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The Difference Lambda-Calculus: A Language for Difference Categories
[article]

2020
*
arXiv
*
pre-print

In this work, we construct

arXiv:2011.14476v1
fatcat:m5jt5is6svgkll2xtibxsqzkny
*a*simply-typed calculus in*the*spirit of*the*differential*lambda*-*calculus*equipped with syntactic infinitesimals and show how its models correspond to*difference*lambda-*categories*... Cartesian*difference**categories*are*a*recent generalisation of Cartesian differential*categories*which introduce*a*notion of "infinitesimal" arrows satisfying an analogue of*the*Kock-Lawvere axiom, with ... ]) • x, u , v, 0*Difference*λ-*Categories*In order to give*a*semantics*for**the*differential λ-calculus, it does not suffice to ask*for**a*Cartesian differential*category*equipped with exponentials -*the*...##
###
The Difference λ-Calculus: A Language for Difference Categories

2020
*
International Conference on Formal Structures for Computation and Deduction
*

Cartesian

doi:10.4230/lipics.fscd.2020.32
dblp:conf/fscd/Alvarez-Picallo20
fatcat:evasmukchveobbwbip2tdqbo5e
*difference**categories*are*a*recent generalisation of Cartesian differential*categories*which introduce*a*notion of "infinitesimal" arrows satisfying an analogue of*the*Kock-Lawvere axiom, with ... In this work, we construct*a*simply-typed calculus in*the*spirit of*the*differential λ-calculus equipped with syntactic "infinitesimals" and show how its models correspond to*difference*λ-*categories*,*a*... Linear-algebraic*lambda*-*calculus*: higher-order, encodings and confluence in*A*. Voronkov, Rewriting techniques and applications. ...##
###
Lambda-calculus schemata

1993
*
LISP and Symbolic Computation
*

*A*

*lambda*-

*calculus*schema is an expression of

*the*

*lambda*

*calculus*augmented by uninterpreted constant and operator symbols. ... When given an interpretation

*for*its constant and operator symbols, certain schemata, called lambda abstractions, naturally define partial functions over

*the*domain of interpretation. ... We thank Carolyn Talcott

*for*several helpful comments and

*for*pointing out

*a*serious error in an earlier draft, and we thank an anonymous referee

*for*several helpful suggestions. ...

##
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The dagger lambda calculus

2014
*
Electronic Proceedings in Theoretical Computer Science
*

Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and

doi:10.4204/eptcs.172.15
fatcat:efqb72jntje2xljatkehfqlmd4
*the**language*is shown to be an internal*language**for*dagger compact*categories*. ... Our construction extends*the*linear typed*lambda**calculus*with*a*linear negation of "trivialised" De Morgan duality. ... Acknowledgements I would like to thank Samson Abramsky, Bob Coecke, Prakash Panangaden, Jonathan Barrett, and*the*anonymous reviewers*for*their invaluable comments and insights. ...##
###
Lambda calculus schemata

1972
*
SIGPLAN notices
*

*A*

*lambda*-

*calculus*schema is an expression of

*the*

*lambda*

*calculus*augmented by uninterpreted constant and operator symbols. ... When given an interpretation

*for*its constant and operator symbols, certain schemata, called lambda abstractions, naturally define partial functions over

*the*domain of interpretation. ... We thank Carolyn Talcott

*for*several helpful comments and

*for*pointing out

*a*serious error in an earlier draft, and we thank an anonymous referee

*for*several helpful suggestions. ...

##
###
The lambda calculus is algebraic

2002
*
Journal of functional programming
*

This paper serves as

doi:10.1017/s0956796801004294
fatcat:n34nmya5tfbbjddyepeomcwgjq
*a*self-contained, tutorial introduction to combinatory models of*the*untyped*lambda**calculus*. We focus particularly on*the*interpretation of free variables. ... In particular, it solves*the*problem of*the*notorious ξ-rule, which asserts that equations should be preserved under binders, and which fails to be sound*for**the*usual interpretation. ... Acknowledgments I would like to thank*the*three anonymous referees*for*their valuable suggestions. ...##
###
Mikrokosmos: an educational lambda calculus interpreter

2018
*
Journal of Open Source Education
*

*The*author would like to thank his advisor

*for*this grant program, professor Pedro

*A*. García-Sánchez,

*for*his patient supervision, advice and encouragement. ... , Cultura y Deporte at

*the*Departamento de Álgebra of

*the*Universidad de Granada. ...

*For*

*the*untyped

*lambda*-

*calculus*, Mikrokosmos works as

*a*Turingcomplete programming

*language*. ...

