@article{ehrhard_differentials_2019,
title = {Differentials and distances in probabilistic coherence spaces},
url = {http://arxiv.org/abs/1902.04836},
abstract = {In probabilistic coherence spaces, a denotational model of probabilistic functional languages, mor-phisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful.},
urldate = {2019-11-28},
journal = {arXiv:1902.04836 [cs]},
author = {Ehrhard, Thomas},
month = feb,
year = {2019},
note = {ZSCC: 0000000
arXiv: 1902.04836},
keywords = {Coherence spaces, Denotational semantics, Differential Linear Logic, Differentiation, Linear logic, Probabilistic programming, Programming language theory}
}
@article{ehrhard_introduction_2016,
title = {An introduction to {Differential} {Linear} {Logic}: proof-nets, models and antiderivatives},
shorttitle = {An introduction to {Differential} {Linear} {Logic}},
url = {http://arxiv.org/abs/1606.01642},
abstract = {Differential Linear Logic enriches Linear Logic with additional logical rules for the exponential connectives, dual to the usual rules of dereliction, weakening and contraction. We present a proof-net syntax for Differential Linear Logic and a categorical axiomatization of its denotational models. We also introduce a simple categorical condition on these models under which a general antiderivative operation becomes available. Last we briefly describe the model of sets and relations and give a more detailed account of the model of finiteness spaces and linear and continuous functions.},
urldate = {2019-11-28},
journal = {arXiv:1606.01642 [cs]},
author = {Ehrhard, Thomas},
month = jun,
year = {2016},
note = {ZSCC: 0000002
arXiv: 1606.01642},
keywords = {Denotational semantics, Differential Linear Logic, Differentiation, Linear logic}
}
@article{fiore_cartesian_2008,
title = {The cartesian closed bicategory of generalised species of structures},
volume = {77},
issn = {00246107},
url = {http://doi.wiley.com/10.1112/jlms/jdm096},
doi = {10/bd2mr9},
abstract = {The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These deﬁnitions encompass most notions of combinatorial species considered in the literature—including of course Joyal’s original notion—together with their associated substitution operation. Our ﬁrst main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudo-comonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed.},
language = {en},
number = {1},
urldate = {2019-11-28},
journal = {Journal of the London Mathematical Society},
author = {Fiore, M. and Gambino, N. and Hyland, M. and Winskel, G.},
month = feb,
year = {2008},
note = {ZSCC: 0000067},
keywords = {Denotational semantics, Differential Linear Logic, Differentiation, Linear logic},
pages = {203--220}
}
@article{ehrhard_differential_2003,
title = {The differential lambda-calculus},
volume = {309},
issn = {0304-3975},
url = {http://www.sciencedirect.com/science/article/pii/S030439750300392X},
doi = {10/bf3b8v},
abstract = {We present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.},
language = {en},
number = {1},
urldate = {2019-11-24},
journal = {Theoretical Computer Science},
author = {Ehrhard, Thomas and Regnier, Laurent},
month = dec,
year = {2003},
note = {ZSCC: 0000307},
keywords = {Differentiation, Linear logic, Programming language theory},
pages = {1--41}
}