scholarly journals Lambda calculus with algebraic simplification for reduction parallelisation: Extended study

2021 ◽  
Vol 31 ◽  
Author(s):  
AKIMASA MORIHATA
Keyword(s):  
Parallel Programming ◽  
Type System ◽  
Lambda Calculus ◽  
Extended Study ◽  
Equational Reasoning ◽  
Typed Lambda Calculus ◽  
Parallel Reduction ◽  
Algebraic Properties ◽  
Algebraic Simplification

Abstract Parallel reduction is a major component of parallel programming and widely used for summarisation and aggregation. It is not well understood, however, what sorts of non-trivial summarisations can be implemented as parallel reductions. This paper develops a calculus named λAS, a simply typed lambda calculus with algebraic simplification. This calculus provides a foundation for studying a parallelisation of complex reductions by equational reasoning. Its key feature is δ abstraction. A δ abstraction is observationally equivalent to the standard λ abstraction, but its body is simplified before the arrival of its arguments using algebraic properties such as associativity and commutativity. In addition, the type system of λAS guarantees that simplifications due to δ abstractions do not lead to serious overheads. The usefulness of λAS is demonstrated on examples of developing complex parallel reductions, including those containing more than one reduction operator, loops with conditional jumps, prefix sum patterns and even tree manipulations.

Operations on records

1991 ◽  
Vol 1 (1) ◽  
pp. 3-48 ◽  
Author(s):  
Luca Cardelli ◽  
John C. Mitchell
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Object Oriented ◽  
Order Type ◽  
Second Order ◽  
Typed Lambda Calculus ◽  
Semantic Models ◽  
Object Oriented Languages

We define a simple collection of operations for creating and manipulating record structures, where records are intended as finite associations of values to labels. A second-order type system over these operations supports both subtyping and polymorphism. We provide typechecking algorithms and limited semantic models.Our approach unifies and extends previous notions of records, bounded quantification, record extension, and parametrization by row-variables. The general aim is to provide foundations for concepts found in object-oriented languages, within a framework based on typed lambda-calculus.

scholarly journals A Complete Calculus of Monotone and Antitone Higher-Order Functions

10.29007/3n54 ◽  
2018 ◽  
Author(s):  
Thomas Icard ◽  
Lawrence Moss
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Higher Order ◽  
Typed Lambda Calculus ◽  
Higher Order Functions

This paper adds monotonicity and antitonicity information to the typed lambda calculus, thereby providing a foundation for the Monotonicity Calculus first developed by van Benthem and others. We establish properties of the type system, propose a syntax, semantics, and proof calculus, and prove completeness for the calculus with respect to hierarchies of monotone and antitone functions over base preorders.

Type safe incremental rebinding

2015 ◽  
Vol 27 (2) ◽  
pp. 94-122 ◽  
Author(s):  
DAVIDE ANCONA ◽  
PAOLA GIANNINI ◽  
ELENA ZUCCA
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Typed Lambda Calculus ◽  
Free Variables ◽  
Open Code

We extend the simply-typed lambda-calculus with a mechanism for dynamic and incremental rebinding of code. Fragments of open code which can be dynamically rebound are values. Differently from standard static binding, which is done on a positional basis, rebinding is done on a nominal basis, that is, free variables in open code are associated with names which do not obey α-equivalence. Moreover, rebinding is incremental, that is, just a subset of names can be rebound, making possible code specialization, and rebinding can even introduce new names. Finally, rebindings, which are associations between names and terms, are first-class values, and can be manipulated by operators such as overriding and renaming. We define a type system in which the type for a rebinding, in addition to specify an association between names and types (similarly to record types), is also annotated. The annotation says whether or not the domain of the rebinding having this type may contain more names than the ones that are specified in the type. We show soundness of the type system.

A general method for proving the normalization theorem for first and second order typed λ-calculi

1999 ◽  
Vol 9 (6) ◽  
pp. 719-739 ◽  
Author(s):  
VENANZIO CAPRETTA ◽  
SILVIO VALENTINI
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Second Order ◽  
Typed Lambda Calculus ◽  
Normalization Theorem ◽  
System F ◽  
Dependent Type ◽  
General Method

In this paper we describe a method for proving the normalization property for a large variety of typed lambda calculi of first and second order, which is based on a proof of equivalence of two deduction systems. We first illustrate the method on the elementary example of simply typed lambda calculus, and then we show how to extend it to a more expressive dependent type system. Finally we use it to prove the normalization theorem for Girard's system F.

scholarly journals Reduction Strategies for Program Extraction

2018 ◽  
Vol 6 (1) ◽  
Author(s):  
Maribel Fernandez ◽  
Ian Mackie ◽  
Paula Severi ◽  
Nora Szasz
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
General Setting ◽  
Type Systems ◽  
Program Extraction ◽  
Pure Type ◽  
Typed Lambda Calculus ◽  
Reduction Strategies ◽  
Pure Type Systems ◽  
Reduction Relation

We introduce Pure Type Systems with Pairs generalising earlier work on program extraction in Typed Lambda Calculus. We model the process of program extraction in these systems by means of a reduction relation called o-reduction, and give strategies for Bo-reduction which will be useful for an implementation of a proof assistant. More precisely, we give an algorithm to compute theo-normal form of a term in Pure Type System with Pairs, and show that this defines a prejection from Pure Type Systems with Pairs to standart Pure Type Systems. This result shows that o-reduction is an operational description of aprgram extraction that is independent of the particular Typed Lambda Calculus specified as a Pure Typoe System. For B-reduction, we define weak and strong reduction strategies using Interaction Nets, generalising well-know efficient strategies for the l-calculus to the general setting of Pure Type Systems.

scholarly journals Backpropagation in the simply typed lambda-calculus with linear negation

2020 ◽  
Vol 4 (POPL) ◽  
pp. 1-27 ◽  
Author(s):  
Aloïs Brunel ◽  
Damiano Mazza ◽  
Michele Pagani
Keyword(s):  
Lambda Calculus ◽  
Typed Lambda Calculus

The Simply Typed Lambda Calculus

2013 ◽  
pp. 5-54
Author(s):  
Henk Barendregt ◽  
Wil Dekkers ◽  
Richard Statman
Keyword(s):  
Lambda Calculus ◽  
Typed Lambda Calculus
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