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A Note on Confluence in Typed Probabilistic Lambda Calculi

2022
*
Electronic Proceedings in Theoretical Computer Science
*

We examine

doi:10.4204/eptcs.357.2
fatcat:qsagxymb25aobgnuanp25dl5ly
*the*effect of small modifications in probabilistic rewriting, affine*variables*, and strategies*on**the*overall*confluence*in this strongly normalizing probabilistic*calculus*. ... While working with*a**lambda**calculus*modelling quantum computation, we found*a*system with probabilistic rewriting rules and strongly normalizing terms. ... In this short paper we consider*a*simply typed*lambda**calculus*extended with*a*coin, and show different possibilities*for*achieving some*sort*of*confluence*, without giving preference to any of them. ...##
###
Formalisation in Constructive Type Theory of Barendregt's Variable Convention for Generic Structures with Binders

2018
*
Electronic Proceedings in Theoretical Computer Science
*

We instantiate

doi:10.4204/eptcs.274.2
fatcat:vcjqyzsrkbh7joszbrij3cbn3y
*the*resulting framework so as to obtain*the**Lambda**Calculus*and System F,*for*which we derive substitution operations and substitution lemmas*for*alpha-conversion and substitution composition ... We then define*a*generic alpha-equivalence relation over*the*types of*the*universe based*on**name*-swapping, and derive iteration and induction principles which work modulo alpha-conversion capturing Barendregt's ... In 2002 Pitts and Gabbay introduced*the*Nominal Logic [9] ,*a**first*-*order*many-*sorted*logic with equality, containing primitives*for*renaming via*name*-swapping,*for*freshness of*names*, and*for**name*-binding ...##
###
A note on confluence in typed probabilistic lambda calculi
[article]

2021
*
arXiv
*
pre-print

We examine

arXiv:2106.06633v1
fatcat:cdx6qg2ydzf5dndwayrwrrpnzy
*the*effect of small modifications in probabilistic rewriting, affine*variables*, and strategies*on**the*overall*confluence*in this strongly normalizing probabilistic*calculus*. ... While working with*a**lambda**calculus*modelling quantum computation, we found*a*system with probabilistic rewriting rules and strongly normalizing terms. ... In this short paper we consider*a*simply typed*lambda**calculus*extended with*a*coin, and show different possibilities*for*achieving some*sort*of*confluence*, without giving preference to any of them. ...##
###
Programming language foundations in Agda

2020
*
Science of Computer Programming
*

*For*comments

*on*

*a*draft of this paper, an extra thank you to James McKinna, Ulf Norell, Andreas Abel, and Benjamin Pierce. This research was supported by EPSRC Programme Grant EP/K034413/1. ...

*For*inventing ideas

*on*which PLFA is based, and

*for*handholding, many thanks to Conor McBride, James McKinna, Ulf Norell, and Andreas Abel. ...

*For*

*the*time being, we have decided to omit Hoare logic in

*order*to focus

*on*

*lambda*

*calculus*. ...

##
###
The call-by-need lambda calculus

1998
*
Journal of functional programming
*

*The*call-byneed

*lambda*

*calculus*is confluent, has

*a*notion of standard reduction, and entails

*the*same observational equivalence relation as

*the*call-by-

*name*

*calculus*. ... We present

*a*

*calculus*that captures

*the*operational semantics of call-by-need. ... Turner and

*a*number of anonymous referees

*for*valuable comments and discussions. ...

##
###
Big Step Normalisation for Type Theory

2020
*
Types for Proofs and Programs
*

Most of

doi:10.4230/lipics.types.2019.4
dblp:conf/types/AltenkirchG19
fatcat:zmjtadnf6jhbnl6dmwvw75wa7i
*the**proof*has been*formalised**using*Agda. ... Compared to previous presentations of big step normalisation*for*e.g.*the*simply-typed*lambda*-*calculus*, we*use**a*quotiented syntax of type theory, which crucially reduces*the*syntactic complexity introduced ... Big step normalisation was developed by Chapman and*the**first*author*for**a*combinatory*calculus*[2] , and*for**the*simply-typed*lambda*-*calculus*[3] . ...##
###
Equational Logic and Rewriting
[chapter]

2014
*
Handbook of the History of Logic
*

After Turing's famous work

doi:10.1016/b978-0-444-51624-4.50006-x
fatcat:eqtvr66m65eula7otl7fr6ppaa
*on**the*halting problem, showing*the*existence of*a*problem unsolvable by mechanical means, Church and Turing then proved that*the**lambda**calculus*and*the*Turing machine*used*...*The**first**one*, explicit induction, is*used*in*proof*assistants,*for*instance Nqthm-ACL2 [89] , Coq [21] , Isabelle [122] or Inka [7] . ...##
###
Coq Coq correct! verification of type checking and erasure for Coq, in Coq

2019
*
Proceedings of the ACM on Programming Languages (PACMPL)
*

Coq is built around

doi:10.1145/3371076
fatcat:sdm7suzouzexpdey3o4o5wzkce
*a*well-delimited kernel that perfoms typechecking*for*definitions in*a*variant of*the**Calculus*of Inductive Constructions (CIC). ... We present*a*verified version of this subtle type-and-*proof*erasure step, therefore enabling*the*verified extraction of*a*safe type-checker*for*Coq. ... is not*used*in*the**proof*of*confluence*. ...##
###
A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

