@inproceedings{dal_lago_geometry_2019,
title = {The {Geometry} of {Bayesian} {Programming}},
doi = {10/ggdk85},
author = {Dal Lago, Ugo and Hoshino, Naohiko},
month = jun,
year = {2019},
note = {ZSCC: 0000000},
keywords = {Bayesian inference, Denotational semantics, Linear logic, Probabilistic programming, Programming language theory, Rewriting theory, Transition systems},
pages = {1--13}
}
@inproceedings{kerjean_higher-order_2019,
address = {Cham},
series = {Lecture {Notes} in {Computer} {Science}},
title = {Higher-{Order} {Distributions} for {Differential} {Linear} {Logic}},
isbn = {978-3-030-17127-8},
doi = {10/ggdmrj},
abstract = {Linear Logic was introduced as the computational counterpart of the algebraic notion of linearity. Differential Linear Logic refines Linear Logic with a proof-theoretical interpretation of the geometrical process of differentiation. In this article, we construct a polarized model of Differential Linear Logic satisfying computational constraints such as an interpretation for higher-order functions, as well as constraints inherited from physics such as a continuous interpretation for spaces. This extends what was done previously by Kerjean for first order Differential Linear Logic without promotion. Concretely, we follow the previous idea of interpreting the exponential of Differential Linear Logic as a space of higher-order distributions with compact-support, which is constructed as an inductive limit of spaces of distributions on Euclidean spaces. We prove that this exponential is endowed with a co-monadic like structure, with the notable exception that it is functorial only on isomorphisms. Interestingly, as previously argued by Ehrhard, this still allows the interpretation of differential linear logic without promotion.},
language = {en},
booktitle = {Foundations of {Software} {Science} and {Computation} {Structures}},
publisher = {Springer International Publishing},
author = {Kerjean, Marie and Pacaud Lemay, Jean-Simon},
editor = {Bojańczyk, Mikołaj and Simpson, Alex},
year = {2019},
note = {ZSCC: NoCitationData[s1]},
keywords = {Denotational semantics, Differential Linear Logic, Differentiation, Linear logic},
pages = {330--347}
}
@inproceedings{fiore_differential_2007,
address = {Berlin, Heidelberg},
series = {Lecture {Notes} in {Computer} {Science}},
title = {Differential {Structure} in {Models} of {Multiplicative} {Biadditive} {Intuitionistic} {Linear} {Logic}},
isbn = {978-3-540-73228-0},
doi = {10/c8vgx8},
abstract = {In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.).},
language = {en},
booktitle = {Typed {Lambda} {Calculi} and {Applications}},
publisher = {Springer},
author = {Fiore, Marcelo P.},
editor = {Della Rocca, Simona Ronchi},
year = {2007},
note = {ZSCC: NoCitationData[s1]},
keywords = {Differential Linear Logic, Differentiation, Linear logic},
pages = {163--177}
}
@inproceedings{girard_geometry_1995,
address = {Berlin, Heidelberg},
series = {{NATO} {ASI} {Series}},
title = {On {Geometry} of {Interaction}},
isbn = {978-3-642-79361-5},
doi = {10/fr557p},
abstract = {The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C*-algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponential-free conclusions.},
language = {en},
booktitle = {Proof and {Computation}},
publisher = {Springer},
author = {Girard, Jean-Yves},
editor = {Schwichtenberg, Helmut},
year = {1995},
note = {ZSCC: NoCitationData[s0]},
keywords = {Interactive semantics, Linear logic},
pages = {145--191}
}
@misc{murfet_linear_nodate,
title = {Linear logic and deep learning},
language = {en},
author = {Murfet, Daniel and Hu, Huiyi},
note = {ZSCC: NoCitationData[s0]},
keywords = {Categorical ML, Linear logic, Machine learning, Semantics}
}