Starting from the following list of titles, clicking on the title takes you to the abstract and citation information. From there, clicking again on the title delivers the full paper as a PDF file.

Where to begin? This depends on whether your preferred point of view is recent perspective, mathematical, computational, philosophical, or hands-on.

*INTRODUCTORY:*
Visit http://chu.stanford.edu/ (unless you
just arrived here from there). There you
have the choice of using the Chu calculator to experiment with actual
couples, or of staying there and reading an elementary introduction to the
subject without having to click on anything. Or both: you can move
between the two using your browser's Back and Forward buttons.

For a comprehensive perspective on applications of Chu spaces to
concurrency, see

Transition and Cancellation in Concurrency and
Branching Time. This relatively long (40 pages)
paper extends

Orthocurrence as Both Interaction and Observation
(IJCAI'01 Workshop on Spatial and Temporal Reasoning) and

Higher Dimensional Automata Revisited (MSCS Aug. 2000,
525-548) while tying many process algebra ideas into one coherent
framework.

Another recent paper is from CONCUR 2002,

Event-State Duality: The Enriched Case
which ties up one more loose end originating with categorical enrichment
and

Temporal Structures (1989).

*MATH:*
For a more
mathematical perspective, start with

Chu spaces: Course notes for the School in Category Theory and
Applications.

*CS:*
For the computational perspective, we recommend again

Transition and Cancellation in Concurrency and
Branching Time. Earlier versions of some of these ideas but with a
considerably narrower view are Chu Spaces and their
Interpretation as Concurrent Objects or the longer

Chu Spaces: A Model of Concurrency, Vineet Gupta's
thesis.

*PHILOSOPHY:*
Philosophers may find

Rational Mechanics and Natural Mathematics of interest.

*INTERACTIVE:*
For
hands-on experience with Chu spaces, try

Chu Spaces Live, a web-based tutorial-cum-calculator
for Chu spaces.

The papers are organized into four categories: Introductory, Technical, Applications, and Prehistory.

Chu spaces: Course notes for the School in Category Theory and Applications

Notes for seven lectures at the categories summer school in Coimbra in 1999.

Chu spaces as a semantic bridge between linear logic and mathematics

The linear logic connection, presented at Linear Logic '96 (Tokyo)

Chu Spaces: Complementarity and Uncertainty in Rational Mechanics"

Notes for five lectures at the TEMPUS summer school in Budapest in 1994.

Chu Spaces and their Interpretation as Concurrent Objects

Overview of concurrency aspects of Chu spaces. In LNCS 1000.

Chu Spaces Live

A web-based tutorial-cum-calculator for Chu spaces.

Chu spaces from the representational viewpoint

Parikhfest paper, 1997.

Types as Processes, via Chu spaces

EXPRESS'97 paper.

Event-State Duality: The Enriched Case

Transition and Cancellation in Concurrency and Branching Time

Orthocurrence as Both Interaction and Observation

Higher Dimensional Automata Revisited

Chu Spaces and their Interpretation as Concurrent Objects

Configuration Structures

Gates Accept Concurrent Behavior

Chu Spaces: A Model of Concurrency

Concurrent Kripke Structures

Time and Information in Sequential and Concurrent Computation

The Stone Gamut: A Coordinatization of Mathematics

Chu spaces as a semantic bridge between linear logic and mathematics

Broadening the Denotational Semantics of Linear Logic(Early conference version of the preceding "semantic bridge" paper)

Linear Logic complements Classical Logic

Linear Logic for Generalized Quantum Mechanics

Chu Spaces: Automata with Quantum Aspects

Rational Mechanics and Natural Mathematics

"Full completeness of multiplicative linear logic for Chu spaces

Establishes a bijection between syntactic and semantic proofs for multiplicative linear logic.

"Towards full completeness for the linear logic of Chu spaces

Establishes a bijection between syntactic and semantic proofs for multiplicative linear logic restricted to two occurrences per variable.

On the Representation of Finite Abelian Groups as Chu Spaces

Represents finite abelian groups in terms of their characters.

Notes on the Chu construction and Recursion

Shows that although recursive equations cannot be solved when the language includes linear logic's

"Notes on Event Structures and Chu"

A preliminary study for Plotkin's LICS'95 paper with van Glabbeek.

"Shorter proof of universality of Chu spaces"

Another proof that the category of k-ary relational structures, for any ordinal k, embeds fully and concretely in the category of Chu spaces over 2^k.

"Chu realizes all small concrete categories"

"The Second Calculus of Binary Relations"

Chu spaces as a form of relation algebra.

Develops an algebra of partially ordered multisets as a concurrent programming language.

"Temporal Structures"

Defines time as an ordered monoid, more generally a monoidal category, and defines processes whose schedules use that notion of time as categories enriched in that monoidal structure.

"Modeling Concurrency with Geometry"

Defines a concurrent process to be a complex of cells whose dimension corresponds to the number of concurrently executing processes making up that cell.

"Event Spaces and Their Linear Logic"

Defines a self-dual category of schedules, dual to a category of automata, as a precursor to Chu spaces.

"The Duality of Time and Information"

An introduction to event spaces starting from nonconcurrent nonbranching programs defined as linear orders, which by themselves form a self-dual category.

