Higman’s Lemma and Its Computational Content

Schwichtenberg, Helmut; Seisenberger, Monika; Wiesnet, Franziskus

doi:10.1007/978-3-319-29198-7_11

Cited by 2 publications

(2 citation statements)

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“…become a major topic in proof theory, and today encompasses both proof mining [12,13,14], which focuses on obtaining quantitative information primarily from proofs in areas of mathematical analysis, and the mechanized synthesis of programs from proofs, which has found many concrete applications in discrete mathematics and computer science [3,4,40]. Though as far back as the 1950s Kreisel already discusses the use of proof theoretic techniques to extract quantitative information from proofs in abstract algebra [19], specifically Hilbert's 17th problem together with his Nullstellensatz, to date there are comparatively few formal applications of proof interpretations in algebra, the computational analysis of which is done largely on a case by case basis.…”

Section: Introductionmentioning

confidence: 99%

“…The extraction of programs from proofs has a long and rich history, dating back to Kreisel's pioneering work on the 'unwinding' of proofs [17,18]. In the ensuing decades, the application of proof interpretations in particular has become a major topic in proof theory, and today encompasses both proof mining [12,13,14], which focuses on obtaining quantitative information primarily from proofs in areas of mathematical analysis, and the mechanized synthesis of programs from proofs, which has found many concrete applications in discrete mathematics and computer science [3,4,40].…”

Section: Introductionmentioning

confidence: 99%

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An Algorithmic Approach to the Existence of Ideal Objects in Commutative Algebra

Powell

Schuster

Wiesnet

2019

Logic, Language, Information, and Computation

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The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert's program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel's no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

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