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Finitary Corecursion for the Infinitary Lambda Calculus
[article]

2015
*
arXiv
*
pre-print

This yields a

arXiv:1505.07736v2
fatcat:o4ghpchdwrdrbp4vo77zhr7tka
*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*. ...##
###
Finitary Corecursion for the Infinitary Lambda Calculus *

unpublished

This yields a

fatcat:cltwsjuhmrfs3lslixcwdkr3ea
*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*. ...##
###
Pure type systems with corecursion on streams

2012
*
SIGPLAN notices
*

This extension is called Pure Type Systems with

doi:10.1145/2398856.2364550
fatcat:grd426s77jalrccdvojd6btryy
*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 ...##
###
Pure type systems with corecursion on streams

2012
*
Proceedings of the 17th ACM SIGPLAN international conference on Functional programming - ICFP '12
*

This extension is called Pure Type Systems with

doi:10.1145/2364527.2364550
dblp:conf/icfp/SeveriV12
fatcat:e6eltq5rtraatlcrbb5fmrgaie
*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 ...##
###
An Alpha-Corecursion Principle for the Infinitary Lambda Calculus
[chapter]

2012
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/978-3-642-32784-1_8
fatcat:u43otj2pzbgevbijpu2eclan6e
*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] . ...##
###
A new coinductive confluence proof for infinitary lambda calculus
[article]

2020
*
arXiv
*
pre-print

We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in

arXiv:1808.05481v4
fatcat:kyvjdffsc5g2xotcmjfwssif4y
*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. ...##
###
A new coinductive confluence proof for infinitary lambda calculus

2018
*
Logical Methods in Computer Science
*

We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in

doi:10.23638/lmcs-16(1:31)2020
fatcat:oz5iah3r3baqfe73zacq4lobki
*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. ...##
###
Infinitary Rewriting Coinductively

2013
*
Types for Proofs and Programs
*

As an illustration, we prove

doi:10.4230/lipics.types.2011.16
dblp:conf/types/EndrullisP11
fatcat:duk5p7ipf5a6lnmo4jmviurcby
*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. ...##
###
An Extensional Böhm Model
[chapter]

2002
*
Lecture Notes in Computer Science
*

We show

doi:10.1007/3-540-45610-4_12
fatcat:pkgqegbq3vcsbpadfitmimbs24
*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. ...##
###
A Coinductive Approach to Proof Search

2013
*
Electronic Proceedings in Theoretical Computer Science
*

We introduce a variant of

doi:10.4204/eptcs.126.3
fatcat:pykkjwy7indrliczaqdiyiksau
*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). ...##
###
A Coinductive Approach to Proof Search through Typed Lambda-Calculi
[article]

2020
*
arXiv
*
pre-print

A second,

arXiv:1602.04382v3
fatcat:prkrtwpzt5fxhdl2au2arduf6u
*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. ...##
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Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search
[article]

2017
*
arXiv
*
pre-print

This approach works by exploiting a representation of

arXiv:1604.02086v2
fatcat:tkj4gjuz45fc5eprtarqauatve
*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. ...##
###
Nominal Coalgebraic Data Types with Applications to Lambda Calculus

2013
*
Logical Methods in Computer Science
*

This allows us to define types of infinite data with binding

doi:10.2168/lmcs-9(4:20)2013
fatcat:sskirbapqnhnxltl2ajymj7bqy
*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. ...##
###
Infinitary Lambda Calculi from a Linear Perspective

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. ...

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

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. ...

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