License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2022.80
URN: urn:nbn:de:0030-drops-168789
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16878/
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Simmons, Will ; Kissinger, Aleks

Higher-Order Causal Theories Are Models of BV-Logic

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LIPIcs-MFCS-2022-80.pdf (0.8 MB)


Abstract

The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[š¯’˛] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmiā€™s BV logic. Furthermore, we obtain a natural interpretation for the sequential product as "A can signal to B, but not vice-versa", which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for š¯’˛ recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.

BibTeX - Entry

@InProceedings{simmons_et_al:LIPIcs.MFCS.2022.80,
  author =	{Simmons, Will and Kissinger, Aleks},
  title =	{{Higher-Order Causal Theories Are Models of BV-Logic}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{80:1--80:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16878},
  URN =		{urn:nbn:de:0030-drops-168789},
  doi =		{10.4230/LIPIcs.MFCS.2022.80},
  annote =	{Keywords: Causality, linear logic, categorical logic, probabilistic coherence spaces, quantum channels}
}

Keywords: Causality, linear logic, categorical logic, probabilistic coherence spaces, quantum channels
Collection: 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
Issue Date: 2022
Date of publication: 22.08.2022


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