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Algebras and coalgebras in the light affine Lambda calculus

Marco Gaboardi, Romain Péchoux
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 [  ... 
doi:10.1145/2858949.2784759 fatcat:ezlynnzdxrepfpxnj3qtddwj2q

Galois-Type Extensions and Equivariant Projectivity [article]

Tomasz Brzezinski, Piotr M. Hajac
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.  ... 
arXiv:0901.0141v1 fatcat:63psswyanrfj3k36ntd6ershom

Derivatives of Turing machines in Linear Logic [article]

James Clift, Daniel Murfet
2019 arXiv   pre-print
We show that these derivatives calculate 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".  ... 
arXiv:1805.11813v2 fatcat:jkkhcoq6ozekpalnuw2qmf22z4

A linear-non-linear model for a computational call-by-value lambda calculus (extended abstract) [article]

Peter Selinger Dalhousie University,
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  ... 
arXiv:0801.0813v1 fatcat:ryf5svfuyvh6dfjyq5pogwdf74

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  ...  The author is partially supported by the ANR project 12IS02001 PACE and the ANR project 14CE250005 ELICA.  ... 
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  ...  The author is partially supported by the ANR project 12IS02001 PACE and the ANR project 14CE250005 ELICA.  ... 
arXiv:1604.08248v1 fatcat:wykju4emo5azna5iywpvds536y

The Expectation Monad in Quantum Foundations

Bart Jacobs, Jorik Mandemaker
2012 Electronic Proceedings in Theoretical Computer Science  
It turns out to sit 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).]  ... 
doi:10.4204/eptcs.95.12 fatcat:zcieiml3lnbdlhgilcdythoxw4

The expectation monad in quantum foundations

Bart Jacobs, Jorik Mandemaker, Robert Furber
2016 Information and Computation  
It turns out to sit 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).]  ... 
doi:10.1016/j.ic.2016.02.009 fatcat:uzf5npxh3rauvfy22fwyj7qnku

On the construction of free algebras for equational systems

Marcelo Fiore, Chung-Kil Hur
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  ... 
doi:10.1016/j.tcs.2008.12.052 fatcat:6bjfr4td4rcizmkb4uxvcbcebm

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]

Jonathan Julián Huerta y Munive, Georg Struth
2021 arXiv   pre-print
We introduce 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.  ... 
arXiv:1909.05618v3 fatcat:7lnrnamltvcutgxfezop4akt7q

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version) [article]

Francesco Gavazzo
2018 arXiv   pre-print
This paper studies 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.  ... 
arXiv:1801.09072v3 fatcat:hvpl33atmjaorlruldhne35ine

A Linear-non-Linear Model for a Computational Call-by-Value Lambda Calculus (Extended Abstract) [chapter]

Peter Selinger, Benoît Valiron
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  ... 
doi:10.1007/978-3-540-78499-9_7 dblp:conf/fossacs/SelingerV08 fatcat:sc5e6vnzhrbzbkyjuuxa4lespa

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)

Lieven Desmet, Martin Johns, Benjamin Livshits, Andrei Sabelfeld, Ernst-Erich Doberkat, Alexander Kurz, Manindra Agrawal, Thomas Thierauf, Christopher Umans, James Davis, Bernd Jähne, Andreas Kolb (+9 others)
2012 unpublished
We review 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.  ... 
fatcat:3zuf3grgabettljj7apksbc4lq
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