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A nonstandard standardization theorem

Beniamino Accattoli, Eduardo Bonelli, Delia Kesner, Carlos Lombardi
2014 Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14  
This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in λcalculus 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.  ... 
doi:10.1145/2535838.2535886 dblp:conf/popl/AccattoliBKL14 fatcat:gjgrgun6hbf2nguwm5tf3tyr5a

Weak optimality, and the meaning of sharing

Thibaut Balabonski
2013 SIGPLAN notices  
sequence 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.  ... 
doi:10.1145/2544174.2500606 fatcat:yd7c33g7qfdtvjt6fa3ikcom5m

Weak optimality, and the meaning of sharing

Thibaut Balabonski
2013 Proceedings of the 18th ACM SIGPLAN international conference on Functional programming - ICFP '13  
sequence 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.  ... 
doi:10.1145/2500365.2500606 dblp:conf/icfp/Balabonski13 fatcat:c23lqvqvpravxhm3gvx3yiv4zy

Meaningless terms in rewriting [chapter]

Richard Kennaway, Vincent Oostrom, Fer-Jan Vries
1996 Lecture Notes in Computer Science  
For example, in lambda calculus, we have reductions such as (λx.xx)(λy.y) → (λy.y)(λy.y), in which the subterm xx becomes instantiated to (λy.y)(λy.y).  ...  All the axioms hold except for Axiom 1, as the nonsimply typable terms are clearly not closed under reduction (even in the λI-calculus).  ...  The remaining depth measures are 0 * 1. For these we argue thus: if (λx.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.  ... 
doi:10.1007/3-540-61735-3_17 fatcat:odn4wjrmubcbletpdhtb4toedu

Open Call-by-Value [chapter]

Beniamino Accattoli, Giulio Guerrieri
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).  ... 
doi:10.1007/978-3-319-47958-3_12 fatcat:b2muwpevj5bptd32xvsjnogo4q

Highlights in infinitary rewriting and lambda calculus

Jörg Endrullis, Dimitri Hendriks, Jan Willem Klop
2012 Theoretical Computer Science  
Next we describe 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.  ...  (λx.M(x))N → M(N) β M → N λx.M → λx.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.  ... 
doi:10.1016/j.tcs.2012.08.018 fatcat:ulhkxwoambaj3pjimztxxlz6jm

A unified approach to fully lazy sharing

Thibaut Balabonski
2012 Proceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '12  
We prove that 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  ... 
doi:10.1145/2103656.2103713 dblp:conf/popl/Balabonski12 fatcat:t7gpelrj6rejhk6g3j4bb3msii

A unified approach to fully lazy sharing

Thibaut Balabonski
2012 SIGPLAN notices  
We prove that 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  ... 
doi:10.1145/2103621.2103713 fatcat:qbknbhelljglfnj3qp4sxzr4iu

Standardization and Böhm Trees for Λμ-Calculus [chapter]

Alexis Saurin
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 Λµcalculus 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.  ... 
doi:10.1007/978-3-642-12251-4_11 fatcat:niemqq2cf5aczbwgdvortwjoqa

Sub-λ-calculi, Classified

François-Régis Sinot
2008 Electronical Notes in Theoretical Computer Science  
When sharing is studied in the λ-calculus, some sub-calculi often pop up, for instance λI 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.  ... 
doi:10.1016/j.entcs.2008.03.038 fatcat:d5hf47bq5jgodea5tclbx7rlge

Böhm theorem and Böhm trees for the Λμ-calculus

Alexis Saurin
2012 Theoretical Computer Science  
A β-normal form (resp. βη-normal form) is a 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  ... 
doi:10.1016/j.tcs.2012.02.027 fatcat:7mgc7lwf5bfvbfch5v5ue263by

Call-by-Value -calculus and LJQ

R. Dyckhoff, S. Lengrand
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.  ... 
doi:10.1093/logcom/exm037 fatcat:jcwtamfhvvfz7d3qdawgabflyy

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Beniamino Accattoli, Ugo Dal Lago, Dale Miller
2016 Logical Methods in Computer Science  
Useful evaluation avoids those steps that only unshare 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.  ... 
doi:10.2168/lmcs-12(1:4)2016 fatcat:3gx2u54kxbc4xe6rmzb33gbws4

Discrete Normalization and Standardization in Deterministic Residual Structures [chapter]

Zurab Khasidashvili, John Glauert
1996 Lecture Notes in Computer Science  
Reasonable concepts of meaningless terms in (infinitary) TRSs and the -calculus are studied in [KOV96] .  ...  λ λ λxt λ f A C The DN theorem implies that contains only a finite number of standard reductions, for any finite semi-linear .  ... 
doi:10.1007/3-540-61735-3_9 fatcat:b2vcbcmnljdbzofrpgpvxpc3se

Infinite λ-calculus and non-sensible models [chapter]

Alessandro Berarducci
2017 Logic and algebra  
We set 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 .  ... 
doi:10.1201/9780203748671-18 fatcat:pzzkf3iwdjbnpg2eueosqsmojy
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