Confluence and Strong Normalisation of the Generalised Multiary λ-Calculus

Santo, José Espírito; Pinto, Luís

doi:10.1007/978-3-540-24849-1_13

Cited by 2 publications

(7 citation statements)

References 6 publications

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“…Instead of π + we use the "eager" version π of rule π, already considered in [Espírito Santo and Pinto 2004] in the context of λJ. This rule also performs in a single step sequences of π-reductions, but contrary to π + , π-reduction must carry on while newly generated π-redexes hide µ-redexes.…”

Section: Confluence and Terminationmentioning

confidence: 99%

“…Confluence. Firstly we recall from [Espírito Santo and Pinto 2004] that relations → π , → β and → βπ in λJ m are confluent.…”

Section: Strong Normalisation and Confluence Of Reductionmentioning

confidence: 99%

“…By extending the confluence proofs for ΛJ presented in [Joachimski and Matthes 2000]. See [Espírito Santo and Pinto 2004] for details. Now we consider confluence in the presence of rule µ.…”

Section: Strong Normalisation and Confluence Of Reductionmentioning

confidence: 99%

“…Other proofs of confluence and strong normalisation of relation → βπµ in λJ m are given in [Espírito Santo and Pinto 2004]. There the lifting of the results from ΛJ is achieved through the mapping ν : λJ m → λJ introduced in that paper.…”

Section: Strong Normalisation and Confluence Of Reductionmentioning

confidence: 99%

“…Although not new, strong normalisation and confluence of reduction receive here new proofs that profit from the above new results; in particular, these new proofs depend only on the properties of the mappings into the subsystems, precisely the mappings that calculate the normal forms w.r.t. permutative conversions (whereas the original [Espírito Santo and Pinto 2004] proofs of strong normalisation and confluence of reduction depended on the properties of other mappings).…”

Section: Introductionmentioning

confidence: 99%

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A calculus of multiary sequent terms

Santo

Pinto

2011

ACM Trans. Comput. Logic

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Multiary sequent terms were originally introduced as a tool for proving termination of permutative conversions in cut-free sequent calculus. This work develops the language of multiary sequent terms into a term calculus for the computational (Curry-Howard) interpretation of a fragment of sequent calculus with cuts and cut-elimination rules. The system, named generalised multiary λ-calculus, is a rich extension of the λ-calculus where the computational content of the sequent calculus format is explained through an enlarged form of the application constructor. Such constructor exhibits the features of multiarity (the ability of forming lists of arguments) and generality (the ability of prescribing a kind of continuation). The system integrates in a modular way the multiary λ-calculus and an isomorphic copy of the λ-calculus with generalised application ΛJ (in particular, natural deduction is captured internally up to isomorphism). In addition, the system: (i) comes with permutative conversion rules, whose role is to eliminate the new features of application; (ii) is equipped with reduction rules-either the µ-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary β-rule. This paper establishes the meta-theory of the system, with emphasis on the role of the µ-rule, and including a study of the interaction of reduction and permutative conversions.

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Section: Confluence and Terminationmentioning

confidence: 99%

“…Confluence. Firstly we recall from [Espírito Santo and Pinto 2004] that relations → π , → β and → βπ in λJ m are confluent.…”

Section: Strong Normalisation and Confluence Of Reductionmentioning

confidence: 99%

Section: Strong Normalisation and Confluence Of Reductionmentioning

confidence: 99%

Section: Strong Normalisation and Confluence Of Reductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A calculus of multiary sequent terms

Santo

Pinto

2011

ACM Trans. Comput. Logic

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Structural Proof Theory as Rewriting

Santo

Frade

Pinto

2006

Lecture Notes in Computer Science

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Abstract. The multiary version of the λ-calculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cut-elimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of non-terminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. well-behaved combinations of rules. We identify six of these "combined" normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of "combined" normal forms is the existence of three ways of expressing multiple application in the calculus.

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Confluence and Strong Normalisation of the Generalised Multiary λ-Calculus

Cited by 2 publications

References 6 publications

A calculus of multiary sequent terms

A calculus of multiary sequent terms

Structural Proof Theory as Rewriting

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