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A Note on Confluence in Typed Probabilistic Lambda Calculi

Rafael Romero, Alejandro Díaz-Caro
2022 Electronic Proceedings in Theoretical Computer Science  
We examine 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.  ... 
doi:10.4204/eptcs.357.2 fatcat:qsagxymb25aobgnuanp25dl5ly

Formalisation in Constructive Type Theory of Barendregt's Variable Convention for Generic Structures with Binders

Ernesto Copello, Nora Szasz, Álvaro Tasistro
2018 Electronic Proceedings in Theoretical Computer Science  
We instantiate 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  ... 
doi:10.4204/eptcs.274.2 fatcat:vcjqyzsrkbh7joszbrij3cbn3y

A note on confluence in typed probabilistic lambda calculi [article]

Rafael Romero, Alejandro Díaz-Caro
2021 arXiv   pre-print
We examine 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.  ... 
arXiv:2106.06633v1 fatcat:cdx6qg2ydzf5dndwayrwrrpnzy

Programming language foundations in Agda

Wen Kokke, Jeremy G. Siek, Philip Wadler
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.  ... 
doi:10.1016/j.scico.2020.102440 fatcat:czwjxx35krfvtclrrmor4gwziy

The call-by-need lambda calculus

JOHN MARAIST, MARTIN ODERSKY, PHILIP WADLER
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.  ... 
doi:10.1017/s0956796898003037 fatcat:hzfbo4ki6bb4pjfuhxppuvqqw4

Big Step Normalisation for Type Theory

Thorsten Altenkirch, Colin Geniet, Assia Mahboubi, Marc Bezem
2020 Types for Proofs and Programs  
Most of 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] .  ... 
doi:10.4230/lipics.types.2019.4 dblp:conf/types/AltenkirchG19 fatcat:zmjtadnf6jhbnl6dmwvw75wa7i

Equational Logic and Rewriting [chapter]

Claude Kirchner, Hélène Kirchner
2014 Handbook of the History of Logic  
After Turing's famous work 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] .  ... 
doi:10.1016/b978-0-444-51624-4.50006-x fatcat:eqtvr66m65eula7otl7fr6ppaa

Coq Coq correct! verification of type checking and erasure for Coq, in Coq

Matthieu Sozeau, Simon Boulier, Yannick Forster, Nicolas Tabareau, Théo Winterhalter
2019 Proceedings of the ACM on Programming Languages (PACMPL)  
Coq is built around 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.  ... 
doi:10.1145/3371076 fatcat:sdm7suzouzexpdey3o4o5wzkce

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Stéphane Lengrand, Roy Dyckhoff, James McKinna, Henk Barendregt
2011 Logical Methods in Computer Science  
We introduce 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  ... 
doi:10.2168/lmcs-7(1:6)2011 fatcat:fto35brddvcupdepcvb7gz25dm

Strong Normalisation for a Gentzen-like Cut-Elimination Procedure [chapter]

Christian Urban
2001 Lecture Notes in Computer Science  
We shall establish this property 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".  ... 
doi:10.1007/3-540-45413-6_32 fatcat:nwatdcl5mbdojecblmazk7ekyu

No solvable lambda-value term left behind

Álvaro García-Pérez, Pablo Nogueira, Neil Jones
2016 Logical Methods in Computer Science  
We provide 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.  ... 
doi:10.2168/lmcs-12(2:12)2016 fatcat:mhkhyz2w7bagbcjxdbve624ode

Touring the MetaCoq Project (Invited Paper)

Matthieu Sozeau
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  ... 
doi:10.4204/eptcs.337.2 fatcat:vkzxls6psredfkeo7xwazecbpa

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

Michael Färber
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  ... 
arXiv:2102.08766v1 fatcat:ewgptnfbazaqpe4fkbf7chuevu

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

Alejandro Díaz-Caro
2022 Electronic Proceedings in Theoretical Computer Science  
We give an overview of 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.  ... 
doi:10.4204/eptcs.357.1 fatcat:rg5577pac5eorkven4cc3mnr5u

Maximal sharing in the Lambda calculus with letrec

Clemens Grabmayer, Jan Rochel
2014 Proceedings of the 19th ACM SIGPLAN international conference on Functional programming - ICFP '14  
Instead of defined purely syntactically, this notion is based 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  ... 
doi:10.1145/2628136.2628148 dblp:conf/icfp/GrabmayerR14 fatcat:7hv2qsgk6jcvljqyk535fwfqwa
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