A computational interpretation of open induction

Berger, Ulrich

doi:10.1109/lics.2004.1319627

Cited by 38 publications

(71 citation statements)

References 24 publications

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“…Last but not least, proofs as the ones of Proposition 5.1 and Corollary 5.1 above have, among other things, inspired a technique of proof by induction for not necessarily Noetherian rings (Schuster 2012); see also (Hendtlass and Schuster 2012). This technique is based upon Open Induction (Raoult 1988), a specific form of which (Berger 2004;Coquand 1992) has been used in one of the other constructive proofs (Coquand and Persson 1999) of the Hilbert basis theorem mentioned before.…”

Section: 32mentioning

confidence: 99%

Constructing Gröbner bases for Noetherian rings

Perdry¹,

Schuster²

2013

Math. Struct. Comp. Sci.

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We prove constructively that every finitely generated polynomial ideal has a Gröbner basis provided that the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the otherwise well-known algorithm to compute the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive rereading of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are understood to be coherent, and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement we provide a prime decomposition for commutative rings possessing the finite-depth property.

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Section: 32mentioning

confidence: 99%

Constructing Gröbner bases for Noetherian rings

Perdry¹,

Schuster²

2013

Math. Struct. Comp. Sci.

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“…In [BBC98] the following recursion was considered, which can be viewed as a more efficient 'demand driven' variant of modified bar recursion. As with modified bar recursion we use an auxiliary constant replacing the if-then-else construct used in [BBC98,Ber04a]: Φygs → y(λn.Ψygs(n ∈ dom(s))) ΨygsT → f [n] ΨygsF → gn(λz.Φyg(s * (n, z))) (5.1) where y : (nat → ρ) → nat, g : nat → (ρ → nat) → ρ and s : (nat × ρ) * is to be viewed as the graph of a finite function with n ∈ dom(s) and s[n] having the expected meanings. In [Ber04a] Here ≺ : ρ × ρ → boole is (the graph of) a wellfounded relation, α, γ : nat → ρ, αn = [α0, .…”

Section: Applicationsmentioning

confidence: 99%

“…Both recursions, (5.1) and (5.2) have the proof-theoretic strengths of full second order arithmetic. Their significance rests on the fact that they can be used to give rather direct realisability interpretations of strong classical theories: (5.1) realises classical countable choice [BBC98], while (5.1) realises open induction [Ber04a], a principle closely related to Nash-William minimal-bad-sequence argument [NW63].…”

Section: Applicationsmentioning

confidence: 99%

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Strong normalisation for applied lambda calculi

Berger¹

2005

Logical Methods in Computer Science

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Abstract. We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting ⊥ is strongly normalising provided all its 'stratified approximations' are. From this we derive a general normalisation theorem for applied typed λ-calculi: If all constants have a total value, then all typeable terms are strongly normalising. We apply this result to extensions of Gödel's system T and system F extended by various forms of bar recursion for which strong normalisation was hitherto unknown.

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“…Later on, Berger and Oliva [8] reformulated Berardi, Bezem and Coquand's realiser in terms of some notion of modified bar recursion. Then, in 2004, Berger [7] reduced the termination of these realisers to some variant of open induction called update induction:…”

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confidence: 99%

A Constructive Proof of Dependent Choice, Compatible with Classical Logic

Herbelin

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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Abstract-Martin-Löf's type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With this restriction, we lose the full axiom of choice but, thanks to a lazily-evaluated coinductive representation of quantification, we are still able to constructively prove the axiom of countable choice, the axiom of dependent choice, and a form of bar induction in ways that make each of them computationally compatible with classical logic.

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A computational interpretation of open induction

Abstract: We study the proof-theoretic and computational

Cited by 38 publications

References 24 publications

Constructing Gröbner bases for Noetherian rings

Constructing Gröbner bases for Noetherian rings

Strong normalisation for applied lambda calculi

A Constructive Proof of Dependent Choice, Compatible with Classical Logic

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