Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T

Joachimski, Felix; Matthes, Ralph

doi:10.1007/s00153-002-0156-9

Cited by 74 publications

(62 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Strong Normalisation (SN), for the implicational fragment: an obvious set of rules for reducing non-normal deductions is strongly normalising, i.e. every reduction sequence terminates [12,Sect. 6], [13,36,37].…”

Section: Cmentioning

confidence: 99%

“…3. SN, for the full language: a straightforward extension of the proof of [37] for implication 4 ; also, the proofs for implication "directly carry over" [12] to a system with conjunctions and disjunctions. An argument (using the ordinary elimination rule for implication) is given in [35] for the rules for implication and existential quantification, with the virtue of illustrating in detail how to handle GE rules where the Tait-Martin-Löf method of induction on types familiar from [11] is not available.…”

Section: Cmentioning

confidence: 99%

“…The paper [8] attempted 12 to deal with all these possibilities and carry out a programme of mechanically generating GE-rules from a set of I-rules with results about harmony.…”

Section: Ge-rules In Generalmentioning

confidence: 99%

See 2 more Smart Citations

Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

View full text Add to dashboard Cite

This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular self-referential "constant" • to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on "general elimination rules" and harmony, with a retraction of one of the main ideas of that work, i.e. the use of "flattened" general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

show abstract

Section: Cmentioning

confidence: 99%

Section: Cmentioning

confidence: 99%

See 1 more Smart Citation

Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

View full text Add to dashboard Cite

show abstract

“…As stated in op. cit., this rewrite relation on λ+ is strongly normalizing, as can be shown using the techniques of [34,15] (a full proof for exactly the rewrite relation considered here appears in Appendix A of [35]). …”

Section: Appendicesmentioning

confidence: 81%

The enriched effect calculus: syntax and semantics

Egger

Møgelberg

Simpson

2012

Journal of Logic and Computation

View full text Add to dashboard Cite

This paper introduces the enriched effect calculus, which extends established type theories for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations.The enriched effect calculus is implemented as an extension of a basic effect calculus without linear primitives, which is closely related to Moggi's computational metalanguage, Filinski's effect PCF and Levy's call-by-push-value. We present syntactic results showing: the fidelity of the behaviour of the linear connectives of the enriched effect calculus; the conservativity of the enriched effect calculus over its non-linear core (the effect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched effect calculus.The second half of the paper investigates models for the enriched effect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard effect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.

show abstract

“…Natural deduction with general elimination rules: This system [27] may be presented as a type system for the λ-calculus with generalized application. The latter is the system ΛJ of [18], which we rename here as λg, for the sake of uniformity with the names of other calculi. Terms of λg are given by…”

Section: Natural Deductionmentioning

confidence: 99%

The λ-Calculus and the Unity of Structural Proof Theory

Santo

2009

Theory Comput Syst

View full text Add to dashboard Cite

Abstract. In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cutelimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato's calculus. It is a calculus with modus ponens and primitive substitution; it is also a "coercion calculus", in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a "multiary" calculus, because "applicative terms" may exhibit a list of several arguments. But the combination of "multiarity" and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.

show abstract

Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T

Cited by 74 publications

References 19 publications

Some Remarks on Proof-Theoretic Semantics

Some Remarks on Proof-Theoretic Semantics

The enriched effect calculus: syntax and semantics

The λ-Calculus and the Unity of Structural Proof Theory

Contact Info

Product

Resources

About