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Document
DOI: 10.4230/LIPIcs.CALCO.2021.7
Tensor of Quantitative Equational Theories

Authors: Giorgio Bacci, Radu Mardare, Prakash Panangaden, and Gordon Plotkin

Published in: LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

Abstract
We develop a theory for the commutative combination of quantitative effects, their tensor, given as a combination of quantitative equational theories that imposes mutual commutation of the operations from each theory. As such, it extends the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We show that under certain assumptions on the quantitative theories the free monad that arises from the tensor of two theories is the categorical tensor of the free monads on the theories. As an application, we provide the first algebraic axiomatizations of labelled Markov processes and Markov decision processes. Apart from the intrinsic interest in the axiomatizations, it is pleasing they are obtained compositionally by means of the sum and tensor of simpler quantitative equational theories.
Cite as

Giorgio Bacci, Radu Mardare, Prakash Panangaden, and Gordon Plotkin. Tensor of Quantitative Equational Theories. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bacci_et_al:LIPIcs.CALCO.2021.7,
  author =	{Bacci, Giorgio and Mardare, Radu and Panangaden, Prakash and Plotkin, Gordon},
  title =	{{Tensor of Quantitative Equational Theories}},
  booktitle =	{9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-212-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{211},
  editor =	{Gadducci, Fabio and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.7},
  URN =		{urn:nbn:de:0030-drops-153628},
  doi =		{10.4230/LIPIcs.CALCO.2021.7},
  annote =	{Keywords: Quantitative equational theories, Tensor, Monads, Quantitative Effects}
}
Document
DOI: 10.4230/LIPIcs.CSL.2020.18
Reverse Derivative Categories

Authors: Robin Cockett, Geoffrey Cruttwell, Jonathan Gallagher, Jean-Simon Pacaud Lemay, Benjamin MacAdam, Gordon Plotkin, and Dorette Pronk

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Abstract
The reverse derivative is a fundamental operation in machine learning and automatic differentiation [Martín Abadi et al., 2015; Griewank, 2012]. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by [Blute et al., 2009] for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.
Cite as

Robin Cockett, Geoffrey Cruttwell, Jonathan Gallagher, Jean-Simon Pacaud Lemay, Benjamin MacAdam, Gordon Plotkin, and Dorette Pronk. Reverse Derivative Categories. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cockett_et_al:LIPIcs.CSL.2020.18,
  author =	{Cockett, Robin and Cruttwell, Geoffrey and Gallagher, Jonathan and Lemay, Jean-Simon Pacaud and MacAdam, Benjamin and Plotkin, Gordon and Pronk, Dorette},
  title =	{{Reverse Derivative Categories}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.18},
  URN =		{urn:nbn:de:0030-drops-116611},
  doi =		{10.4230/LIPIcs.CSL.2020.18},
  annote =	{Keywords: Reverse Derivatives, Cartesian Reverse Differential Categories, Categorical Semantics, Cartesian Differential Categories, Dagger Categories, Automatic Differentiation}
}
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