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The lambda calculus is algebraic

2002
*
Journal of functional programming
*

This paper serves as a self-contained, tutorial introduction to combinatory models

doi:10.1017/s0956796801004294
fatcat:n34nmya5tfbbjddyepeomcwgjq
*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. ...##
###
Scott Is Always Simple
[chapter]

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. ...

##
###
On the algebraic models of lambda calculus

2000
*
Theoretical Computer Science
*

Another

doi:10.1016/s0304-3975(00)00059-1
fatcat:szd4aecnmnb5tbpkv4gynyt52u
*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. ...##
###
Easiness in graph models

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*...

##
###
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

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. ...

##
###
Topological incompleteness and order incompleteness of the lambda calculus

2003
*
ACM Transactions on Computational Logic
*

A model

doi:10.1145/772062.772067
fatcat:ptvuocqvcbe5nawqmtj2ztltyu
*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. ...##
###
Order-incompleteness and finite lambda reduction models

2003
*
Theoretical Computer Science
*

Many familiar models

doi:10.1016/s0304-3975(02)00038-5
fatcat:iklf5xdrtvbtvpklntvijhjycu
*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. ...##
###
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]

2016
*
arXiv
*
pre-print

We present a model

arXiv:1603.02133v1
fatcat:nwhasor5mracnf36a4nnze2p4m
*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 ! ...##
###
What is a model of the lambda calculus?

1982
*
Information and Control
*

An elementary, purely

doi:10.1016/s0019-9958(82)80087-9
fatcat:ws7l2zql4bg63jvyxe5w5zhx5y
*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*. ...##
###
Implementing a computer algebra system in Haskell

2007
*
Applied Mathematics and Computation
*

Mixed computation requires speed and safety that

doi:10.1016/j.amc.2007.02.126
fatcat:mqs5h24cjjdgflgqhi2myy5hfu
*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*. ...##
###
Modular domain-specific language components in scala

2010
*
Proceedings of the ninth international conference on Generative programming and component engineering - GPCE '10
*

Programs

doi:10.1145/1868294.1868307
dblp:conf/gpce/HoferO10
fatcat:37fuuoi3wbfchbdn2wamx4dqga
*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 ...
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