Posets With Interfaces as a Model for Concurrency

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Abstract

We introduce posets with interfaces (iposets) and generalise their standard serial composition to a new gluing composition. In the partial order semantics of concurrency, interfaces and gluing allow modelling events that extend in time and across components. Alternativelytaking a decompositional view, interfaces allow cutting through events, while serial composition may only cut through edges of a poset. We show that iposets under gluing composition form a category, which generalises the monoid of posets under serial composition up to isomorphism. They form a 2-category when a subsumption order and a lax tensor in the form of a non-commutative parallel composition are added, which generalises the interchange monoids used for modelling series-parallel posets. We also study the gluing-parallel hierarchy of iposets, which generalises the standard series-parallel one. The class of gluing-parallel iposets contains that of series-parallel posets and the class of interval orders, which are well studied in concurrency theory, too. We also show that it is strictly contained in the class of all iposets by identifying several forbidden substructures.

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Bibtex (lrde.bib)

@Article{	  fahrenberg.22.iandc,
  author	= {Uli Fahrenberg and Christian Johansen and Georg Struth and
		  Krzysztof Ziemia{\'n}ski},
  title		= {Posets With Interfaces as a Model for Concurrency},
  journal	= {Information and Computation},
  volume	= 285,
  number	= {B},
  pages		= 104914,
  year		= 2022,
  url		= {https://doi.org/10.1016/j.ic.2022.104914},
  doi		= {10.1016/j.ic.2022.104914},
  timestamp	= {Tue, 28 Jun 2022 21:08:18 +0200},
  biburl	= {https://dblp.org/rec/journals/iandc/FahrenbergJSZ22.bib},
  bibsource	= {dblp computer science bibliography, https://dblp.org},
  abstract	= {We introduce posets with interfaces (iposets) and
		  generalise their standard serial composition to a new
		  gluing composition. In the partial order semantics of
		  concurrency, interfaces and gluing allow modelling events
		  that extend in time and across components. Alternatively,
		  taking a decompositional view, interfaces allow cutting
		  through events, while serial composition may only cut
		  through edges of a poset. We show that iposets under gluing
		  composition form a category, which generalises the monoid
		  of posets under serial composition up to isomorphism. They
		  form a 2-category when a subsumption order and a lax tensor
		  in the form of a non-commutative parallel composition are
		  added, which generalises the interchange monoids used for
		  modelling series-parallel posets. We also study the
		  gluing-parallel hierarchy of iposets, which generalises the
		  standard series-parallel one. The class of gluing-parallel
		  iposets contains that of series-parallel posets and the
		  class of interval orders, which are well studied in
		  concurrency theory, too. We also show that it is strictly
		  contained in the class of all iposets by identifying
		  several forbidden substructures.}
}