scholarly journals An Adequate Left-Associated Binary Numeral System in the lambda-Calculus (Revised Version)

1996 ◽  
Vol 3 (6) ◽  
Author(s):  
Mayer Goldberg
Keyword(s):  
Functional Programming ◽  
Lambda Calculus ◽  
Successor Function ◽  
Binary Expansion ◽  
Programming Calculi

<p>This paper introduces a sequence of lambda-expressions modelling the binary expansion of integers. We derive expressions computing the test for zero, the successor function, and the predecessor function, thereby showing the sequence to be an adequate numeral system. These functions can be computed efficiently. Their complexity is independent of the order of evaluation.</p><p><br />Keywords: Programming calculi, lambda-calculus, functional programming.</p>

scholarly journals An Adequate Left-Associated Binary Numeral System in the lambda-Calculus

1995 ◽  
Vol 2 (42) ◽  
Author(s):  
Mayer Goldberg
Keyword(s):  
Functional Programming ◽  
Lambda Calculus ◽  
Successor Function ◽  
Programming Calculi

<p>This paper introduces a sequence of lambda-expressions, modelling the binary<br />expansion of integers. We derive expressions computing the test<br />for zero, the successor function, and the predecessor function, thereby<br />showing the sequence to be an adequate numeral system. These functions<br />can be computed efficiently. Their complexity is independent of<br />the order of evaluation.</p><p><br />Keywords: Programming calculi, lambda-calculus, functional programming.</p>

scholarly journals EDITORIAL

1998 ◽  
Vol 8 (4) ◽  
pp. 319-321 ◽  
Author(s):  
SIMON PEYTON JONES ◽  
PHIL WADLER
Keyword(s):  
Functional Programming ◽  
Lambda Calculus ◽  
Language Design ◽  
Functional Languages ◽  
Functional Language ◽  
The Past ◽  
Semantic Models ◽  
Compiled Code ◽  
Design Speed ◽  
Academic Content

In the end, research on functional languages does little good unless they are used to write something other than compilers for functional languages. However, if one scans a typical functional programming conference or journal, one mainly sees papers on twists in language design, speed-ups in compiled code, clever new analyses, or refinements to semantic models. It much less common to see a paper that considers a functional language as a tool to some other practical end. We would like to see this change.The Journal of Functional Programming carries, and will continue to carry, articles on all aspects of functional programming from lambda calculus theory to language design to implementation. But we have specially sought, and will continue to seek, papers on functional programming practice and experience.Research and papers on practice and experience sometimes receive less attention because they are perceived as possessing less academic content. So we want to remind potential authors that we have published a number of papers on this topic in the past, and to spell out the criteria we apply to such papers.

scholarly journals Parametric polymorphism and operational equivalence

2000 ◽  
Vol 10 (3) ◽  
pp. 321-359 ◽  
Author(s):  
ANDREW M. PITTS
Keyword(s):  
Programming Language ◽  
Functional Programming ◽  
Closure Operator ◽  
Lambda Calculus ◽  
Logical Relation ◽  
Data Types ◽  
Mathematical Properties ◽  
Parametric Polymorphism ◽  
Operational Behaviour

Studies of the mathematical properties of impredicative polymorphic types have for the most part focused on the polymorphic lambda calculus of Girard–Reynolds, which is a calculus of total polymorphic functions. This paper considers polymorphic types from a functional programming perspective, where the partialness arising from the presence of fixpoint recursion complicates the nature of potentially infinite (‘lazy’) data types. An approach to Reynolds' notion of relational parametricity is developed that works directly on the syntax of a programming language, using a novel closure operator to relate operational behaviour to parametricity properties of types. Working with an extension of Plotkin's PCF with ∀-types, lazy lists and existential types, we show by example how the resulting logical relation can be used to prove properties of polymorphic types up to operational equivalence.

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.

Linearization of the lambda-calculus and its relation with intersection type systems

2004 ◽  
Vol 14 (5) ◽  
pp. 519-546 ◽  
Author(s):  
MÁRIO FLORIDO ◽  
LUÍS DAMAS
Keyword(s):  
Programming Languages ◽  
Functional Programming ◽  
Type System ◽  
Lambda Calculus ◽  
Type Systems ◽  
Functional Programming Languages

In this paper we present a notion of expansion of a term in the lambda-calculus which transforms terms into linear terms. This transformation replaces each occurrence of a variable in the original term by a fresh variable taking into account non-trivial implications in the structure of the term caused by these simple replacements. We prove that the class of terms which can be expanded is the same of terms typable in an Intersection Type System, i.e. the strongly normalizable terms. We then show that expansion is preserved by weak-head reduction, the reduction considered by functional programming languages.

Sign in / Sign up

Export Citation Format

Share Document