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Formalisation of the computation of the echelon form of a matrix in Isabelle/HOL

Authors:

Jesús Aransay,

Jose DivasónAuthors Info & Claims

Formal Aspects of Computing, Volume 28, Issue 6

Pages 1005 - 1026

https://doi.org/10.1007/s00165-016-0383-1

Published: 01 November 2016 Publication History

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Abstract

In this contribution we present a formalised algorithm in the Isabelle/HOL proof assistant to compute echelon forms, and, as a consequence, characteristic polynomials of matrices. We have proved its correctness over Bézout domains, but its executability is only guaranteed over Euclidean domains, such as the integer ring and the univariate polynomials over a field. This is possible since the algorithm has been parameterised by a (possibly non-computable) operation that returns the Bézout coefficients of a pair of elements of a ring. The echelon form is also used to compute determinants and inverses of matrices. As a by-product, some algebraic structures have been implemented (principal ideal domains, Bézout domains, etc.). In order to improve performance, the algorithm has been refined to immutable arrays inside of Isabelle and code can be generated to functional languages as well.

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Cited By

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Divasón JThiemann R(2022)A Formalization of the Smith Normal Form in Higher-Order LogicJournal of Automated Reasoning10.1007/s10817-022-09631-566:4(1065-1095)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s10817-022-09631-5
Shi ZWu AYang XGuan YLi YSong X(2018)Formal analysis of the kinematic Jacobian in screw theoryFormal Aspects of Computing10.1007/s00165-018-0468-030:6(739-757)Online publication date: 1-Nov-2018
https://dl.acm.org/doi/10.1007/s00165-018-0468-0
Lambán LMartín-Mateos FRubio JRuiz-Reina J(2017)Using Abstract Stobjs in ACL2 to Compute Matrix Normal FormsInteractive Theorem Proving10.1007/978-3-319-66107-0_23(354-370)Online publication date: 26-Sep-2017
https://dl.acm.org/doi/10.1007/978-3-319-66107-0_23

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Formalisation of the computation of the echelon form of a matrix in Isabelle/HOL

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Reviews

Reviewer: Jacques Carette

Formalization is rapidly evolving from being highly desirable but thought too difficult to be realistic, to being quite feasible and well on its way to being expected. The raw technology of theorem provers certainly is ready. What is still evolving are the best techniques to apply, and the libraries of "basic" material to support more advanced applications. Having said that, the task is still not just pure development. Long-term reusability of formal libraries requires sufficient generality, which does not always coincide with easy executability. A careful path of compromise is needed-which is exactly what is done here. While matrix computations are well understood, preexisting work in mathematics (too abstract, so not computable) and in computer science (too low level, albeit extremely efficient, for reuse) is not helpful. A new path is needed. Here, the authors carefully lay the groundwork for parts of a verified linear algebra library in Isabelle/HOL. Then, they further document how a small network of theories (17, of which 11 are new and described here) help bridge the gap between fields and semirings in such a way that matrices over these domains have good computational and provability properties. Although the authors are aiming for verified efficient code, they really only obtain half: verified code. By using functional (also known as immutable) data structures, the resulting algorithms are much too memory hungry to be realistic. Granted, what is obtained is considerably faster than what used to be feasible for code verified in this manner-but there is a still a nontrivial gap to bridge. This is a clear, well-written paper, showing the relentless march forward of science. It's not a breakthrough, but a carefully constructed building block that can be relied on, with all choices on a twisting, not obvious route documented and justified. Online Computing Reviews Service

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Published In

Formal Aspects of Computing Volume 28, Issue 6

Nov 2016

169 pages

ISSN:0934-5043

EISSN:1433-299X

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 November 2016

Accepted: 09 June 2016

Received: 19 November 2015

Published in FAC Volume 28, Issue 6

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Divasón JThiemann R(2022)A Formalization of the Smith Normal Form in Higher-Order LogicJournal of Automated Reasoning10.1007/s10817-022-09631-566:4(1065-1095)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s10817-022-09631-5
Shi ZWu AYang XGuan YLi YSong X(2018)Formal analysis of the kinematic Jacobian in screw theoryFormal Aspects of Computing10.1007/s00165-018-0468-030:6(739-757)Online publication date: 1-Nov-2018
https://dl.acm.org/doi/10.1007/s00165-018-0468-0
Lambán LMartín-Mateos FRubio JRuiz-Reina J(2017)Using Abstract Stobjs in ACL2 to Compute Matrix Normal FormsInteractive Theorem Proving10.1007/978-3-319-66107-0_23(354-370)Online publication date: 26-Sep-2017
https://dl.acm.org/doi/10.1007/978-3-319-66107-0_23

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Recommendations

Formalizing Axiomatic Systems for Propositional Logic in Isabelle/HOL

Programming Language Semantics with Isabelle/HOL

A Formalization and Proof Checker for Isabelle’s Metalogic

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