Strong Normalization of the Dual Classical Sequent Calculus

Dougherty, Daniel J.; Ghilezan, Silvia; Lescanne, Pierre; Likavec, Silvia

doi:10.1007/11591191_13

Cited by 7 publications

(6 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof rests on a reducibility argument. Similar approaches for the propositional fragment can be found in the literature [10,20]; however, the biggest inf luence on our proof was the one by Parigot for the second-order extension of the Symmetric Lambda-Calculus [19].…”

Section: Strong Normalization Of the Second-order Dual Calculussupporting

confidence: 61%

Classical Logic with Mendler Induction

Campos

Fiore

2015

Logical Foundations of Computer Science

View full text Add to dashboard Cite

We investigate (co-) induction in classical logic under the propositions-as-types paradigm, considering propositional, secondorder and (co-) inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-) iterator and show that it is strongly normalizing. We prove this using a reducibility argument. 1 Introduction The Curry-Howard isomorphism. The interplay between logic and computer science has a long and rich history. In particular, the Curry-Howard isomorphism, the correspondence between types and propositions, and between typings and proofs, is a long established bridge through which results in one field can fruitfully migrate to the other. Classical calculi. One such example-motivating of the research presented herein-is the use of programming calculi based on Gentzen's sequent calculus LK [11]. At its core, LK is a calculus of the dual concepts of necessary assumptions and possible conclusions-concepts that map neatly, on the computer science side, to required inputs (or computations) and possible outputs (or continuations). As an example of the kind of analysis that can be done by focusing on this separation of concepts, Curien and Herbelin [6] and Wadler [22] devised LK-based calculi that showed, syntacticly, the duality of the two most common evaluations strategies: call-by-name and call-by-value. While originally such classical calculi included only propositional types-i.e. conjunction, disjunction, negation, implication and subtraction (the dual connective of implication)-they were later extended with second-order types [15, 20] and also with positive (co-) inductive types [15]. The latter fundamentally depended on the existence of a map operation of the underlying type. Here we do away with this restriction. Mendler induction. We take as primitive a more general induction scheme based on the work of Mendler [17]. He proposed an induction operator for System F that operated in a way akin to the callby-name fix-point combinator-thus avoiding direct use of said mapping operations-but ensured termination by clever use of polymorphic types. Further adding to its generality, it was later shown that with Mendler's iterator one could in fact induct on data-types of arbitrary variance-i.e. datatypes whose induction variable may also appear negatively [16, 21]. Due to its generality, Mendler induction has been applied in a number of different contexts, such as higher-order recursive types [1, 2] and automated theorem proving [13]. Classical logic and Mendler induction. The central question of this paper is whether Mendler induction can be lifted to non-functional settings without introducing ill-effects-specifically, non-Vol. 00, No.

show abstract

Section: Strong Normalization Of the Second-order Dual Calculussupporting

confidence: 61%

Classical Logic with Mendler Induction

Campos

Fiore

2015

Logical Foundations of Computer Science

View full text Add to dashboard Cite

show abstract

“…This proof will appear in Battyanyi's PhD thesis [2] who will also consider the dual calculus. Note that Dougherty & all [12] have shown the strong normalization of this calculus by the reducibility method using the technique of the fixed point construction.…”

Section: Introductionmentioning

confidence: 92%

Arithmetical Proofs of Strong Normalization Results for the Symmetric λμ-Calculus

David

Nour

2005

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. The symmetric λµ-calculus is the λµ-calculus introduced by Parigot in which the reduction rule µ ′ , which is the symmetric of µ, is added. We give arithmetical proofs of some strong normalization results for this calculus. We show (this is a new result) that the µµ ′ -reduction is strongly normalizing for the un-typed calculus. We also show the strong normalization of the βµµ ′ -reduction for the typed calculus: this was already known but the previous proofs use candidates of reducibility where the interpretation of a type was defined as the fix point of some increasing operator and thus, were highly non arithmetical.

show abstract

“…In the classical setting, there are several term calculi based on classical sequent logic, in which terms unambiguously encode sequent derivations and reduction corresponds to cut elimination: Barbanera and Berardi's Symmetric Calculus [3], Curien-Herbelin's λµ µ-calculus [7], Urban-Bierman's calculus [37], Wadler's Dual Calculus [45]. In contrast to natural deduction proof systems, sequent calculi exhibit inherent symmetries in proof structures which create technical difficulties in analyzing the reduction properties of these calculi [12,11,16].…”

Section: Introductionmentioning

confidence: 99%

Computational interpretation of classical logic with explicit structural rules

Ghilezan¹,

Lescanne²,

Žunić³

2012

Preprint

Self Cite

View full text Add to dashboard Cite

We present a calculus providing a Curry-Howard correspondence to classical logic represented in the sequent calculus with explicit structural rules, namely weakening and contraction. These structural rules introduce explicit erasure and duplication of terms, respectively. We present a type system for which we prove the type-preservation under reduction. A mutual relation with classical calculus featuring implicit structural rules has been studied in detail. From this analysis we derive strong normalisation property.

show abstract

Strong Normalization of the Dual Classical Sequent Calculus

Cited by 7 publications

References 26 publications

Classical Logic with Mendler Induction

Classical Logic with Mendler Induction

Arithmetical Proofs of Strong Normalization Results for the Symmetric λμ-Calculus

Computational interpretation of classical logic with explicit structural rules

Contact Info

Product

Resources

About