## Chu spaces and their interpretation as concurrent objects (2005)

Citations: | 21 - 0 self |

### BibTeX

@MISC{Pratt05chuspaces,

author = {Vaughan Pratt},

title = {Chu spaces and their interpretation as concurrent objects },

year = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

A Chu space is a binary relation = | from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of particular interest for computer science is their interpretation as computational processes, which takes A to be a schedule of events distributed in time, X to be an automaton of states forming an information system in the sense of Scott, and the pairs (a, x) in the = | relation to be the individual transcriptions of the making of history. The traditional homogeneous binary relations of transition on X and precedence on A are recovered as respectively the right and left residuals of the heterogeneous binary relation = | with itself. The natural algebra of Chu spaces is that of linear logic, made a process algebra by the process interpretation.

### Citations

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(Show Context)
Citation Context ...ra. This motivates the development of process algebras forming a suitable basis for concurrent programming languages. Noteworthy such algebras include Hoare's Communicating Sequential Processes (CSP) =-=[Hoa78]-=-, Milner's Calculus of Communicating Systems (CCS) [Mil89], and Bergstra and Klop's Algebra of Concurrent Processes (ACP) [BK84, BK89]. The token game does not lend itself well to algebra, a limitatio... |

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Citation Context ...ified by the category SProc [GN95], and of Milner on the ��-calculus [MPW92] and more recently action calculi [Mil93]. Moreover its conception of network is more the channel-connected modules of K=-=ahn [Kah74] than the -=-alternating places and transitions of Petri [Pet62]. Petri nets express the duality of events and states in terms of a "token game" played on a bipartite graph. This bipartiteness is the dis... |

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(Show Context)
Citation Context ...redated Lafont and Streicher's "game." An advantage of "Chu space" is that it requires no disambiguating qualification to uniquely identify it, unlike "game." By analogy =-=with categories enriched in V [Kel82]-=- one might refer to the objects of the general Chu construction Chu(V; k) as V -enriched Chu spaces. Acknowledgements The application of Chu spaces to computation was developed in collaboration with m... |

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(Show Context)
Citation Context ...st noticed by Seely [See89], furnishing Girard's linear logic [Gir87] with a natural constructive semantics. Barr then proposed the Chu construction as a source of constructive models of linear logic =-=[Bar91]-=-. The case V = Set is important for its combination of simplicity and generality. This case was first treated explicitly by Lafont and Streicher [LS91], where they treated its connections with von Neu... |

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(Show Context)
Citation Context ... ties together three long-established computational frameworks, it neglects more recent developments such as the recent work of Abramsky on interaction categories as exemplified by the category SProc =-=[GN95], an-=-d of Milner on the ��-calculus [MPW92] and more recently action calculi [Mil93]. Moreover its conception of network is more the channel-connected modules of Kahn [Kah74] than the alternating place... |

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(Show Context)
Citation Context ... objects of the general Chu construction Chu(V; k) as V -enriched Chu spaces. Acknowledgements The application of Chu spaces to computation was developed in collaboration with my student Vineet Gupta =-=[Gup94]-=-. Rob van Glabbeek and Gordon Plotkin have separately been invaluable sounding-boards and sources of ideas, and jointly of results [VGP95]. Carolyn Brown, with Doug Gurr, was the first to explore the ... |

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(Show Context)
Citation Context ...en invaluable sounding-boards and sources of ideas, and jointly of results [VGP95]. Carolyn Brown, with Doug Gurr, was the first to explore the connection between the Chu construction and concurrency =-=[BG90]-=-, and has been a useful source of insights. Dusko Pavlovic has developed a formal sense in which Chu spaces are universal for mathematics, more correctly their couniversality. I am also grateful to H.... |

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(Show Context)
Citation Context ...al spaces [LS91]. These representations can be combined to represent topological relational structures such as topological groups, topological vector spaces (the main *-autonomous category studied in =-=[Bar79]-=-), ordered Stone spaces [Pri70], and so on. More recently (note in preparation) we have shown that every small concrete category C is realizable in ChuK where K is the disjoint union of the underlying... |

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(Show Context)
Citation Context ...t us from delving into here. To summarize briefly, distances are organized as a quantale [Ros90] or complete semilattice monoid. This combines our earlier investigations of purely temporal structures =-=[CCMP91]-=- with our more recent work on time-information duality and Chu spaces in a way that permits the temporal-structure results to be lifted directly, via duality, to information structures such as automat... |

21 |
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(Show Context)
Citation Context ...parably simple structure. The process interpretation of Chu spaces makes this structure a process algebra interpretation of linear logic. This reinforces the connections previously noticed by Asperti =-=[Asp87]-=- and Gehlot and Gunter [GG89] between linear logic and concurrency in the Petri net setting (Section 4). 2 Definitions Definition 1 A Chu space over a set K is a triple A = (A; =j; X) consisting of se... |

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Citation Context ...process interpretation of Chu spaces makes this structure a process algebra interpretation of linear logic. This reinforces the connections previously noticed by Asperti [Asp87] and Gehlot and Gunter =-=[GG89]-=- between linear logic and concurrency in the Petri net setting (Section 4). 2 Definitions Definition 1 A Chu space over a set K is a triple A = (A; =j; X) consisting of sets A and X and a function =j ... |

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