems provide a chance of increasing the reliability of software systems, namely Computer Algebra systems.  ... 
doi:10.1007/978-3-642-27549-4_15 fatcat:ynkiskuvrvapne5ogt72torv2e

Mechanical Verification of Interactive Programs Specified by Use Cases

Guillaume Claret, Yann Regis-Gianas
2015 2015 IEEE/ACM 3rd FME Workshop on Formal Methods in Software Engineering  
To this end we introduce the notion of scenarios, well-typed schema of interactions between an environment and a program. We design and certify a blog system as an illustration.  ...  Some ideas coming from functional programming languages have been successful to improve the way we write safer programs, compared to traditional imperative languages, but these ideas mostly apply to code  ...  This debugger relies on the existing tactics mode [6] of Coq, using this mode to explore the execution paths of an interactive computation; • a blog system implemented and certified using our method.  ... 
doi:10.1109/formalise.2015.17 dblp:conf/icse/ClaretR15 fatcat:2hots6ulgjcvzi3luddoxhesry

Proof Assistant Decision Procedures for Formalizing Origami [chapter]

Cezary Kaliszyk, Tetsuo Ida
2011 Lecture Notes in Computer Science  
Such properties have been shown with the help of computer algebra systems.  ...  Proofs performed with computer algebra systems can be accompanied by proof documents, still they lack complete mathematical rigorousity, like the one provided by proof assistant checked proofs.  ...  We have built a prototype computer algebra like input-response-loop inside HOL Light [13] , with the user interface designed close to the interfaces of popular computer algebra systems.  ... 
doi:10.1007/978-3-642-22673-1_4 fatcat:n3dfqj73afgphagwrc3umn4yhi

Exact Solutions in Log-Concave Maximum Likelihood Estimation [article]

Alexandros Grosdos, Alexander Heaton, Kaie Kubjas, Olga Kuznetsova, Georgy Scholten, Miruna-Stefana Sorea
2020 arXiv   pre-print
Even in the case of two equations, very little is known about solutions to these systems.  ...  As an alternative, we use Smale's alpha-theory to refine approximate numerical solutions and to certify solutions to log-concave density estimation.  ...  This project started at the Summer School on Geometric and Algebraic Combinatorics at Sorbonne University in June 2019.  ... 
arXiv:2003.04840v1 fatcat:ffnd7eq5jvd6thlsus4x4memq4

Automated Certified Proofs with CiME3

Évelyne Contejean, Pierre Courtieu, Julien Forest, Olivier Pons, Xavier Urbain, Marc Herbstritt
2011 International Conference on Rewriting Techniques and Applications  
Thus, these features open the way for using CiME3 to add automation to proofs of termination or confluence in a formal development in the Coq proof assistant.  ...  in certification problem format in order to certify them with a sceptical proof assistant like Coq.  ...  Acknowledgements The authors would like to thank the anonymous referees for their fruitful comments and their help in improving the presentation of this article.  ... 
doi:10.4230/lipics.rta.2011.21 dblp:conf/rta/ContejeanCFPU11 fatcat:ecma23vs75egpnhbc7ntfnfdli

Page 2064 of Mathematical Reviews Vol. 58, Issue 3 [page]

1979 Mathematical Reviews  
A system of linear algebraic equations of the form x=Ax+f is given, where A is a positive matrix of norm less than 1. The system is solved by a modification of the Ritz method.  ...  Other sections of the guide discuss the validation procedures used for testing EISPACK, report execution times of the EISPACK sub- routines on several machines, advertise the certified status and 58 #  ... 

Calcium: computing in exact real and complex fields [article]

Fredrik Johansson
2020 arXiv   pre-print
The system combines efficient field operations with automatic discovery and certification of algebraic relations, resulting in a practical computational model of ℝ and ℂ in which equality is rigorously  ...  ,a_n) where the extensions numbers a_k may be algebraic or transcendental.  ...  Symbolic and algebraic systems Most computer algebra systems arguably belong to one of two paradigms.  ... 
arXiv:2011.01728v1 fatcat:exo2wsaqgba4xencqfv6c5hmry

A3PAT, an approach for certified automated termination proofs

Évelyne Contejean, Andrei Paskevich, Xavier Urbain, Pierre Courtieu, Olivier Pons, Julien Forest
2010 Proceedings of the ACM SIGPLAN 2010 workshop on Partial evaluation and program manipulation - PEPM '10  
This paper presents the approach developed in Project A3PAT to discover and moreover certify, with full automation, termination proofs for term rewriting systems.  ...  Software engineering, automated reasoning, rule-based programming or specifications often use rewriting systems for which termination, among other properties, may have to be ensured.  ...  We plan on adding automation to proof assistants on these topics in the near future.  ... 
doi:10.1145/1706356.1706370 dblp:conf/pepm/ContejeanPUCPF10 fatcat:3ugihraiezd3ppcchabilbmiri

The Reliable Algorithmic Software Challenge RASC Dedicated to Thomas Ottmann on the Occasion of His 60th Birthday [chapter]

Kurt Mehlhorn
2003 Lecture Notes in Computer Science  
On input A and b the program returns either ¡ "the system is solvable" and a vector x such that A The certifying program is easy to check.  ...  It aims to exploit the fact that computations with algebraic numbers can be performed exactly.  ... 
doi:10.1007/3-540-36477-3_19 fatcat:ojxhcvqsrbhihkpab6535oghaq
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