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Finitary Corecursion for the Infinitary Lambda Calculus [article]

Stefan Milius, Thorsten Wißmann
2015 arXiv   pre-print
This yields a corecursion principle that allows the definition of operations such as substitution on rational λ-trees.  ...  Kurz et al. have recently shown that infinite λ-trees with finitely many free variables modulo α-equivalence form a final coalgebra for a functor on the category of nominal sets.  ...  In this sense we have achieved finitary corecursion for the infinitary λ-calculus.  ... 
arXiv:1505.07736v2 fatcat:o4ghpchdwrdrbp4vo77zhr7tka

Finitary Corecursion for the Infinitary Lambda Calculus *

Stefan Milius, Thorsten Wißmann
unpublished
This yields a corecursion principle that allows the definition of operations such as substitution on rational λ-trees. 1 Introduction One of the most important concepts in computer science is the λ-calculus  ...  Gabbay and Pitts then consider the functor L α X = V + [V]X + X × X expressing the type of the term constructors of the λ-calculus (note that the abstraction functor [V]X is a quotient of V × X modulo  ...  In this sense we have achieved finitary corecursion for the infinitary λ-calculus.  ... 
fatcat:cltwsjuhmrfs3lslixcwdkr3ea

Pure type systems with corecursion on streams

Paula G. Severi, Fer-Jan J. de Vries
2012 SIGPLAN notices  
This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calculus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS.  ...  In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq.  ...  Acknowledgments We would like to acknowledge Alexander Kurz, Tadeusz Litak and Daniela Petrişan for discussing the papers by Krishnaswami  ... 
doi:10.1145/2398856.2364550 fatcat:grd426s77jalrccdvojd6btryy

Pure type systems with corecursion on streams

Paula G. Severi, Fer-Jan J. de Vries
2012 Proceedings of the 17th ACM SIGPLAN international conference on Functional programming - ICFP '12  
This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calculus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS.  ...  In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq.  ...  Acknowledgments We would like to acknowledge Alexander Kurz, Tadeusz Litak and Daniela Petrişan for discussing the papers by Krishnaswami  ... 
doi:10.1145/2364527.2364550 dblp:conf/icfp/SeveriV12 fatcat:e6eltq5rtraatlcrbb5fmrgaie

An Alpha-Corecursion Principle for the Infinitary Lambda Calculus [chapter]

Alexander Kurz, Daniela Petrişan, Paula Severi, Fer-Jan de Vries
2012 Lecture Notes in Computer Science  
We show that the terms of the infinitary lambda-calculus form the final coalgebra for the same functor.  ...  Gabbay and Pitts proved that lambda-terms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets.  ...  Preliminaries on Infinitary Lambda Calculus We assume familiarity with basic notions and notations of the finite λ-calculus [5] .  ... 
doi:10.1007/978-3-642-32784-1_8 fatcat:u43otj2pzbgevbijpu2eclan6e

A new coinductive confluence proof for infinitary lambda calculus [article]

Łukasz Czajka
2020 arXiv   pre-print
We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result.  ...  The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.  ...  PROOF FOR INFINITARY LAMBDA CALCULUS 31:11 Vol. 16 :1 A NEW COINDUCTIVE CONFLUENCE PROOF FOR INFINITARY LAMBDA CALCULUS 31:13 β O} and U = H ∪ O.  ... 
arXiv:1808.05481v4 fatcat:kyvjdffsc5g2xotcmjfwssif4y

A new coinductive confluence proof for infinitary lambda calculus

Łukasz Czajka
2018 Logical Methods in Computer Science  
We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result.  ...  The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.  ...  PROOF FOR INFINITARY LAMBDA CALCULUS 31:11 Vol. 16 :1 A NEW COINDUCTIVE CONFLUENCE PROOF FOR INFINITARY LAMBDA CALCULUS 31:13 β O} and U = H ∪ O.  ... 
doi:10.23638/lmcs-16(1:31)2020 fatcat:oz5iah3r3baqfe73zacq4lobki

Infinitary Rewriting Coinductively

Jörg Endrullis, Andrew Polonsky, Marc Herbstritt
2013 Types for Proofs and Programs  
As an illustration, we prove the existence part of the infinitary standardization theorem. The proof is fully formalized in Coq using coinductive types.  ...  The paper concludes with a characterization of infinite lambda terms which reduce to themselves in a single beta step.  ...  The only finite λ-term M such that M → M via a root step is Ω ≡ (λx. xx)(λx. xx). In infinitary lambda calculus, the situation becomes more involved.  ... 
doi:10.4230/lipics.types.2011.16 dblp:conf/types/EndrullisP11 fatcat:duk5p7ipf5a6lnmo4jmviurcby

