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Algebras and coalgebras in the light affine Lambda calculus

2015
*
SIGPLAN notices
*

*Algebra*

*and*

*coalgebra*are widely used to model data types

*in*functional programming languages

*and*proof assistants. ... Their use permits to better structure

*the*computations

*and*also to enhance

*the*expressivity of a language or of a proof system. ...

*The*

*Light*

*Affine*

*Lambda*

*Calculus*

*The*

*Light*

*Affine*

*Lambda*

*Calculus*is

*the*

*affine*version of

*the*

*Light*Linear

*Lambda*

*Calculus*[39]

*and*provide a concrete syntax for Intuitionistic

*Light*

*Affine*Logic [ ...

##
###
Galois-Type Extensions and Equivariant Projectivity
[article]

2009
*
arXiv
*
pre-print

*The*theory of general Galois-type extensions is presented, including

*the*interrelations between

*coalgebra*extensions

*and*

*algebra*(co)extensions, properties of corresponding (co)translation maps,

*and*rudiments ... At

*the*same time, to bring together K-theory

*and*general Galois theory,

*the*equivariant projectivity of extensions is assumed resulting

*in*

*the*centrepiece concept of a principal extension. ...

*In*a non-universal

*calculus*case,

*and*

*in*particularly nice examples (e.g.,

*in*

*the*case of

*the*quantum Hopf fibering with

*the*3D-

*calculus*discussed

*in*[20] ),

*the*above equalities can be obtained. ...

##
###
Derivatives of Turing machines in Linear Logic
[article]

2019
*
arXiv
*
pre-print

We show that these derivatives calculate

arXiv:1805.11813v2
fatcat:jkkhcoq6ozekpalnuw2qmf22z4
*the*rate of change of probabilities naturally arising*in**the*Sweedler semantics of linear logic proofs. ...*The*resulting theory is applied to*the*problem of synthesising Turing machines by gradient descent. ...*The*differential*lambda**calculus*is a system introduced by Ehrhard*and*Regnier [17]*in*which arbitrary algorithms (that is,*lambda*terms) may be "differentiated". ...##
###
A linear-non-linear model for a computational call-by-value lambda calculus (extended abstract)
[article]

2008
*
arXiv
*
pre-print

*The*"!" operator gives rise to a comonad, as

*in*

*the*linear logic models of Seely, Bierman,

*and*Benton.

*The*side-effects give rise to a monad, as

*in*Moggi's computational

*lambda*

*calculus*. ... Such a

*lambda*

*calculus*was used by Selinger

*and*Valiron as

*the*backbone of a functional programming language for quantum computation. ...

*In*

*light*of our present work, we can conclude that a model of

*the*quantum

*lambda*

*calculus*consists of a linear category for duplication (C, L, T, ⊸), such that

*the*associated category of computations C ...

##
###
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 ...

*The*author is partially supported by

*the*ANR project 12IS02001 PACE

*and*

*the*ANR project 14CE250005 ELICA. ...

##
###
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 ...

*The*author is partially supported by

*the*ANR project 12IS02001 PACE

*and*

*the*ANR project 14CE250005 ELICA. ...

##
###
The Expectation Monad in Quantum Foundations

2012
*
Electronic Proceedings in Theoretical Computer Science
*

It turns out to sit

doi:10.4204/eptcs.95.12
fatcat:zcieiml3lnbdlhgilcdythoxw4
*in*between known monads: on*the*one hand*the*distribution*and*ultrafilter monad,*and*on*the*other hand*the*continuation monad. ... This expectation monad is used*in*two probabilistic analogues of fundamental results of Manes*and*Gelfand for*the*ultrafilter monad:*algebras*of*the*expectation monad are convex compact Hausdorff spaces ... [Here we use*the*"*lambda*" notation from*the**lambda**calculus*[7] :*the*expression λ x. · · · is used for*the*function x → · · · . We also use*the*associated application rule (λ x. f (x))(y) = f (y).] ...##
###
The expectation monad in quantum foundations

2016
*
Information and Computation
*

It turns out to sit

doi:10.1016/j.ic.2016.02.009
fatcat:uzf5npxh3rauvfy22fwyj7qnku
*in*between known monads: on*the*one hand*the*distribution*and*ultrafilter monad,*and*on*the*other hand*the*continuation monad. ... This expectation monad is used*in*two probabilistic analogues of fundamental results of Manes*and*Gelfand for*the*ultrafilter monad:*algebras*of*the*expectation monad are convex compact Hausdorff spaces ... [Here we use*the*"*lambda*" notation from*the**lambda**calculus*[7] :*the*expression λ x. · · · is used for*the*function x → · · · . We also use*the*associated application rule (λ x. f (x))(y) = f (y).] ...##
###
On the construction of free algebras for equational systems

2009
*
Theoretical Computer Science
*

*The*purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate

*and*develop

*the*finitary

*and*transfinite construction of free

*algebras*... for equational systems;

*and*to illustrate

*the*use of equational systems as needed

*in*modern applications. where v ∈ V , o is an operator of arity k,

*and*t i ∈ T Σ (V ) for all i = 1, . . . , k. ...

