A universal algorithm for Krull's theorem

Powell, Thomas; Schuster, Peter; Wiesnet, Franziskus

doi:10.1016/j.ic.2021.104761

Cited by 8 publications

(3 citation statements)

References 56 publications

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“…For example, in the case of the Bolzano-Weierstrass theorem [OP15b], our state might record information of the form x n ∈ I, collecting information about the location of sequence elements. For applications in algebra [PSW22], one might instead store information about a particular maximal ideal.…”

Section: :20 T Powellmentioning

confidence: 99%

Proofs as stateful programs: A first-order logic with abstract Hoare triples, and an interpretation into an imperative language

Powell

2024

Logical Methods in Computer Science

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We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the assertions about the state compose in the correct way. The main result of the paper is a realizability interpretation of our logic that extracts programs into a mixed functional/imperative language. All programs expressible in this language act on the state in a sequential manner, and we make this intuition precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is very general, and our intention is that it can be instantiated and extended in a variety of different ways. We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded while rule, and conclude by outlining several other directions for future work.

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Section: :20 T Powellmentioning

confidence: 99%

Proofs as stateful programs: A first-order logic with abstract Hoare triples, and an interpretation into an imperative language

Powell

2024

Logical Methods in Computer Science

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“…An important part of constructive algebra is to devise tools to import proofs from classical commutative algebra into the constructive setting. 6 The following two statements are established test cases exploring the power of such tools [50,51,40,42,56,52,4,45,14,15].…”

Section: On the Intersection Of All Prime Idealsmentioning

confidence: 99%

Maximal ideals in countable rings, constructively

Blechschmidt¹,

Schuster²

2022

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“…The exploration of the scope and limits of program extraction is an active field of research that has led to many interesting results regarding the partial inclusion of classical logic [9][10][11] and mathematical principles such as various forms of choice [12][13][14][15][16] and induction principles [15,[17][18][19]. The benefit of program extraction is not only that extracted programs come automatically with formal correctness proofs but also that important additional properties of programs concerning computational complexity, totality and typability can be obtained.…”

Section: Introductionmentioning

confidence: 99%