##
###
Arrays and The Lambda Calculus
[chapter]

1991
*
Arrays, Functional Languages, and Parallel Systems
*

Much further work is necessary to realize

doi:10.1007/978-1-4615-4002-1_1
fatcat:x3ilmdqjxfhbnernbxp63ekob4
*the*potential power of*the**lambda**calculus*as*a*machine*language**for*actual use. ... But what is*the*higher level*language*with respect to*the**lambda**calculus*, which is in itself*a*higher level*language*? ...##
###
Lineal: A linear-algebraic Lambda-calculus
[article]

2017
*
arXiv
*
pre-print

This

arXiv:quant-ph/0612199v5
fatcat:f2gtq7b63bhcpnakhm5rde4764
*language*extends*the**Lambda*-*calculus*with*the*possibility to make arbitrary linear combinations of terms alpha.t + beta.u. ... We describe how to "execute" this*language*in terms of*a*few rewrite rules, and justify them through*the*two fundamental requirements that*the**language*be*a**language*of linear operators, and that it be ... Acknowledgments*The*authors would like to thank Alejandro Díaz-Caro, Evelyne Contejean, Philippe Jorrand, Jean-Pierre Jouannaud, Claude Marché, Simon Perdrix, Benoît Valiron and Lionel Vaux*for*some enlightening ...##
###
A Lambda Calculus with Forms
[chapter]

2005
*
Lecture Notes in Computer Science
*

More precisely, we study

doi:10.1007/11550679_7
fatcat:hmrkjlea3vd7dbi44uhcgsow6m
*a*substitution-free variant of*the**lambda*-*calculus*, called λF , where names are replaced with forms and parameter passing is modeled using explicit contexts and show how*the*λF ... In this paper, we develop*a*formal form-based framework*for**the*definition of contractual specifications. ...*The*author thanks Jean-Guy Schneider, Oscar Nierstrasz, and*the*anonymous reviewers*for*their valuable comments and discussions. ...##
###
Semantics of a Typed Algebraic Lambda-Calculus

2010
*
Electronic Proceedings in Theoretical Computer Science
*

In this paper we propose

doi:10.4204/eptcs.26.14
fatcat:vvlq5y2tozbgvblqnzf3jglaaq
*a*semantic analysis of*a*general simply-typed*lambda*-*calculus*endowed with*a*structure of vector space. We sketch*the*relation with two established vectorial lambda-calculi. ... Then we study*the*problems arising from*the*addition of*a*fixed point combinator and how to modify*the*equational theory to solve them. ... Acknowledgments I would like to thank Gilles Dowek*for*introducing me to algebraic calculi. I would also like to thank Pablo Arrighi and*the*research group CAPP in Grenoble*for*helpful discussions. ...##
###
Full abstraction for the quantum lambda-calculus

2019
*
Proceedings of the ACM on Programming Languages (PACMPL)
*

Our first contribution is

doi:10.1145/3371131
fatcat:fqe6ykkygvaqjlzioee6qmh264
*a*full abstraction result*for**the*games model of*the*quantum λ-calculus. ... In particular,*the*quantum λ-calculus is*a*higher-order*language*with quantum primitives, mixing quantum data and classical control. ... ACKNOWLEDGMENTS We are grateful to*the*anonymous reviewers*for*their helpful comments and suggestions. ...##
###
The differential lambda-calculus

2003
*
Theoretical Computer Science
*

We state and prove some basic results (con uence, strong normalization in

doi:10.1016/s0304-3975(03)00392-x
fatcat:4cbn3lmgnjefhcp7ogdau22ml4
*the*typed case), and also*a*theorem relating*the*usual Taylor series of analysis to*the*linear head reduction of*lambda*-*calculus*... We present an extension of*the**lambda*-*calculus*with di erential constructions. ... aspects of*the*di erential*lambda*-*calculus*. ...##
###
Intersection Types for the Computational lambda-Calculus
[article]

2020
*
arXiv
*
pre-print

Moving from

arXiv:1907.05706v3
fatcat:bqtmqcmayfhjthbiysjbfuecqi
*the*abstract definition of monads, we introduce*a*version of*the*call-by-value computational*lambda*-*calculus*based on Wadler's variant with unit and bind combinators, and without let. ... We then introduce an intersection type system inspired to Barendregt, Coppo and Dezani system*for*ordinary untyped*lambda*-*calculus*, establishing type invariance under conversion, and provide models of ...*The*filter model we have obtained is*for**a*generic monad; we conjecture that*the*model is initial in*a*suitable*category*, to which all its instances belong. ...##
###
A typed, algebraic, computational lambda-calculus

2013
*
Mathematical Structures in Computer Science
*

We first develop

doi:10.1017/s0960129512000205
fatcat:kigcwq3rhzhgrg5to247jyw3y4
*a*categorical analysis of*a*general simply-typed*lambda*-*calculus*endowed with*a*structure of module. ...*The*categorical analysis carries nicely over to*the*modified*language*. We provide various concrete models, both*for**the*case without fixpoints and*the*case with fixpoints. ... making*the**language*fall back on*a*classical*lambda*-*calculus*with pointers. ...
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