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A nonstandard standardization theorem

2014
*
Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14
*

This paper focuses on standardization for

doi:10.1145/2535838.2535886
dblp:conf/popl/AccattoliBKL14
fatcat:gjgrgun6hbf2nguwm5tf3tyr5a
*the*linear substitution*calculus*, a*calculus*with ES capable of mimicking reduction*in**λ**calculus*and linear logic proof-nets. ... Contrary to other extant calculi with ES, our system can be equipped with a residual theory*in**the*sense of Lévy, which is used to prove a left-to-right standardization theorem for*the**calculus*with ES ... This work was partially funded by*the*French-Argentinian Laboratory*in*Computer Science INFINIS (www.infinis.org) and by*the*Qatar National Research Fund under grant NPRP 09-1107-1-168. ...##
###
Weak optimality, and the meaning of sharing

2013
*
SIGPLAN notices
*

sequence

doi:10.1145/2544174.2500606
fatcat:yd7c33g7qfdtvjt6fa3ikcom5m
*in**the*weak*λ*-*calculus*without sharing. ... We consider*the*weak*λ*-*calculus*as a basis of functional programming languages, and we adapt to this setting*the*concepts of optimal reductions that were defined for*the*full*λ*-*calculus*. ... Thanks also to François Pottier, Mike Rainey, and*the*ICFP reviewers for their helpful comments and suggestions. ...##
###
Weak optimality, and the meaning of sharing

2013
*
Proceedings of the 18th ACM SIGPLAN international conference on Functional programming - ICFP '13
*

sequence

doi:10.1145/2500365.2500606
dblp:conf/icfp/Balabonski13
fatcat:c23lqvqvpravxhm3gvx3yiv4zy
*in**the*weak*λ*-*calculus*without sharing. ... We consider*the*weak*λ*-*calculus*as a basis of functional programming languages, and we adapt to this setting*the*concepts of optimal reductions that were defined for*the*full*λ*-*calculus*. ... Thanks also to François Pottier, Mike Rainey, and*the*ICFP reviewers for their helpful comments and suggestions. ...##
###
Meaningless terms in rewriting
[chapter]

1996
*
Lecture Notes in Computer Science
*

For example,

doi:10.1007/3-540-61735-3_17
fatcat:odn4wjrmubcbletpdhtb4toedu
*in*lambda*calculus*, we have reductions such as (*λ*x.xx)(*λ*y.y) → (*λ*y.y)(*λ*y.y),*in*which*the*subterm xx becomes instantiated to (*λ*y.y)(*λ*y.y). ... All*the*axioms hold except for Axiom 1, as*the*nonsimply typable terms*are*clearly not closed under reduction (even*in**the**λ*I-*calculus*). ...*The*remaining depth measures*are*0 * 1. For these we argue thus: if (*λ*x.s)t is reducible to a 0 * 1-*stable*term, then so is s[x := t]. By Axiom 3 (proved below), this implies that s is also. ...##
###
Open Call-by-Value
[chapter]

2016
*
Lecture Notes in Computer Science
*

*The*elegant theory of

*the*call-by-value lambda-

*calculus*relies on weak evaluation and closed terms, that

*are*natural hypotheses

*in*

*the*study of programming languages. ... To model proof assistants, however, strong evaluation and open terms

*are*required, and it is well known that

*the*operational semantics of call-by-value becomes problematic

*in*this case, as first pointed ...

*in*which

*redexes*

*are*evaluated, but evaluation never gets stuck, as every β -

*redex*will eventually become a β v -

*redex*and be fired (unless evaluation diverges). ...

##
###
Highlights in infinitary rewriting and lambda calculus

2012
*
Theoretical Computer Science
*

Next we describe

doi:10.1016/j.tcs.2012.08.018
fatcat:ulhkxwoambaj3pjimztxxlz6jm
*the*fundamental trichotomy*in**the*semantics of*λ*-*calculus*: Böhm trees, Lévy-Longo trees, and Berarducci trees. ... We present a recent observation concerning*the*continuity of infinitary rewriting.*In**the*second section we present an excursion to*the*infinitary*λ*-*calculus*. ... (*λ*x.M(x))N → M(N) β M → N*λ*x.M →*λ*x.N d M → N MZ → NZ l M → N ZM → ZN r patterns*are*on*the*spine*are*spine*redexes*.*The*uppermost one is*the*head*redex*. ...##
###
A unified approach to fully lazy sharing

2012
*
Proceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '12
*

We prove that

doi:10.1145/2103656.2103713
dblp:conf/popl/Balabonski12
fatcat:t7gpelrj6rejhk6g3j4bb3msii
*the*known implementations of full laziness*are*all equivalent*in*terms of*the*number of β-reductions performed, although they behave differently regarding*the*duplication of terms. ... We establish a link between*the*optimality theories of weak*λ*-calculi and first-order rewriting systems by expressing fully lazy*λ*-lifting*in*our framework, thus emphasizing*the*first-order essence of ... Acknowledgements*The*author would like to thank Delia Kesner for her continuous support, Vincent van Oostrom for a mysterious but insightful remark that triggered this investigation, and*the*anonymous ...##
###
A unified approach to fully lazy sharing

