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.CALCO.2021.12
URN: urn:nbn:de:0030-drops-153678
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15367/
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Castelnovo, Davide ; Miculan, Marino

Closure Hyperdoctrines

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LIPIcs-CALCO-2021-12.pdf (0.9 MB)

Abstract

(Pre)closure spaces are a generalization of topological spaces covering also the notion of neighbourhood in discrete structures, widely used to model and reason about spatial aspects of distributed systems.
In this paper we present an abstract theoretical framework for the systematic investigation of the logical aspects of closure spaces. To this end, we introduce the notion of closure (hyper)doctrines, i.e. doctrines endowed with inflationary operators (and subject to suitable conditions). The generality and effectiveness of this concept is witnessed by many examples arising naturally from topological spaces, fuzzy sets, algebraic structures, coalgebras, and covering at once also known cases such as Kripke frames and probabilistic frames (i.e., Markov chains). By leveraging general categorical constructions, we provide axiomatisations and sound and complete semantics for various fragments of logics for closure operators. Hence, closure hyperdoctrines are useful both for refining and improving the theory of existing spatial logics, and for the definition of new spatial logics for new applications.

BibTeX - Entry

@InProceedings{castelnovo_et_al:LIPIcs.CALCO.2021.12,
  author =	{Castelnovo, Davide and Miculan, Marino},
  title =	{{Closure Hyperdoctrines}},
  booktitle =	{9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)},
  pages =	{12:1--12:21},
  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.dagstuhl.de/opus/volltexte/2021/15367},
  URN =		{urn:nbn:de:0030-drops-153678},
  doi =		{10.4230/LIPIcs.CALCO.2021.12},
  annote =	{Keywords: categorical logic, topological semantics, closure operators, spatial logic}
}

Keywords: categorical logic, topological semantics, closure operators, spatial logic
Collection: 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)
Issue Date: 2021
Date of publication: 08.11.2021

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