An Extensional Böhm Model [chapter]

Paula Severi, Fer-Jan de Vries
2002 Lecture Notes in Computer Science  
We show the existence of an infinitary confluent and normalising extension of the finite extensional lambda calculus with beta and eta.  ...  The eta rule has not been considered before in infinitary lambda calculus mainly because of a counterexample [12, 11] showing that arbitrary transfinite βηreductions can not be compressed into reductions  ...  Acknowledgements The first author thanks the Department of Mathematics and Computer Science of Leicester University for their hospitality.  ... 
doi:10.1007/3-540-45610-4_12 fatcat:pkgqegbq3vcsbpadfitmimbs24

A Coinductive Approach to Proof Search

José Espírito Santo, Ralph Matthes, Luís Pinto
2013 Electronic Proceedings in Theoretical Computer Science  
We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the "solution spaces" (called B\"ohm forests), which are a  ...  We need to study "co-contraction" of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point  ...  Ralph Matthes thanks the Centro de Matemática of Universidade do Minho for funding research visits to José Espírito Santo and Luís Pinto to start this research (2011/2012).  ... 
doi:10.4204/eptcs.126.3 fatcat:pykkjwy7indrliczaqdiyiksau

A Coinductive Approach to Proof Search through Typed Lambda-Calculi [article]

José Espírito Santo, Ralph Matthes, Luís Pinto
2020 arXiv   pre-print
A second, finitary representation is proposed, where the lambda-calculus that represents proofs is extended with a formal greatest fixed point.  ...  lambda-term to each finitary term.  ...  Ralph Matthes had been funded by the Climt project (ANR-11-BS02-016 of the French Agence Nationale de la Recherche). All authors had been partially funded by COST action CA15123 EUTYPES.  ... 
arXiv:1602.04382v3 fatcat:prkrtwpzt5fxhdl2au2arduf6u

Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search [article]

José Espírito Santo, Ralph Matthes, Luís Pinto
2017 arXiv   pre-print
This approach works by exploiting a representation of the search space generated by a given inhabitation problem, which is in terms of a lambda-calculus for proof search that the authors developed recently  ...  The representation may be seen as extending the Curry-Howard representation of proofs by lambda-terms, staying within the methods of lambda-calculus and type systems.  ...  Note that summands cannot be lambda-abstractions.  ... 
arXiv:1604.02086v2 fatcat:tkj4gjuz45fc5eprtarqauatve

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

Alexander Kurz, Daniela Petrisan, Paula Severi, Fer-Jan de Vries, Jan Rutten
2013 Logical Methods in Computer Science  
This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.  ...  Acknowledgements We are grateful to Andy Pitts for suggesting us to treat the infinitely many free variables of a term as constants.  ...  We would also like to thank Christian Urban for helpful discussions. Finally, we acknowledge insightful improvements suggested by the referees.  ... 
doi:10.2168/lmcs-9(4:20)2013 fatcat:sskirbapqnhnxltl2ajymj7bqy

Infinitary Lambda Calculi from a Linear Perspective

Ugo Dal Lago
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
The obtained calculus embeds the infinitary applicative λ-calculus and is universal for computations over infinite strings.  ...  We introduce a linear infinitary λ-calculus, called Λ∞, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively  ...  Acknowledgment The author would like to thank Patrick Baillot, Marco Gaboardi and Olivier Laurent for useful discussions about the topics of this paper.  ... 
doi:10.1145/2933575.2934505 dblp:conf/lics/Lago16 fatcat:om7mpkcnkzf2zcypjjpyukthgm

Infinitary λ-Calculi from a Linear Perspective (Long Version) [article]

Ugo Dal Lago
2016 arXiv   pre-print
The obtained calculus embeds the infinitary applicative λ-calculus and is universal for computations over infinite strings.  ...  We introduce a linear infinitary λ-calculus, called ℓΛ_∞, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted  ...  Acknowledgment The author would like to thank Patrick Baillot, Marco Gaboardi and Olivier Laurent for useful discussions about the topics of this paper.  ... 
arXiv:1604.08248v1 fatcat:wykju4emo5azna5iywpvds536y
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