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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

Scott Is Always Simple [chapter]

Antonino Salibra
2012 Lecture Notes in Computer Science  
In this paper we give an outline of recent algebraic results concerning theories and models of the untyped lambda calculus.  ...  Indeed, a λ-theory may correspond to a possible operational semantics of lambda calculus, as well as it may be induced by a model of lambda calculus through the kernel congruence relation of the interpretation  ...  The connection between the syntax and the semantics of lambda calculus is established by the completeness theorem of lambda calculus: every λ-theory is the equational theory of some λ-model.  ... 
doi:10.1007/978-3-642-32589-2_3 fatcat:jqadnhzb5jcudfucqdvxbirasa

On the algebraic models of lambda calculus

Antonino Salibra
2000 Theoretical Computer Science  
Another result of the paper is an algebraic proof of consistency of the inÿnitary lambda calculus. Finally, some algebraic constructions by Krivine are generalized to lambda abstraction algebras.  ...  The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus.  ...  Acknowledgements The author is grateful to the referees for many helpful suggestions and for pointing out numerous places where the text in the original version could be improved.  ... 
doi:10.1016/s0304-3975(00)00059-1 fatcat:szd4aecnmnb5tbpkv4gynyt52u

Easiness in graph models

C. Berline, A. Salibra
2006 Theoretical Computer Science  
In this paper we concentrate on the semantics G of lambda calculus given in terms of graph models, graph semantics for short.  ...  These models, isolated in the seventies by Plotkin et al. [37] within the continuous semantics, have proved useful for giving proofs of consistency of extensions of lambda calculus and for studying operational  ...  The equations defining L, which make the lambda calculus consistent with the lattice operations of join and meet, are used to define lattice term operations on the term algebra L , the quotient of by the  ... 
doi:10.1016/j.tcs.2005.11.005 fatcat:367kbdjdh5cerfzoybtblycmvi

Page 26 of Mathematical Reviews Vol. , Issue 99a [page]

1991 Mathematical Reviews  
{For the entire collection see MR 98f:68017.} 99a:03020 03B40 68N15 68Q55 Lavatelli, Carolina (F-ENS-I; Paris) Algebraic interpretation of lambda calculus with resources.  ...  The constant ¢ is interpreted by the element of M that corre- sponds to a continuous function that has value T on some (open) subset ® and | on the rest of M.  ... 

A Note on Absolutely Unorderable Combinatory Algebras

S. Lusin
2003 Journal of Logic and Computation  
In this paper we prove that a wide class of combinatory algebras admits extensions with a non-trivial compatible partial order.  ...  Plotkin [16] has conjectured that there exists an absolutely unorderable combinatory algebra, namely a combinatory algebra which cannot be embedded in another combinatory algebra admitting a non-trivial  ...  prevent variables in lambda calculus from operating as real algebraic variables.  ... 
doi:10.1093/logcom/13.4.481 fatcat:fn2nyncmcvdt7e3uue2xdic6gu

Topological incompleteness and order incompleteness of the lambda calculus

Antonino Salibra
2003 ACM Transactions on Computational Logic  
A model of the untyped lambda calculus univocally induces a lambda theory (i.e., a congruence relation on λ-terms closed under α-and β-conversion) through the kernel congruence relation of the interpretation  ...  In this paper we introduce a new technique to prove in a uniform way the incompleteness of all denotational semantics of lambda calculus which have been proposed so far, including the strongly stable one  ...  ACKNOWLEDGMENTS The author wishes to thank Gordon Plotkin and the referees for helpful comments and suggestions.  ... 
doi:10.1145/772062.772067 fatcat:ptvuocqvcbe5nawqmtj2ztltyu

Order-incompleteness and finite lambda reduction models

Peter Selinger
2003 Theoretical Computer Science  
Many familiar models of the untyped lambda calculus are constructed by order-theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus.  ...  Equivalently, the open and closed term algebras of the untyped lambda calculus cannot be non-trivially partially ordered.  ...  Acknowledgements I would like to thank Gordon Plotkin for introducing me to the problem of partial orders on term models, and for many stimulating discussions.  ... 
doi:10.1016/s0304-3975(02)00038-5 fatcat:iklf5xdrtvbtvpklntvijhjycu

Page 2828 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
Author summary: “Church’s lambda-calculus is modified by in- troducing a new mechanism, the lambda-bar operator, which neu- tralizes the effect of one preceding lambda binding.  ...  This paper concerns some shortcomings in E. J. Lemmon’s discus- sion of the interpretation of common modal systems [Aristotelian society proceedings, Suppl.  ... 

