A verified Common Lisp implementation of Buchberger’s algorithm in ACL2

Medina‐Bulo, Inmaculada; Palomo-Lozano, Francisco; Ruiz–Reina, José–Luis

doi:10.1016/j.jsc.2009.07.002

Cited by 12 publications

(8 citation statements)

References 15 publications

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“…We will call the elements of this ring simplicial polynomials. The ACL2 formalization of simplicial polynomials presented here is similar to the formalization of polynomials over the rational field developed in [20].…”

Section: Formalization Issuesmentioning

confidence: 95%

“…In our example: ((3 2) (1 2 5)). Let us call such pairs simplicial terms, using a terminology borrowed from algebraic polynomial theory (see, for instance, the formalization in [20]). Note that although a simplicial term is a simplicial operator, we call it in a special way to emphasize the fact that it is in canonical form.…”

Section: Formalization Issuesmentioning

confidence: 99%

“…The ring of simplicial polynomials is just a particular instance of this generic theory, obtained using encapsulation in combination with the functional instantiation inference rule of ACL2. (A related development for polynomials in commutative algebra can be found in [20]. )…”

Section: Simplicial Polynomials and Ring Propertiesmentioning

confidence: 99%

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Formalization of a normalization theorem in simplicial topology

Lambán

Martín-Mateos

Rubio

et al. 2012

Ann Math Artif Intell

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In this paper we present a complete formalization of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. Even if the Normalization Theorem is usually stated as a higher-order result (with a Category Theory flavor) we manage to give a first-order proof of it. To this aim it is instrumental the introduction of an algebraic data structure called simplicial polynomial. As a demonstration of the validity of our techniques we developed a formal proof in the ACL2 theorem prover. This work is dedicated to our colleague and friend Mirian Andrés. She started this research but passed away at the age of only 29 due to a car accident. Mirian, the best friend for your friends, we do not forget you.

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Section: Formalization Issuesmentioning

confidence: 95%

Section: Formalization Issuesmentioning

confidence: 99%

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Formalization of a normalization theorem in simplicial topology

Lambán

Martín-Mateos

Rubio

et al. 2012

Ann Math Artif Intell

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“…Another formalization that focuses very much on the actual computation of Gröbner bases is that of Medina-Bulo, Palomo-Lozano, Alonso-Jiménez and Ruiz-Reina [26] in ACL2 [22], dating back to 2010. There, however, the representation of power-products and polynomials is fixed to ordered lists of exponents and monomials, respectively, owing to the limited expressiveness of the underlying system.…”

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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“…To the best of our knowledge, reduction rings have never been the subject of formal theory exploration in any software system so far; Gröbner bases in polynomial rings over fields have already been formalized in ACL2 [10], Coq and OCaml [16,6] and Mizar [13], though, and a formalization in Isabelle by the author of this paper is currently in progress. Moreover, the purely algorithmic aspect (no theorems and proofs) of a variation of reduction rings was implemented in Theorema in [4].…”

Section: Introductionmentioning

confidence: 99%

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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A verified Common Lisp implementation of Buchberger’s algorithm in ACL2

Cited by 12 publications

References 15 publications

Formalization of a normalization theorem in simplicial topology

Formalization of a normalization theorem in simplicial topology

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

Mathematical Theory Exploration in Theorema: Reduction Rings

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