Certifying properties of an efficient functional program for computing Gröbner bases

Jorge, J. Santiago; Gulías, Víctor M.; Freire, José Luis

doi:10.1016/j.jsc.2007.07.016

Cited by 8 publications

(7 citation statements)

References 12 publications

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“…P provides the additional functions and relations needed to prove that it belongs to the class of reduction polynomial domains, and thus is a reduction ring thanks to the key result in Polynomials.nb. 6 The proof of this claim is part of the theory, of course.…”

Section: Reduction Ring Theorymentioning

confidence: 98%

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Mathematical Theory Exploration in Theorema: Reduction Rings

Maletzky¹

2016

Lecture Notes in Computer Science

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Abstract. In this paper we present the first-ever computer formalization of the theory of Gröbner bases in reduction rings, which is an important theory in computational commutative algebra, in Theorema. Not only the formalization, but also the formal verification of all results has already been fully completed by now; this, in particular, includes the generic implementation and correctness proof of Buchberger's algorithm in reduction rings. Thanks to the seamless integration of proving and computing in Theorema, this implementation can now be used to compute Gröbner bases in various different domains directly within the system. Moreover, a substantial part of our formalization is made up solely by "elementary theories" such as sets, numbers and tuples that are themselves independent of reduction rings and may therefore be used as the foundations of future theory explorations in Theorema. In addition, we also report on two general-purpose Theorema tools we developed for an efficient and convenient exploration of mathematical theories: an interactive proving strategy and a "theory analyzer" that already proved extremely useful when creating large structured knowledge bases.

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Section: Reduction Ring Theorymentioning

confidence: 98%

“…The first attempt in[3] was erroneous 6. Once again, this is only true if R is a reduction ring and T is a domain of commutative power-products.…”

mentioning

confidence: 99%

Mathematical Theory Exploration in Theorema: Reduction Rings

Maletzky¹

2016

Lecture Notes in Computer Science

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“…Moreover, it is possible to automatically extract executable, certified OCaml code for computing Gröbner bases from the formalization. In 2009, Jorge, Guilas and Freire [21] took the reverse direction: they first implemented an efficient version of Buchberger's algorithm directly in OCaml and then proved it correct, making use of the underlying formal theory in Coq.…”

Section: Related Workmentioning

confidence: 99%

Gröbner Bases of Modules and Faugère’s $$F_4$$F4 Algorithm in Isabelle/HOL

Maletzky

Immler

2018

Lecture Notes in Computer Science

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We present an elegant, generic and extensive formalization of Gröbner bases in Isabelle/HOL. The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger's algorithm, Buchberger's criteria for avoiding useless pairs), but also includes more advanced features like reduced Gröbner bases. Particular highlights are the first-time formalization of Faugère's matrix-based F4 algorithm and the fact that the entire theory is formulated for modules and submodules rather than rings and ideals. All formalized algorithms can be translated into executable code operating on concrete data structures, enabling the certified computation of (reduced) Gröbner bases and syzygy modules.

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“…Hence, if the value is successfully written, then the device is not from "S3." This final check is done by line (11), when the value written to the device, is compared to the value read from the device. A bug in the code could result from the wrong initial value being loaded, or the wrong bits of a register being tested.…”

Section: Background and Motivationmentioning

confidence: 99%

“…Equational reasoning has been applied to many problems in software analysis, such as certifying properties of a functional program [11], linking first class primitive modules [12], and rewriting Haskell fragments [13].…”

Section: Related Workmentioning

confidence: 99%

Equational Reasoning on x86 Assembly Code

Coogan

Debray

2011

2011 IEEE 11th International Working Conference on Source Code Analysis and Manipulation

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Abstract-Analysis of software is essential to addressing problems of correctness, efficiency, and security. Existing source code analysis tools are very useful for such purposes, but there are many instances where high-level source code is not available for software that needs to be analyzed. A need exists for tools that can analyze assembly code, whether from disassembled binaries or from handwritten sources. This paper describes an equational reasoning system for assembly code for the ubiquitous Intel x86 architecture, focusing on various problems that arise in low-level equational reasoning, such as register-name aliasing, memory indirection, conditioncode flags, etc. Our system has successfully been applied to the problem of simplifying execution traces from obfuscated malware executables.

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Certifying properties of an efficient functional program for computing Gröbner bases

Cited by 8 publications

References 12 publications

Mathematical Theory Exploration in Theorema: Reduction Rings

Mathematical Theory Exploration in Theorema: Reduction Rings

Gröbner Bases of Modules and Faugère’s $$F_4$$F4 Algorithm in Isabelle/HOL

Equational Reasoning on x86 Assembly Code

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