Explicit substitution calculi with de Bruijn indices and intersection type systems

2014 ◽  
Vol 23 (2) ◽  
pp. 295-340 ◽  
Author(s):  
D. L. Ventura ◽  
F. Kamareddine ◽  
M. Ayala-Rincon
Keyword(s):  
Type Systems ◽  
Explicit Substitution ◽  
De Bruijn ◽  
De Bruijn Indices

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.

Pure Type Systems with de Bruijn Indices

2002 ◽  
Vol 45 (2) ◽  
pp. 187-201 ◽  
Author(s):  
F. Kamareddine
Keyword(s):  
Type Systems ◽  
Pure Type ◽  
Pure Type Systems ◽  
De Bruijn ◽  
De Bruijn Indices

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 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 Distribution of Variables in Lambda-Terms with Restrictions on De Bruijn Indices and De Bruijn Levels

10.37236/8579 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Bernhard Gittenberger ◽  
Isabella Larcher
Keyword(s):  
The Other ◽  
Bounded Number ◽  
Mean And Variance ◽  
De Bruijn ◽  
De Bruijn Indices ◽  
Strange Phenomenon ◽  
Normally Distributed

We consider two special subclasses of lambda-terms that are restricted by a bound on the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound on the nesting levels of abstractions, i.e., the number of De Bruijn levels, respectively. We show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. For the class of lambda-terms with bounded De Bruijn index we derive closed formulas for the constant. For the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we show quantitative and distributional results on the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called "unary profile" that attains a very interesting shape.  Our results give a combinatorial explanation of an earlier discovered strange phenomenon exhibited by the counting sequence of this particular class of lambda-terms. 

scholarly journals A unified treatment of syntax with binders

2012 ◽  
Vol 22 (4-5) ◽  
pp. 614-704 ◽  
Author(s):  
NICOLAS POUILLARD ◽  
FRANÇOIS POTTIER
Keyword(s):  
Natural Number ◽  
Data Structures ◽  
Transformation Function ◽  
Natural Numbers ◽  
Programming Error ◽  
Free Variables ◽  
De Bruijn ◽  
Abstract Interface ◽  
Representation Techniques ◽  
De Bruijn Indices

AbstractAtoms and de Bruijn indices are two well-known representation techniques for data structures that involve names and binders. However, using either technique, it is all too easy to make a programming error that causes one name to be used where another was intended. We propose an abstract interface to names and binders that rules out many of these errors. This interface is implemented as a library in Agda. It allows defining and manipulating term representations in nominal style and in de Bruijn style. The programmer is not forced to choose between these styles: on the contrary, the library allows using both styles in the same program, if desired. Whereas indexing the types of names and terms with a natural number is a well-known technique to better control the use of de Bruijn indices, we index types with worlds. Worlds are at the same time more precise and more abstract than natural numbers. Via logical relations and parametricity, we are able to demonstrate in what sense our library is safe, and to obtain theorems for free about world-polymorphic functions. For instance, we prove that a world-polymorphic term transformation function must commute with any renaming of the free variables. The proof is entirely carried out in Agda.

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 Proof-relevant π-calculus: a constructive account of concurrency and causality

2017 ◽  
Vol 28 (9) ◽  
pp. 1541-1577 ◽  
Author(s):  
ROLY PERERA ◽  
JAMES CHENEY
Keyword(s):  
Formal Definition ◽  
Meta Data ◽  
Transition Relation ◽  
Π Calculus ◽  
De Bruijn ◽  
Target States ◽  
Definition Of ◽  
De Bruijn Indices ◽  
Proof Terms

We present a formalisation in Agda of the theory of concurrent transitions, residuation and causal equivalence of traces for the π-calculus. Our formalisation employs de Bruijn indices and dependently typed syntax, and aligns the ‘proved transitions’ proposed by Boudol and Castellani in the context of CCS with the proof terms naturally present in Agda's representation of the labelled transition relation. Our main contributions are proofs of the ‘diamond lemma’ for the residuals of concurrent transitions and a formal definition of equivalence of traces up to permutation of transitions.In the π-calculus, transitions represent propagating binders whenever their actions involve bound names. To accommodate these cases, we require a more general diamond lemma where the target states of equivalent traces are no longer identical, but are related by abraidingthat rewires the bound and free names to reflect the particular interleaving of events involving binders. Our approach may be useful for modelling concurrency in other languages where transitions carry meta-data sensitive to particular interleavings, such as dynamically allocated memory addresses.

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