Page 5090 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews  
95i:03030 The interest of this article lies in the extension of the result on storage operators to the classical second-order lambda-calculus (in which there is a special constant c: V¥(-=~X — X)).  ...  The main result is that, if a term T is of type Vx(-Int[x] — —Int*[x]), then it is also a storage operator for a term @ of type Int[s”0] in the classical second-order lambda-calculus.  ... 

Page 6384 of Mathematical Reviews Vol. , Issue 91M [page]

1991 Mathematical Reviews  
The author contrasts his result with previously known results for operations on integers and strings and asks whether these can be put together in a common formulation for operations on general free algebraic  ...  In the classical A-calculus the operation of substitution is de- fined by means of the metalanguage.  ... 

Von Neumann Algebras form a Model for the Quantum Lambda Calculus [article]

Kenta Cho, Abraham Westerbaan
2016 arXiv   pre-print
We present a model of Selinger and Valiron's quantum lambda calculus based on von Neumann algebras, and show that the model is adequate with respect to the operational semantics.  ...  Although they gave the definition of concrete models of the quantum lambda calculus, results on them (e.g. how to interpret the quantum lambda calculus; adequacy of models) have never been given.  ...  In our model of the quantum lambda calculus the von Neumann algebras of the form ∞ (X) serve as the interpretation of the duplicable types (of the form !  ... 
arXiv:1603.02133v1 fatcat:nwhasor5mracnf36a4nnze2p4m

What is a model of the lambda calculus?

Albert R. Meyer
1982 Information and Control  
An elementary, purely algebraic definition of model for the untyped lambda calculus is given. This definition is shown to be equivalent to the natural semantic definition based on environments.  ...  These definitions of model are consistent with, and yield a completeness theorem for, the standard axioms for lambda convertibility. A simple construction of models for lambda calculus is reviewed.  ...  In the final section we cite some additional results connecting the model theory and proof theory of lambda calculus.  ... 
doi:10.1016/s0019-9958(82)80087-9 fatcat:ws7l2zql4bg63jvyxe5w5zhx5y

Implementing a computer algebra system in Haskell

José Romildo Malaquias, Carlos Roberto Lopes
2007 Applied Mathematics and Computation  
Mixed computation requires speed and safety that interpreted computer algebra cannot provide.  ...  Symbolic algorithms are mostly written for interpreters in untyped languages. Therefore, symbolic mathematics is usually slow, and bug ridden.  ...  It is possible to identify terms in the lambda calculus, which, when suitably interpreted, behave like the number 2 and like the multiplication operator.  ... 
doi:10.1016/j.amc.2007.02.126 fatcat:mqs5h24cjjdgflgqhi2myy5hfu

Modular domain-specific language components in scala

Christian Hofer, Klaus Ostermann
2010 Proceedings of the ninth international conference on Generative programming and component engineering - GPCE '10  
Programs in domain-specific embedded languages (DSELs) can be represented in the host language in different ways, for instance implicitly as libraries, or explicitly in the form of abstract syntax trees  ...  Traditional designs for DSELs fix the form of representation, which means that it is not possible to choose the best representation for a particular interpretation or transformation.  ...  Acknowledgments The authors would like to thank Adriaan Moors, Michael Achenbach, and the anonymous reviewers for their insightful comments and suggestions that helped improve the presentation of the paper  ... 
doi:10.1145/1868294.1868307 dblp:conf/gpce/HoferO10 fatcat:37fuuoi3wbfchbdn2wamx4dqga
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