scholarly journals Lambda calculus with algebraic simplification for reduction parallelization by equational reasoning

2019 ◽  
Vol 3 (ICFP) ◽  
pp. 1-25
Author(s):  
Akimasa Morihata
Keyword(s):  
Lambda Calculus ◽  
Equational Reasoning ◽  
Algebraic Simplification

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.

Lambda Calculus with Types

2009 ◽  
Author(s):  
Henk Barendregt ◽  
Wil Dekkers ◽  
Richard Statman
Keyword(s):  
Lambda Calculus

A lambda calculus with naive substitution

1979 ◽  
Vol 28 (3) ◽  
pp. 269-282 ◽  
Author(s):  
John Staples
Keyword(s):  
Lambda Calculus ◽  
Classical Case ◽  
Syntactic Theory ◽  
Alternative Approach ◽  
New Formulation

AbstractAn alternative approach is proposed to the basic definitions of the lassical lambda calculus. A proof is sketched of the equivalence of the approach with the classical case. The new formulation simplifies some aspects of the syntactic theory of the lambda calculus. In particular it provides a justification for omitting in syntactic theory discussion of changes of bound variable.

scholarly journals Reasoning with the HERMIT: tool support for equational reasoning on GHC core programs

2016 ◽  
Vol 50 (12) ◽  
pp. 23-34 ◽  
Author(s):  
Andrew Farmer ◽  
Neil Sculthorpe ◽  
Andy Gill
Keyword(s):  
Tool Support ◽  
Equational Reasoning

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

scholarly journals Algebras and coalgebras in the light affine Lambda calculus

2015 ◽  
Vol 50 (9) ◽  
pp. 114-126 ◽  
Author(s):  
Marco Gaboardi ◽  
Romain Péchoux
Keyword(s):  
Lambda Calculus

scholarly journals The Limits of Computation

Axiomathes ◽  
2021 ◽  
Author(s):  
Andrew Powell
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Lambda Calculus ◽  
Second Order ◽  
Functional Interpretation ◽  
Arbitrary Type ◽  
Computable Functions

AbstractThis article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions.

scholarly journals Maximal sharing in the Lambda calculus with letrec

2014 ◽  
Vol 49 (9) ◽  
pp. 67-80 ◽  
Author(s):  
Clemens Grabmayer ◽  
Jan Rochel
Keyword(s):  
Lambda Calculus

Embedding a full linear Lambda calculus in Haskell

2016 ◽  
Vol 50 (12) ◽  
pp. 177-188
Author(s):  
Jeff Polakow
Keyword(s):  
Lambda Calculus
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