Strong normalization of classical natural deduction with disjunctions

Nakazawa, Koji; Tatsuta, Makoto

doi:10.1016/j.apal.2008.01.003

Cited by 5 publications

(3 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…the implicative intuitionistic logic but his proof contains an error as Matthes pointed out in [8]. Nakazawa and Tatsuta corrected de Groote's proof in [12] by using the notion of augmentations.…”

Section: Introductionmentioning

confidence: 99%

Strong normalization results by translation

David¹,

Nour²

2010

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

We prove the strong normalization of full classical natural deduction (i.e. with conjunction, disjunction and permutative conversions) by using a translation into the simply typed λµ-calculus. We also extend Mendler's result on recursive equations to this system. The systemsDefinition 2.1 Let V and W be disjoint sets of variables.1. The set of λ-terms is defined by the following grammar2. The set of λµ-terms is defined by the following grammar3. The set of λµ →∧∨ -terms is defined by the following grammarNote that, for the λµ-calculus, we have adopted here the so-called de Groote calculus which is the extension of Parigot's calculus where the distinction between named and un-named terms is forgotten. In this calculus, µα is not necessarily followed by [β]. We also write (α M ) instead of [α]M .1 Recently, we have been aware of a paper by Wojdyga [18] who uses the same kind of translations but where all the atomic types are collapsed to ⊥. Our translation allows us to extend trivially Mendler's result whereas the one of Wojdyga, of course, does not.

show abstract

Section: Introductionmentioning

confidence: 99%

Strong normalization results by translation

David¹,

Nour²

2010

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

show abstract

“…We also thank the referees of the conference version [12] for, among other things, pointing out the work of Nakazawa and Tatsuta, whom we thank for an advanced copy of [30]. …”

Section: Acknowledgementsmentioning

confidence: 99%

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

Santo

Matthes

Pinto

2007

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. The intuitionistic fragment of the call-by-name version of Curien and Herbelin's λµμ-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed λ-calculus. Our embedding is a continuation-and-garbagepassing style translation, the inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's λµ-calculus. The embedding strictly simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need "units of garbage" to be passed. We apply the same method to other calculi, namely successive extensions of the simply-typed λ-calculus leading to our intuitionistic system, and already for the simplest extension we consider (λ-calculus with generalised application), this yields the first proof of strong normalisation through a reduction-preserving embedding. The results obtained extend to second and higher-order calculi.

show abstract

“…The solution, first used by de Groote ([2], [3]), for first-order logic, is a CPS-translation. Our proof is similar to de Groote's but the version of CPS we use is based on Nakazawa and Tatsuta [6].…”

mentioning

confidence: 91%

Short Proofs of Strong Normalization

Wojdyga

Lecture Notes in Computer Science

View full text Add to dashboard Cite

This paper presents simple, syntactic strong normalization proofs for the simply-typed λ-calculus and the polymorphic λ-calculus (system F) with the full set of logical connectives, and all the permutative reductions. The normalization proofs use translations of terms and types of λ →,∧,∨,⊥ to terms and types of λ→ and from F ∀,∃,→,∧,∨,⊥ to F ∀,→ . I didn't find a proof really nice, and taking little space [4, p. 130]. For instance, many proofs, like these in [7,9,10] are based on the computability method, or (in the polymorphic case) candidates of reducibility. This requires re-doing each time the same argument, but in a more complex way, due to the increased complexity of the language.We believe that methodologically the most adequate approach is by reducing the question of strong normalization of the extended systems to the known strong normalization of the base systems, involving only implication and the universal quantifier. We propose two such proofs in what follows.The first proof reduces the calculus λ →,∧,∨,⊥ with connectives ∧, ∨, →, ⊥ to the calculus λ → . Here we use the strong normalization of λ → with beta-etareductions. The proof is based on composing the ordinary reduction of classical connectives to implication and absurdity with Ong's translation of the λµ-calculus to the ordinary λη-calculus, as described e.g. in [8, Chapter 6]. To our knowledge this is the most direct way of showing SN for system λ →,∧,∨,⊥ .The above method does not however extend to the polymorphic case. Indeed, the translation is strictly type-driven and requires an a priori knowledge of all types a given expression can obtain by polymorphic instantiation. Also the well known definition of logical connectives in system F:is not adequate. The translation preserves beta-conversion, but not the permutations. The solution, first used by de Groote ([2], [3]), for first-order logic, is a CPS-translation. Our proof is similar to de Groote's but the version of CPS we use is based on Nakazawa and Tatsuta [6].

show abstract

Strong normalization of classical natural deduction with disjunctions

Cited by 5 publications

References 29 publications

Strong normalization results by translation

Strong normalization results by translation

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

Short Proofs of Strong Normalization

Contact Info

Product

Resources

About