“Bigraph”的意思、由来-开放百科全书

A bigraph (often used in the plural bigraphs) can be modelled as the superposition of a graph (the link graph) and a set of trees (the place graph).^[1]^[2]

Each node of the bigraph is part of a graph and also part of some tree that describes how the nodes are nested. Bigraphs can be conveniently and formally displayed as diagrams.^[1] They have applications in the modelling of distributed systems for ubiquitous computing and can be used to describe mobile interactions. They have also been used by Robin Milner in an attempt to subsume Calculus of Communicating Systems (CCS) and π-calculus.^[2] They have been studied in the context of category theory.^[3]

Anatomy of a bigraph

Aside from nodes and (hyper-)edges, a bigraph may have associated with it one or more regions which are roots in the place forest, and zero or more holes in the place graph, into which other bigraph regions may be inserted. Similarly, to nodes we may assign controls that define identities and an arity (the number of ports for a given node to which link-graph edges may connect). These controls are drawn from a bigraph signature. In the link graph we define inner and outer names, which define the connection points at which coincident names may be fused to form a single link.

Foundations

where

is a set of nodes,

is a set of edges,

is the control map that assigns controls to nodes,

is the parent map that defines the nesting of nodes, and

is the link map that defines the link structure.

The notation

indicates that the bigraph has

holes (sites) and a set of inner names

and

regions, with a set of outer names

. These are respectively known as the inner and outer interfaces of the bigraph.

Formally speaking, each bigraph is an arrow in a symmetric partial monoidal category (usually abbreviated spm-category) in which the objects are these interfaces.^[4] As a result, the composition of bigraphs is definable in terms of the composition of arrows in the category.

Extensions and variants

Bigraphs with sharing

Bigraphs with sharing^[5] are a generalisation of Milner's formalisation that allows for a straightforward representation of overlapping or intersecting spatial locations. In bigraphs with sharing, the place graph is defined as a directed acyclic graph (DAG), i.e.

is a binary relation instead of a map. The definition of link graph is unaffected by the introduction of sharing. Note that standard bigraphs are a sub-class of bigraphs with sharing.

Areas of application of bigraphs with sharing include wireless networking protocols,^[6] real-time management of domestic wireless networks^[7] and mixed reality systems.^[8]

Implementations

See also

Bibliography

References

1. ^¹A Brief Introduction To Bigraphs, IT University of Copenhagen, Denmark.
2. ^¹Milner, Robin. The Bigraphical Model, University of Cambridge Computer Laboratory, UK.
3. ^{{cite journal|first=Robin|last=Milner|title=Bigraphs and Their Algebra|journal=Electronic Notes in Theoretical Computer Science|volume=209|pages=5–19|year=2008|doi=10.1016/j.entcs.2008.04.002}}
4. ^{{cite conference|first=Robin|last=Milner|title=Bigraphical Categories|series=Lecture Notes in Computer Science|volume=5710|pages=30–36|year=2009|booktitle=CONCUR 2009 - Concurrency Theory|publisher=Springer-Verlag|doi=10.1007/978-3-642-04081-8_3}}
5. ^{{cite journal|first1=Michele|last1=Sevegnani|first2=Muffy|last2=Calder|title=Bigraphs with sharing|journal=Theoretical Computer Science|volume=577|pages=43–73|year=2015|doi=10.1016/j.tcs.2015.02.011}}
6. ^{{cite journal|first1=Muffy|last1=Calder|first2=Michele|last2=Sevegnani|title=Modelling IEEE 802.11 CSMA/CA RTS/CTS with stochastic bigraphs with sharing|journal=Formal Aspects of Computing|volume=26|issue=3|pages=537–561|year=2014|doi=10.1007/s00165-012-0270-3}}
7. ^{{cite journal|first1=Muffy|last1=Calder|first2=Alexandros|last2=Koliousis|first3=Michele|last3=Sevegnani|first4=Joseph|last4=Sventek|title=Real-time verification of wireless home networks using bigraphs with sharing|journal=Science of Computer Programming|volume=80|pages=288–310|year=2014|doi=10.1016/j.scico.2013.08.004}}
8. ^{{Cite journal|last=Benford|first=Steve|last2=Calder|first2=Muffy|last3=Rodden|first3=Tom|last4=Sevegnani|first4=Michele|date=2016-05-01|title=On Lions, Impala, and Bigraphs: Modelling Interactions in Physical/Virtual Spaces|journal=ACM Trans. Comput.-Hum. Interact.|volume=23|issue=2|pages=9:1–9:56|doi=10.1145/2882784|issn=1073-0516|url=http://eprints.nottingham.ac.uk/39044/1/main_savannah-accepted.pdf}}
9. ^{{Cite book|title=Computer Aided Verification|last=Sevegnani|first=Michele|last2=Calder|first2=Muffy|date=2016-07-17|publisher=Springer International Publishing|isbn=9783319415390|editor-last=Chaudhuri|editor-first=Swarat|series=Lecture Notes in Computer Science|pages=494–501|language=en|doi=10.1007/978-3-319-41540-6_27|editor-last2=Farzan|editor-first2=Azadeh|url = http://eprints.gla.ac.uk/119384/13/119384.pdf}}