2012
*
SIGPLAN notices
*

We prove that

doi:10.1145/2103621.2103713
fatcat:qbknbhelljglfnj3qp4sxzr4iu
*the*known implementations of full laziness*are*all equivalent*in*terms of*the*number of β-reductions performed, although they behave differently regarding*the*duplication of terms. ... We establish a link between*the*optimality theories of weak*λ*-calculi and first-order rewriting systems by expressing fully lazy*λ*-lifting*in*our framework, thus emphasizing*the*first-order essence of ... Acknowledgements*The*author would like to thank Delia Kesner for her continuous support, Vincent van Oostrom for a mysterious but insightful remark that triggered this investigation, and*the*anonymous ...##
###
Standardization and Böhm Trees for Λμ-Calculus
[chapter]

2010
*
Lecture Notes in Computer Science
*

*In*

*the*present paper, we study solvability and investigate Böhm trees for Λµ-

*calculus*. Moreover, we make clear

*the*connections between Λµ

*calculus*and innitary

*λ*-calculi. ...

*the*failure of separation

*in*Parigot's original λµ-

*calculus*. ... Acknowledgments:

*The*author wishes to thank Simona Ronchi della Rocca as well as Kazushige Terui for helpful discussions on

*the*material of this paper. ...

##
###
Sub-λ-calculi, Classified

2008
*
Electronical Notes in Theoretical Computer Science
*

When sharing is studied

doi:10.1016/j.entcs.2008.03.038
fatcat:d5hf47bq5jgodea5tclbx7rlge
*in**the**λ*-*calculus*, some sub-calculi often pop up, for instance*λ*I or*the*linear*λ*-*calculus*. ... Such a definition only makes sense when*the*subcalculi*are**stable*by β-reduction. ...*In*this paper, focused on*the**λ*-*calculus*, this last choice is better-suited: we do not want to mess up with β-reduction, and, for instance, have β-*redexes*which*are*not*redexes**in**the*subsystem. ...##
###
Böhm theorem and Böhm trees for the Λμ-calculus

2012
*
Theoretical Computer Science
*

A β-normal form (resp. βη-normal form) is a

doi:10.1016/j.tcs.2012.02.027
fatcat:7mgc7lwf5bfvbfch5v5ue263by
*λ*-term containing no β-*redex*(resp. βη-*redex*). ... Separation is done internally to*the**λ*-*calculus*itself: no part of a normal*λ*-term is inaccessible nor irrelevant for computation and only constructions of*the**λ*-*calculus**are*required to achieve this program ... Acknowledgements*The*author wishes to thank Pierre-Louis Curien, Dale Miller, Simona Ronchi della Rocca and Hugo Herbelin for their advice and guidance as well as*the*anonymous reviewers for their relevant ...##
###
Call-by-Value -calculus and LJQ

2007
*
Journal of Logic and Computation
*

*in*lambda

*calculus*, establishing a connection between LJQ and

*the*CBV

*calculus*

*λ*C of Moggi. ...

*in*

*the*other. ...

*The*fragments

*are*

*stable*under

*the*reductions

*in*Fig. 5 and

*are*sufficient to simulate β V and η V through

*the*CPS-translations, as we shall see

*in*section 3.4. ...

##
###
(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

2016
*
Logical Methods in Computer Science
*

Useful evaluation avoids those steps that only unshare

doi:10.2168/lmcs-12(1:4)2016
fatcat:3gx2u54kxbc4xe6rmzb33gbws4
*the*output without contributing to beta-*redexes*, i.e.*the*steps that cause*the*blow-up*in*size. ... Such a theorem cannot be proved by directly relating lambda-*calculus*with Turing machines or random access machines, because of*the*size explosion problem: there*are*terms that*in*a linear number of steps ... solving*the*problem. (2) Complexity Classes*in**the**λ*-*Calculus*. ...##
###
Discrete Normalization and Standardization in Deterministic Residual Structures
[chapter]

1996
*
Lecture Notes in Computer Science
*

Reasonable concepts of meaningless terms

doi:10.1007/3-540-61735-3_9
fatcat:b2vcbcmnljdbzofrpgpvxpc3se
*in*(infinitary) TRSs and*the*-*calculus**are*studied*in*[KOV96] . ...*λ**λ**λ*xt*λ*f A C*The*DN theorem implies that contains only a finite number of standard reductions, for any finite semi-linear . ...##
###
Infinite λ-calculus and non-sensible models
[chapter]

2017
*
Logic and algebra
*

We set

doi:10.1201/9780203748671-18
fatcat:pzzkf3iwdjbnpg2eueosqsmojy
*the*foundations for such a*calculus*, which might turn out to be a useful tool for*the*study of non-sensible models of*λ*-*calculus*. ... This allows us to strengthen some results of Jacopini and Venturini Zilli concerning easy*λ*-terms. Our results depend on an infinitary version of*λ*-*calculus*. ... During our attempts to solve*the*problem Intrigila suggested to expand*the*language of*λ*-*calculus*with a new constant δ together with*the*rule δ → δω 3 . ...
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