Generalizing a Mathematical Analysis Library in Isabelle/HOL

Aransay, Jesús; Divasón, Jose

doi:10.1007/978-3-319-17524-9_30

Cited by 9 publications

(11 citation statements)

References 5 publications

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“…This limitation had been pointed out in some previous developments over this Library (see, for instance, ). This will enable us to carry out computations over any field, such as F p , Q, R and C. This generalisation is thoroughly described in Aransay & Divasón (2015). It has involved the introduction of new structures (locales and type classes), the reproduction of some previous definitions and proofs in those structures, and also in some particular cases the formalisation of different proofs for results that hold both in a generic field F and in R, but whose previous proofs in R made use of specific properties of such a structure.…”

Section: Hol Multivariate Analysismentioning

confidence: 99%

Formalisation in higher-order logic and code generation to functional languages of the Gauss-Jordan algorithm

Aransay

Divasón

2015

J. Funct. Prog.

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In this paper, we present a formalisation in a proof assistant, Isabelle/HOL, of a naive version of the Gauss-Jordan algorithm, with explicit proofs of some of its applications; and, additionally, a process to obtain versions of this algorithm in two different functional languages (SML and Haskell) by means of code generation techniques from the verified algorithm. The aim of this research is not to compete with specialised numerical implementations of Gauss-like algorithms, but to show that formal proofs in this area can be used to generate usable functional programs. The obtained programs show compelling performance in comparison to some other verified and functional versions, and accomplish some challenging tasks, such as the computation of determinants of matrices of big integers and the computation of the homology of matrices representing digital images.

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Section: Hol Multivariate Analysismentioning

confidence: 99%

Formalisation in higher-order logic and code generation to functional languages of the Gauss-Jordan algorithm

Aransay

Divasón

2015

J. Funct. Prog.

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“…2 500 lines in each SML and Haskell) is considerable and covers a wide range of applications in Linear Algebra. Its formalisation (available in [5]) took 15 000 lines of Isabelle code. The HMA Library infrastructure reduced significatively the amount of mathematical results to be formalised.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

“…The source files of the development are available from [5]; they have been developed under the Isabelle 2013-2 version. The previous web site also includes the SML and Haskell code generated from the Isabelle specifications, and also the input matrices that have been used in the benchmarks presented in Section 5.…”

Section: Introductionmentioning

confidence: 99%

Applications of the Gauss-Jordan algorithm, done right

Aransay,

Divasón

2014

Preprint

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Computer Algebra systems are widely spread because of some of their remarkable features such as their ease of use and performance. Nonetheless, this focus on performance sometimes leads to unwanted consequences: algorithms and computations are implemented and carried out in a way which is sometimes not transparent to the users, and that can lead to unexpected failures. In this paper we present a formalisation in a proof assistant system of a naive version of the Gauss-Jordan algorithm, with explicit proofs of some of its applications, and additionally a process to obtain versions of this algorithm in two different functional languages (SML and Haskell) by means of code generation techniques from the verified algorithm. The obtained programs are then applied to test cases, which, despite the simplicity of the original algorithm, have shown remarkable features in comparison to some Computer Algebra systems, such as Mathematica R (where some of these computations are even incorrect), or Sage (in comparison to which the generated programs show a compelling performance). The aim of the paper is to show that, with the current technology in Theorem Proving, formalising Linear Algebra procedures is a challenging but rewarding task, which provides programs that can be compared in some aspects to state of the art procedures in Computer Algebra systems, and whose correctness is formally proved.

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“…Formalizations of real vectors and real matrices have been proposed in higher-order-logic theorem provers ( Harrison, 2013 ), HOL4 ( Shi et al, 2014 ; Shi, Guan & Li, 2020 ), PVS ( Herencia-Zapana et al, 2012 ), Isabelle ( Aransay & Divasón, 2015 ), Mizar ( Bancerek et al, 2018 ) and Coq ( Boldo, Lelay & Melquiond, 2015 ; Mahboubi & Tassi, 2017 ; Dénes & Bertot, 2011 ) and automated theorem prover ACL2 ( Gamboa, Cowles & Baalen, 2003 ). Also, the complex vectors, bivectors and some complex matrix arithmetic is formalized in ( Afshar et al, 2014 ).…”

Section: Introductionmentioning

confidence: 99%

Formal verification of Matrix based MATLAB models using interactive theorem proving

Gauhar

Rashid

Hasan

et al. 2021

PeerJ Computer Science

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MATLAB is a software based analysis environment that supports a high-level programing language and is widely used to model and analyze systems in various domains of engineering and sciences. Traditionally, the analysis of MATLAB models is done using simulation and debugging/testing frameworks. These methods provide limited coverage due to their inherent incompleteness. Formal verification can overcome these limitations, but developing the formal models of the underlying MATLAB models is a very challenging and time-consuming task, especially in the case of higher-order-logic models. To facilitate this process, we present a library of higher-order-logic functions corresponding to the commonly used matrix functions of MATLAB as well as a translator that allows automatic conversion of MATLAB models to higher-order logic. The formal models can then be formally verified in an interactive theorem prover. For illustrating the usefulness of the proposed library and approach, we present the formal analysis of a Finite Impulse Response (FIR) filter, which is quite commonly used in digital signal processing applications, within the sound core of the HOL Light theorem prover.

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Generalizing a Mathematical Analysis Library in Isabelle/HOL

Cited by 9 publications

References 5 publications

Formalisation in higher-order logic and code generation to functional languages of the Gauss-Jordan algorithm

Formalisation in higher-order logic and code generation to functional languages of the Gauss-Jordan algorithm

Applications of the Gauss-Jordan algorithm, done right

Formal verification of Matrix based MATLAB models using interactive theorem proving

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