λβ′ — A λ-calculus with a generalized β-reduction rule

1995

We provide a new characterization of L evy's redex-families in thecalculus 11] as suitable paths in the initial term of the derivation. The idea is that redexes in a same family are created by \contraction" (via -reduction) of a unique common path in the initial term. This fact gives new evidence about the \common nature" of redexes in a same family, and about the possibility of sharing their reduction. In general, paths seem to provide a very friendly and intuitive tool for reasoning about redex-families, as well in theory (using paths, we shall provide a remarkably simple proof of the equivalence between extraction 11] and labeling) as in practice (our characterization underlies all recent works on optimal graph reduction techniques for the -calculus 9, 6, 7, 1], providing an original and intuitive understanding of optimal implementations).

Theoretical Computer Science, 2004

The coinductive-calculus co arises by a coinductive interpretation of the grammar of the standard-calculus and contains non-well-founded-terms. An appropriate notion of reduction is analyzed and proven to be con uent by means of a detailed analysis of the usual Tait/Martin-L of style development argument. This yields bounds for the lengths of those joining reduction sequences that are guaranteed to exist by con uence. These bounds also apply to the well-founded-calculus, thus adding quantitative information to the classic result.

Archive for Mathematical Logic, 2007

We introduce new proof systems G[β] and G ext[β], which are equivalent to the standard equational calculi of λβ- and λβη- conversion, and which may be qualified as ‘analytic’ because it is possible to establish, by purely proof-theoretical methods, that in both of them the transitivity rule admits effective elimination. This key feature, besides its intrinsic conceptual significance, turns out to provide a common logical background to new and comparatively simple demonstrations—rooted in nice proof-theoretical properties of transitivity-free derivations—of a number of well-known and central results concerning β- and βη-reduction. The latter include the Church–Rosser theorem for both reductions, the Standardization theorem for β- reduction, as well as the Normalization (Leftmost reduction) theorem and the Postponement of η-reduction theorem for βη-reduction

Term Rewriting and Applications, 2006

We present an extension of the ()-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover the expressiveness of ML-style pattern matching. We then prove that this system enjoys the Church-Rosser property using a semi-automatic`divide and conquer' technique by which we determine all the pairs of commuting subsystems of the formalism (considering all the possible combinations of the nine primitive reduction rules). Finally, we prove a separation theorem similar to B ohm's theorem for the whole formalism.

Mathematical Problems of Computer Science, 2018

In this paper the canonical notions of δ-reduction for typed λ-terms are considered. Typed λ-terms use variables of any order and constants of order ≤ 1, where constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The canonical notions of δ-reduction are the notions of δ-reduction that are used in the implementations of functional programming languages. It is shown that for the main canonical notion of δ-reduction the notion of βδ-reduction has the Church-Rosser property. It is also shown that there exists a canonical notion of δ-reduction such that the notion of βδ-reduction does not have Church-Rosser property.

1991

In Computer Science, Lambda Calculus has been mainly used as the skeleton of functional programming languages. It has also been used as a higher order parameterization mechanism in some specification languages. In this paper we view -calculus as both the applicative structure of a ...

Theory of Computing Systems / Mathematical Systems Theory, 2009

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 cut-elimination 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: normalisation, 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.

Theoretical Computer Science, 1993

A model-theoretic operation is characterised that preserves the property of being a model of typed λ-calculus. (i.e., the result of applying it to a model of typed λ-calculus is another model of typed λ-calculus.) An expression is well-typed iff the class of its models is closed under this operation.

Typed Lambda Calculus and Applications, 2011

We introduce an intersection type assignment system for the pure λµ-calculus, which is invariant under subject reduction and expansion. The system is obtained by describing Streicher and Reus's denotational model of continuations in the category of ω-algebraic lattices via Abramsky's domain logic approach. This provides a tool for showing the completeness of the type assignment system with respect to the continuation models via a filter model construction. We also show that typed λµ-terms in Parigot's system have a non-trivial intersection typing in our system. * Γ, x: Take δ ′ = δ 1 ∧δ 2 ; then by weakening and (→E), we get Γ, x: then, by Lem. 5.3 there exists δ ′′ , κ, ρ such that δ = δ ′′ ×κ→π, Γ, x:δ ′ ⊢ M 1 : κ→ρ | ∆ and Γ, x:δ ′ ⊢ M 2 : δ ′′ | ∆. Then, by induction, we have Γ ⊢ M 1 [L/x] : δ ′′ ×κ→π | ∆ and Γ ⊢ M 2 [L/x] : δ ′′ | ∆; the result follows by (→E).

Rewriting Techniques and …, 2002