Perpetual Reductions inλ-Calculus

Raamsdonk, Femke van; Severi, Paula; Sørensen, Morten Heine; Xi, Hongwei

doi:10.1006/inco.1998.2750

Cited by 43 publications

(20 citation statements)

References 17 publications

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“…(This is denoted ϕ(S) in [7,35].) The following is analogous to the Lemma 2.3.14 in [7] -notice, however, a difference in the last point:…”

Section: The Uniformity Theoremmentioning

confidence: 81%

“…[6,16]), which in turn is a consequence of the Finiteness of Developments Theorem. A nice proof of the latter can be found in [35], following some of the steps of the realizability technique. However, since our aim here is to provide a "syntactic" proof of strong normalization, we give a purely combinatory proof of this theorem, by adapting a technique of de Vrijer [42].…”

Section: The Uniformity Theoremmentioning

confidence: 96%

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On strong normalization and type inference in the intersection type discipline

Boudol

2008

Theoretical Computer Science

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We introduce a new unification procedure for the type inference problem in the intersection type discipline. It is well known that type inference in this case should succeed exactly for the strongly normalizing expressions. We give a proof for the strong normalization result in the intersection type discipline, which we obtain by putting together some well-known results and proof techniques. Our proof uses a variant of Klop's extended λ-calculus, for which it is shown that strong normalization is equivalent to weak normalization. This is proved here by means of a finiteness of developments theorem, obtained following de Vrijer's combinatory technique. The main property of this extended calculus is uniformity, i.e. weak and strong normalizabilities coincide. The strong normalizability result is therefore a consequence of the fact, first established by Coppo and Dezani (for the λ-calculus) that typability implies weak normalizability. We then show that the unification process which is the basis for type inference exactly corresponds to reduction in the extended λ-calculus. Finally we show that our notion of unification allows us to compute a principal typing for any typable λ-expression.

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“…(This is denoted ϕ(S) in [7,35].) The following is analogous to the Lemma 2.3.14 in [7] -notice, however, a difference in the last point:…”

Section: The Uniformity Theoremmentioning

confidence: 81%

Section: The Uniformity Theoremmentioning

confidence: 96%

On strong normalization and type inference in the intersection type discipline

Boudol

2008

Theoretical Computer Science

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“…data sn (n : N) {a Γ} (t : Tm Γ a) : Set where acc : (∀ {t'} → t n ⇒β t' → sn n t') → sn n t Van Raamsdonk et al (1999) pioneered a more explicit characterization of strongly normalizing terms SN, namely the least set closed under introductions, formation of neutral (=stuck) terms, and weak head expansion. We adapt their technique from lambdacalculus to λ ; herein, it is crucial to work with well-typed terms to avoid junk like fst (λ x. x) which does not exist in pure lambda-calculus.…”

Section: Strong Normalizationmentioning

confidence: 99%

“…For this, we utilize Agda's new copattern feature (Abel et al, 2013). The set of strongly normalizing terms is defined inductively by distinguishing on the shape of terms, following van Raamsdonk et al (1999) and Joachimski and Matthes (2003). The first author has formalized this technique before in Twelf (Abel, 2008); in this work we extend these results by a proof of equivalence to the standard notion of strong normalization.…”

Section: Introductionmentioning

confidence: 99%

A Formalized Proof of Strong Normalization for Guarded Recursive Types

Abel

Vezzosi

2014

Programming Languages and Systems

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Abstract. We consider a simplified version of Nakano's guarded fixed-point types in a representation by infinite type expressions, defined coinductively. Smallstep reduction is parametrized by a natural number "depth" that expresses under how many guards we may step during evaluation. We prove that reduction is strongly normalizing for any depth. The proof involves a typed inductive notion of strong normalization and a Kripke model of types in two dimensions: depth and typing context. Our results have been formalized in Agda and serve as a case study of reasoning about a language with coinductive type expressions.

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“…‖M‖ B denotes the Barendregt's norm of M, that is, the length of the perpetual reduction sequence from M, if M ∈ β-SN; or ω, otherwise. → B is important because of two properties: (i) M is β-SN iff the perpetual reduction from M is finite [1]; (ii) ‖M‖ β = ‖M‖ B [10,12]. When this norm is meant, we may drop the subscript.…”

Section: Introductionmentioning

confidence: 97%

A note on preservation of strong normalisation in the λ-calculus

Santo

2011

Theoretical Computer Science

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a b s t r a c tAn auxiliary notion of reduction ρ on the λ-terms preserves strong normalisation if all strongly normalising terms for β are also strongly normalising for β∪ρ. We give a sufficient condition for ρ to preserve strong normalisation. As an example of application, we check easily the sufficient condition for Regnier's σ -reduction rules and the ''assoc''-reduction rule inspired by calculi with let-expressions. This gives the simplest proof so far that the union of all these rules preserves strong normalisation.

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Perpetual Reductions inλ-Calculus

Cited by 43 publications

References 17 publications

On strong normalization and type inference in the intersection type discipline

On strong normalization and type inference in the intersection type discipline

A Formalized Proof of Strong Normalization for Guarded Recursive Types

A note on preservation of strong normalisation in the λ-calculus

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