scholarly journals Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices

2010 ◽  
Vol 15 ◽  
pp. 69-82 ◽  
Author(s):  
Daniel Ventura ◽  
Mauricio Ayala-Rincón ◽  
Fairouz Kamareddine
Keyword(s):  
Normal Forms ◽  
Type System ◽  
De Bruijn ◽  
De Bruijn Indices

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 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

scholarly journals de Bruijn notation as a nested datatype

1999 ◽  
Vol 9 (1) ◽  
pp. 77-91 ◽  
Author(s):  
RICHARD S. BIRD ◽  
ROSS PATERSON
Keyword(s):  
Natural Number ◽  
Type System ◽  
Sir Arthur Conan Doyle ◽  
Sherlock Holmes ◽  
Arthur Conan Doyle ◽  
De Bruijn ◽  
Definition Of ◽  
System Programming

“I have no data yet. It is a capital mistake to theorise before one has data.” Sir Arthur Conan Doyle The Adventures of Sherlock Holmesde Bruijn notation is a coding of lambda terms in which each occurrence of a bound variable x is replaced by a natural number, indicating the ‘distance’ from the occurrence to the abstraction that introduced x. One might suppose that in any datatype for representing de Bruijn terms, the distance restriction on numbers would have to be maintained as an explicit datatype invariant. However, by using a nested (or non-regular) datatype, we can define a representation in which all terms are well-formed, so that the invariant is enforced automatically by the type system. Programming with nested types is only a little more difficult than programming with regular types, provided we stick to well-established structuring techniques. These involve expressing inductively defined functions in terms of an appropriate fold function for the type, and using fusion laws to establish their properties. In particular, the definition of lambda abstraction and beta reduction is particularly simple, and the proof of their associated properties is entirely mechanical.

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.

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.

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 Counting and generating lambda terms

2013 ◽  
Vol 23 (5) ◽  
pp. 594-628 ◽  
Author(s):  
KATARZYNA GRYGIEL ◽  
PIERRE LESCANNE
Keyword(s):  
Functional Programming ◽  
Complex Analysis ◽  
Lambda Calculus ◽  
Typical Structure ◽  
Combinatorial Properties ◽  
Combinatorial Interpretations ◽  
Random Generators ◽  
De Bruijn ◽  
The Church ◽  
De Bruijn Indices

AbstractLambda calculus is the basis of functional programming and higher order proof assistants. However, little is known about combinatorial properties of lambda terms, in particular, about their asymptotic distribution and random generation. This paper tries to answer questions like: How many terms of a given size are there? What is a ‘typical’ structure of a simply typable term? Despite their ostensible simplicity, these questions still remain unanswered, whereas solutions to such problems are essential for testing compilers and optimizing programs whose expected efficiency depends on the size of terms. Our approach toward the aforementioned problems may be later extended to any language with bound variables, i.e., with scopes and declarations. This paper presents two complementary approaches: one, theoretical, uses complex analysis and generating functions, the other, experimental, is based on a generator of lambda terms. Thanks to de Bruijn indices (de Bruijn, N. (1972) Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagat. Math.34(5), 381–392), we provide three families of formulas for the number of closed lambda terms of a given size and we give four relations between these numbers which have interesting combinatorial interpretations. As a by-product of the counting formulas, we design an algorithm for generating λ-terms. Performed tests provide us with experimental data, like the average depth of bound variables and the average number of head lambdas. We also create random generators for various sorts of terms. Thereafter, we conduct experiments that answer questions like: What is the ratio of simply typable terms among all terms? (Very small!) How are simply typable lambda terms distributed among all lambda terms? (A typable term almost always starts with an abstraction.) In this paper, abstractions and applications have size 1 and variables have size 0.

scholarly journals Exploring Neural Methods for Parsing Discourse Representation Structures

2018 ◽  
Vol 6 ◽  
pp. 619-633 ◽  
Author(s):  
Rik van Noord ◽  
Lasha Abzianidze ◽  
Antonio Toral ◽  
Johan Bos
Keyword(s):  
Formal Semantics ◽  
High Accuracy ◽  
Training Data ◽  
Semantic Parsing ◽  
Discourse Representation ◽  
De Bruijn ◽  
Better Than ◽  
De Bruijn Indices

Neural methods have had several recent successes in semantic parsing, though they have yet to face the challenge of producing meaning representations based on formal semantics. We present a sequence-to-sequence neural semantic parser that is able to produce Discourse Representation Structures (DRSs) for English sentences with high accuracy, outperforming traditional DRS parsers. To facilitate the learning of the output, we represent DRSs as a sequence of flat clauses and introduce a method to verify that produced DRSs are well-formed and interpretable. We compare models using characters and words as input and see (somewhat surprisingly) that the former performs better than the latter. We show that eliminating variable names from the output using De Bruijn indices increases parser performance. Adding silver training data boosts performance even further.

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