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

Mario Alvarez-Picallo, C.-H. Luke Ong
2020 arXiv   pre-print
In this work, we construct 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  ... 
arXiv:2011.14476v1 fatcat:m5jt5is6svgkll2xtibxsqzkny

The Difference λ-Calculus: A Language for Difference Categories

Mario Alvarez-Picallo, C.-H. Luke Ong, Zena M. Ariola
2020 International Conference on Formal Structures for Computation and Deduction  
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  ...  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.  ... 
doi:10.4230/lipics.fscd.2020.32 dblp:conf/fscd/Alvarez-Picallo20 fatcat:evasmukchveobbwbip2tdqbo5e

Lambda-calculus schemata

Michael J. Fischer
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.  ... 
doi:10.1007/bf01019461 fatcat:z36lhsq5gjcqbplghyagyvrhzy

The dagger lambda calculus

Philip Atzemoglou
2014 Electronic Proceedings in Theoretical Computer Science  
Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and 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.  ... 
doi:10.4204/eptcs.172.15 fatcat:efqb72jntje2xljatkehfqlmd4

Lambda calculus schemata

Michael J. Fischer
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.  ... 
doi:10.1145/942578.807077 fatcat:avd3myyi6rdahm5xowg3pujoiq

The lambda calculus is algebraic

PETER SELINGER
2002 Journal of functional programming  
This paper serves as 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.  ... 
doi:10.1017/s0956796801004294 fatcat:n34nmya5tfbbjddyepeomcwgjq

Mikrokosmos: an educational lambda calculus interpreter

Mario Román
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.  ... 
doi:10.21105/jose.00029 fatcat:vquganejs5cepgvbs4qynbdb6a

Arrays and The Lambda Calculus [chapter]

Klaus Berkling
1991 Arrays, Functional Languages, and Parallel Systems  
Much further work is necessary to realize 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?  ... 
doi:10.1007/978-1-4615-4002-1_1 fatcat:x3ilmdqjxfhbnernbxp63ekob4

Lineal: A linear-algebraic Lambda-calculus [article]

Pablo Arrighi, Gilles Dowek
2017 arXiv   pre-print
This 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  ... 
arXiv:quant-ph/0612199v5 fatcat:f2gtq7b63bhcpnakhm5rde4764

A Lambda Calculus with Forms [chapter]

Markus Lumpe
2005 Lecture Notes in Computer Science  
More precisely, we study 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.  ... 
doi:10.1007/11550679_7 fatcat:hmrkjlea3vd7dbi44uhcgsow6m

Semantics of a Typed Algebraic Lambda-Calculus

Benoît Valiron
2010 Electronic Proceedings in Theoretical Computer Science  
In this paper we propose 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.  ... 
doi:10.4204/eptcs.26.14 fatcat:vvlq5y2tozbgvblqnzf3jglaaq

Full abstraction for the quantum lambda-calculus

Pierre Clairambault, Marc de Visme
2019 Proceedings of the ACM on Programming Languages (PACMPL)  
Our first contribution is 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.  ... 
doi:10.1145/3371131 fatcat:fqe6ykkygvaqjlzioee6qmh264

The differential lambda-calculus

Thomas Ehrhard, Laurent Regnier
2003 Theoretical Computer Science  
We state and prove some basic results (con uence, strong normalization in 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.  ... 
doi:10.1016/s0304-3975(03)00392-x fatcat:4cbn3lmgnjefhcp7ogdau22ml4

Intersection Types for the Computational lambda-Calculus [article]

Ugo de'Liguoro, Riccardo Treglia
2020 arXiv   pre-print
Moving from 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.  ... 
arXiv:1907.05706v3 fatcat:bqtmqcmayfhjthbiysjbfuecqi

A typed, algebraic, computational lambda-calculus

BENOÎT VALIRON
2013 Mathematical Structures in Computer Science  
We first develop 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.  ... 
doi:10.1017/s0960129512000205 fatcat:kigcwq3rhzhgrg5to247jyw3y4
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