2011
*
Logical Methods in Computer Science
*

We introduce

doi:10.2168/lmcs-7(1:6)2011
fatcat:fto35brddvcupdepcvb7gz25dm
*the*idea of Pure Type Sequent*Calculus*with meta-*variables*(PTSCalpha), by enriching*the*syntax of*a*permutation-free sequent*calculus**for*propositional logic due to Herbelin, which is strongly ... Finally, we consider how to*use**the*explicitly scoped meta-*variables*of PTSCalpha to represent partial*proof*-terms, and*use*them to analyse interactive*proof*construction. ... Acknowledgements*The*authors are grateful to Delia Kesner, Gilles Dowek, Hugo Herbelin, Arnaud Spiwack, Vincent Siles, Alex Simpson and David Pym*for*their helpful remarks and comments and*for*pointing ...##
###
Strong Normalisation for a Gentzen-like Cut-Elimination Procedure
[chapter]

2001
*
Lecture Notes in Computer Science
*

We shall establish this property

doi:10.1007/3-540-45413-6_32
fatcat:nwatdcl5mbdojecblmazk7ekyu
*using**the*recursive path*ordering*by Dershowitz. ... Inspired by work of Bloo and Geuvers concerning*the*λx-*calculus*, we shall show that*a*simple trick allows*us*to preserve this property in our cut-elimination procedure. ... Notice however that*names*and co-*names*are not*the*same notions as*a**variable*in*the**lambda*-*calculus*: whilst*a*term can be substituted*for**a**variable*,*a**name*or*a*co-*name*can only be "renamed". ...##
###
No solvable lambda-value term left behind

2016
*
Logical Methods in Computer Science
*

We provide

doi:10.2168/lmcs-12(2:12)2016
fatcat:mhkhyz2w7bagbcjxdbve624ode
*a*definition of solvability*for**the**lambda*-value*calculus*that does capture operational relevance and such that*a*consistent*proof*-theory can be constructed where unsolvables are equated attending ... There is*a*definition of solvability*for**the**lambda*-value*calculus*, called v-solvability, but it is not synonymous with operational relevance, some*lambda*-value normal forms are unsolvable, and unsolvables ... Acknowledgement*A*preliminary version of this work was presented at*the*11th International Workshop*on*Domain Theory and Applications (Domains XI), Paris, 10th September 2014. ...##
###
Touring the MetaCoq Project (Invited Paper)

2021
*
Electronic Proceedings in Theoretical Computer Science
*

*for*PCUIC;

*a*sound type and

*proof*erasure procedure from PCUIC to untyped

*lambda*-

*calculus*, i.e.,

*the*core of

*the*extraction mechanism of Coq. ... MetaCoq is

*a*collaborative project that aims to tackle these difficulties to provide

*the*

*first*fully-certified realistic implementation of

*a*type checker

*for*

*the*full

*calculus*underlying

*the*Coq

*proof*... It corresponds closely to COQ's internal syntax, handling local

*variables*(tRel, with

*a*de Bruijn index), free

*named*

*variables*(tVar), existential

*variables*(tEvar),

*sorts*(tSort),

*the*type-cast construct ...

##
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Small, Fast, Concurrent Proof Checking for the lambda-Pi Calculus Modulo Rewriting
[article]

2021
*
arXiv
*
pre-print

*The*

*proof*checker implements

*the*

*lambda*-Pi

*calculus*modulo rewriting, which is an established framework to uniformly express

*a*multitude of logical systems. ...

*The*

*proof*checker is faster than

*the*reference

*proof*checker

*for*this

*calculus*, Dedukti,

*on*all of five evaluated datasets obtained from

*proof*assistants and interactive theorem provers. ... Furthermore, I would like to thank Gaspard Ferey, Guillaume Genestier and Gabriel Hondet

*for*explaining to me

*the*inner workings of Dedukti, and Emilie Grienenberger

*for*providing me with

*the*Dedukti export ...

##
###
A Quick Overview on the Quantum Control Approach to the Lambda Calculus

2022
*
Electronic Proceedings in Theoretical Computer Science
*

We give an overview of

doi:10.4204/eptcs.357.1
fatcat:rg5577pac5eorkven4cc3mnr5u
*the*quantum control line of research within*the**lambda**calculus*, ranging from untyped calculi up to categorical and realisability models. ... This is*a*summary of*the*last 10+ years of research in this area, starting from Arrighi and Dowek's seminal work until today. ... Acknowledgements*The*author wishes to thank Eduardo Bonelli and Mauricio Ayala-Rincón*for**the*inclusion of this overview in LSFA 2021 as an invited talk, and*for*their*useful*comments. ...##
###
Maximal sharing in the Lambda calculus with letrec

2014
*
Proceedings of the 19th ACM SIGPLAN international conference on Functional programming - ICFP '14
*

Instead of defined purely syntactically, this notion is based

doi:10.1145/2628136.2628148
dblp:conf/icfp/GrabmayerR14
fatcat:7hv2qsgk6jcvljqyk535fwfqwa
*on**a*graph semantics.*lambda*-letrec-terms are interpreted as*first*-*order*term graphs so that unfolding equivalence between terms is preserved ... We show how*a*maximal degree of sharing can be obtained*for*programs expressed as terms in*the**lambda**calculus*with letrec. ... Acknowledgment We want to thank Vincent van Oostrom*for*extensive feedback*on**a*draft, Doaitse Swierstra and Dimitri Hendriks*for*helpful comments, and Jeroen Keiren*for**a*suggestion concerning restricting ...
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