*Lambda*-

*calculus*

*algebras*As a concrete example of

*algebras*with monoid structure, we start by considering

*the*syntax of

*the*λ-

*calculus*, with models given as certain Σ λ -monoids on

*the*presheaf category ...

##
###
Page 1929 of Mathematical Reviews Vol. , Issue Subject Index
[page]

*
Mathematical Reviews
*

On

*the*number of fixed points of a combinator*in**lambda**calculus*. ... (English summary) 2001¢:03107 Saibi, Amokrane see Huet, Gérard, 20014:03024 Salibra, Antonino On*the**algebraic*models of*lambda**calculus*. ...##
###
Page 98 of Mathematical Reviews Vol. 32, Issue Index
[page]

*
Mathematical Reviews
*

(English summary) 2000d:03024
Wolter, Uwe (with Martini, Alfio) Shedding new

*light**in**the*world of logical systems. (English summary) 2000b:03230 — A*coalgebraic*introduction to CSP. ... (English summary) 2000i:03093 Roversi, Luca Concrete syntax for intuitionistic*light**affine*logic with polymorphic type assignment. ...##
###
Predicate Transformer Semantics for Hybrid Systems: Verification Components for Isabelle/HOL
[article]

2021
*
arXiv
*
pre-print

We introduce

arXiv:1909.05618v3
fatcat:7lnrnamltvcutgxfezop4akt7q
*the*semantic foundations of this framework*and*summarise their Isabelle formalisation as well as*the*resulting verification components. ... It supports reasoning about*the*temporal evolutions of hybrid programs*in**the*style of differential dynamic logic modelled by flows or invariant sets for vector fields. ... of*the*RAMiCS 2018 conference*and*our Oslo lecture series on Isabelle/HOL for fruitful discussions. ...##
###
Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version)
[article]

2018
*
arXiv
*
pre-print

This paper studies

arXiv:1801.09072v3
fatcat:hvpl33atmjaorlruldhne35ine
*the*quantitative refinements of Abramsky's applicative similarity*and*bisimilarity*in**the*context of a generalisation of Fuzz, a call-by-value λ-*calculus*with a linear type system that ... can express programs sensitivity, enriched with*algebraic*operations à la Plotkin*and*Power. ... Acknowledgments*The*author would like to thank Ugo Dal Lago, Raphaëlle Crubillé,*and*Paul Levy for*the*many useful comments*and*suggestions. ...##
###
A Linear-non-Linear Model for a Computational Call-by-Value Lambda Calculus (Extended Abstract)
[chapter]

*
Foundations of Software Science and Computational Structures
*

*The*"!" operator gives rise to a comonad, as

*in*

*the*linear logic models of Seely, Bierman,

*and*Benton.

*The*side-effects give rise to a monad, as

*in*Moggi's computational

*lambda*

*calculus*. ... Such a

*lambda*

*calculus*was used by Selinger

*and*Valiron as

*the*backbone of a functional programming language for quantum computation. ...

*In*

*light*of our present work, we can conclude that a model of

*the*quantum

*lambda*

*calculus*consists of a linear category for duplication (C, L, T, ), such that

*the*associated category of computations C ...

##
###
Coalgebraic Logics (Dagstuhl Seminar 12411) Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421) Time-of-Flight Imaging: Algorithms, Sensors and Applications (Dagstuhl Seminar 12431) Foundations and Challenges of Change and Evolution in Ontologies (Dagstuhl Seminar 12441) Requirements Management-Novel Perspectives and Challenges (Dagstuhl Seminar 12442)

2012
unpublished

We review

fatcat:3zuf3grgabettljj7apksbc4lq
*the*requirements problem as defined by Jackson*and*Zave [2]. We then discuss how computational complexity creeps*in**and*how to cope with it. ...*In*addition, we sketch some approaches for dealing with requirements evolution, adopted from*the*PhD thesis of Neil Ernst [1]. ... Applications include an alpha-corecursion principle for*the*infinitary*lambda**calculus**and*corecursive definitions of infinite normal forms. ...
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