Chu spaces: Course notes for the School in Category Theory and Applications

Chu Spaces Live
....

A web-based (Java) tutorial-cum-calculator for operating on Chu spaces with the
operations of both process algebra and linear logic (which overlap,
notably at plus and times). Caveat: your browser must support the
current version of Java (1.1), which Javasoft and Internet Explorer do
but Netscape unfortunately has fallen behind and currently only beta
prereleases of Netscape (4.05) can run Chu Spaces Live.

Chu spaces from the representational viewpoint PDF version.... HTML version

Types as Processes, via Chu spaces

Chu Spaces: Complementarity and Uncertainty in Rational Mechanics

Chu Spaces and their Interpretation as Concurrent Objects

Event-State Duality: The Enriched Case

Transition and Cancellation in Concurrency and Branching Time

Orthocurrence as both Interaction and Observation

Higher Dimensional Automata Revisited

Gates Accept Concurrent Behavior

Chu Spaces: A Model of Concurrency

We provide several equivalent definitions of Chu spaces, including two pictorial representations. Chu spaces represent processes as automata or schedules, and Chu duality gives a simple way of converting between schedules and automata. We show that Chu spaces can represent various concurrency concepts like conflict, temporal precedence and internal and external choice, and they distinguish between causing and enabling events.

We present a process algebra for Chu spaces including the standard combinators like parallel composition, sequential composition, choice, interaction, restriction, and show that the various operational identities between these hold for Chu spaces. The solution of recursive domain equations is possible for most of these operations, giving us an expressive specification and programming language. We define a history preserving equivalence between Chu spaces, and show that it preserves the causal structure of a process.

We define a process algebra of event Kripke structures, showing how to combine them in the usual ways---parallel composition, sequential composition, choice, interaction and iteration. Various properties of these connectives like associativity and distributivity are proved. We then show that Winskel's event structures can be embedded in the class of event Kripke structures, and define partial synchronous composition, the primary connective for event structures, for event Kripke structures, and show its equivalence to Winskel's definition.

Time and Information in Sequential and Concurrent Computation

The Stone Gamut: A Coordinatization of Mathematics

Chu spaces as a semantic bridge between linear logic and mathematics

Broadening the Denotational Semantics of Linear Logic

Linear Logic complements Classical Logic

Linear Logic for Generalized Quantum Mechanics

*This paper is really prehistory, in that it hints
at Chu spaces only at the end, where it calls them partial distributive
lattices.
*

Chu Spaces: Automata with Quantum Aspects

Rational Mechanics and Natural Mathematics

This paper addresses the chief stumbling block for Descartes' 17th-century philosophy of mind-body dualism, how can the fundamentally dissimilar mental and physical planes causally interact with each other? We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. We use Chu spaces and residuation to derive all causal interaction, both between and within the two planes, from a uniform and algebraically rich theory of between-plane interaction alone. Lifting the two-valued Boolean logic of binary relations to the complex-valued fuzzy logic of quantum mechanics transforms residuation into a natural generalization of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the Heisenberg-Schr"odinger quantum-mechanical solution to analogous problems of causal interaction in physics.

On the Representation of Finite Abelian Groups as Chu Spaces

Notes on the Chu construction and Recursion

Notes on Event Structures and Chu

Shorter proof of universality of Chu spaces

Chu realizes all small concrete categories

The Second Calculus of Binary Relations

Towards Full Completeness for the Linear Logic of Chu spaces (or)

Full Completeness of the Linear Logic of Chu Spaces

Modeling Concurrency with Partial Orders

Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a natural class of models of concurrent processes. The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets or pomsets. Just as sets of strings form languages, so do sets of pomsets form processes. We introduce a number of operations useful for specifying concurrent processes and demonstrate their utility on some basic examples. Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components. The general benefits of the approach are that it is conceptually straightforward, involves fewer artificial constructs than many competing models of concurrency, yet is applicable to a considerably wider range of types of systems, including systems with buses and ethernets, analog systems, and real-time systems.

Modeling Concurrency with Geometry

Event Spaces and Their Linear Logic

Here we achieve both via the notion of an event space as a poset with all nonempty joins representing concurrence and a top representing the unreachable event. The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state. The algebra is that of a parallel programming language expanded to the language of full linear logic, Girard's axiomatization of which is satisfied by the event space interpretation of this language.

Event spaces resemble finite-dimensional vector spaces in distinguishing tensor product from direct product and in being isomorphic to their double dual, but differ from them in distinguishing direct product from direct sum and tensor product from tensor sum.

The Duality of Time and Information

To accommodate flexible distributed computing systems we then bring in choice and concurrency and pass to partially ordered time and information, the formal basis for this extension being Birkhoff-Stone dualtiy. A degree of freedom in how this is done permits a perfectly symmetric logic of computation amounting to Girard's full linear logic, which we view as the natural logic of computation when equal importance is attached to choice and concurrency.

We conclude with an assessment of the prospects for extending the duality to other organizations of time and information besides partial orders in order to accommodate real time, nonmonotonic logic, and automata that can forget, and speculate on the philosophical significance of the duality.