scholarly journals Swapping: a natural bridge between named and indexed explicit substitution calculi

2011 ◽  
Vol 49 ◽  
pp. 1-15
Author(s):  
Ariel Mendelzon ◽  
Alejandro Ríos ◽  
Beta Ziliani
Keyword(s):  
Explicit Substitution

Pure type systems with explicit substitution

2001 ◽  
Vol 11 (1) ◽  
pp. 3-19 ◽  
Author(s):  
ROEL BLOO
Keyword(s):  
Type Systems ◽  
Pure Type ◽  
Reduction Property ◽  
Explicit Substitution ◽  
Pure Type Systems

We define an extension of pure type systems with explicit substitution. We show that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker, though still useful, subject reduction property is established. A more complicated extension is proposed for which subject reduction does hold in general.

scholarly journals ON STEPWISE EXPLICIT SUBSTITUTION

1993 ◽  
Vol 04 (03) ◽  
pp. 197-240 ◽  
Author(s):  
FAIROUZ KAMAREDDINE ◽  
ROB NEDERPELT
Keyword(s):  
Lambda Calculus ◽  
Object Language ◽  
Explicit Substitution ◽  
Reduction Strategies ◽  
Partial Unfolding

This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of β-reduction including local and global β-reduction. In other words substitution, thanks to the new notation, can be easily formalised as an object language notion rather than remaining a meta language one. Such formalisation will have advantages with respect to various areas including functional application and the partial unfolding of definitions. Moreover our substitution is, we believe, the most general to date. This is shown by the fact that our framework can accommodate most of the known reduction strategies, which range from local to global. Finally, we show how the calculus of substitution of Abadi et al., can be embedded into our calculus. We show moreover that many of the rules of Abadi et al. are easily derivable in our calculus.

Pure type systems with explicit substitutions

2015 ◽  
Vol 25 ◽  
Author(s):  
DANIEL FRIDLENDER ◽  
MIGUEL PAGANO
Keyword(s):  
Type Systems ◽  
Pure Type ◽  
Explicit Substitution ◽  
Explicit Substitutions ◽  
Pure Type Systems ◽  
De Bruijn ◽  
New Formulation ◽  
De Bruijn Indices ◽  
Meta Theory

AbstractWe introduce a new formulation of pure type systems (PTSs) with explicit substitution and de Bruijn indices and formally prove some of its meta-theory. Using techniques based on Normalisation by Evaluation, we prove that untyped conversion can be typed for predicative PTSs. Although this equivalence was settled by Siles and Herbelin for the conventional presentation of PTSs, we strongly conjecture that our proof method can also be applied to PTSs with η.

THE SOUNDNESS OF EXPLICIT SUBSTITUTION WITH NAMELESS VARIABLES

1998 ◽  
Vol 09 (03) ◽  
pp. 321-349
Author(s):  
FAIROUZ KAMAREDDINE
Keyword(s):  
Classical Calculus ◽  
Explicit Substitution ◽  
De Bruijn ◽  
De Bruijn Indices

We show the soundness of a λ-calculus ℬ where de Bruijn indices are used, substitution is explicit, and reduction is step-wise. This is done by interpreting ℬ in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and step-wise reduction and the first clear account of exactly when α-reduction is needed.

scholarly journals Modeling, Sharing, and Recursion for Weak Reduction Strategies using Explicit Substitution

1996 ◽  
Vol 3 (56) ◽  
Author(s):  
Zine-El-Abidine Benaissa ◽  
Pierre Lescanne ◽  
Kristoffer H. Rose
Keyword(s):  
Programming Languages ◽  
Data Structures ◽  
Functional Programming ◽  
Explicit Substitution ◽  
Reduction Strategies ◽  
Recursive Data Structures ◽  
Recursive Data ◽  
Parallel Evaluation ◽  
Major Reduction ◽  
Weak Reduction

<p>We present the lambda sigma^a_w calculus, a formal synthesis of the concepts of<br />sharing and explicit substitution for weak reduction. We show how<br />lambda sigma^a_w can be used as a foundation of implementations of functional<br />programming languages by modelling the essential ingredients of such<br />implementations, namely weak reduction strategies, recursion, space<br />leaks, recursive data structures, and parallel evaluation, in a uniform way.<br />First, we give a precise account of the major reduction strategies<br />used in functional programming and the consequences of choosing <br /> lambda-graph-reduction vs. environment-based evaluation. Second, we show<br />how to add constructors and explicit recursion to give a precise account<br />of recursive functions and data structures even with respect to<br />space complexity. Third, we formalize the notion of space leaks in lambda sigma^a_w<br />and use this to define a space leak free calculus; this suggests optimisations<br />for call-by-need reduction that prevent space leaking and enables<br />us to prove that the "trimming" performed by the STG machine does<br />not leak space.<br />In summary we give a formal account of several implementation<br />techniques used by state of the art implementations of functional programming<br />languages.</p><p>Keywords. Implementation of functional programming, lambda<br />calculus, weak reduction, explicit substitution, sharing, recursion, space<br />leaks.</p>

scholarly journals A λ-calculus with explicit weakening and explicit substitution

2001 ◽  
Vol 11 (1) ◽  
pp. 169-206 ◽  
Author(s):  
RENÉ DAVID ◽  
BRUNO GUILLAUME
Keyword(s):  
Strong Normalization ◽  
Explicit Substitution ◽  
Explicit Substitutions ◽  
De Bruijn ◽  
De Bruijn Indices

Since Melliès showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented.

scholarly journals Explicit substitution on the edge of strong normalization

1999 ◽  
Vol 211 (1-2) ◽  
pp. 375-395 ◽  
Author(s):  
Roel Bloo ◽  
Herman Geuvers
Keyword(s):  
Strong Normalization ◽  
Explicit Substitution
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