## 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.

This work was supported by ONR under grant number N00014-92-J-1974.

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## References

A. Asperti. A logic for concurrency. Manuscript, November 1987.

M. Barr. *

*-Autonomous categories*, volume 752 of*Lecture Notes in Mathematics*. Springer-Verlag, 1979.M. Barr. *-Autonomous categories and linear logic.

*Math Structures in Comp. Sci.*, 1(2):159â€“178, 1991.C. Brown and D. Gurr. A categorical linear framework for Petri nets. In J. Mitchell, editor,

*Logic in Computer Science*, pages 208â€“218. IEEE Computer Society, June 1990.J.A. Bergstra and J.W. Klop. Process algebra for synchronous communication.

*Information and Control*, 60:109â€“137, 1984.J.A. Bergstra and J.W. Klop. Process theory based on bisimulation semantics. In

*Proc. REX School/Workshop on Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency*, pages 50â€“122, Springer-Verlag, 1989.R.T Casley, R.F. Crew, J. Meseguer, and V.R. Pratt. Temporal structures.

*Math. Structures in Comp. Sci.*, 1(2):179â€“213, July 1991.C. A. Gunter and V. Gehlot. Nets as tensor theories. (preliminary report). In G. De Michelis, editor,

*Applications of Petri Nets*, pages 174â€“191, 1989. Also Univ. of Pennsylvania, Logic and Computation Report Nr 17.J.-Y. Girard. Linear logic.

*Theoretical Computer Science*, 50:1â€“102, 1987.S. Gay and R. Nagarajan. A typed calculus of synchronous processes. In

*Logic in Computer Science*, pages 210â€“220. IEEE Computer Society, June 1995.V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In

*Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci.*, pages 62â€“71, 1993.V. Gupta. Concurrent Kripke structures. In

*North American Process Algebra Workshop*, Proceedings, Cornell CS-TR-93-1369, August 1993.V. Gupta.

*Chu Spaces: A Model of Concurrency*. PhD thesis, Stanford University, September 1994. Tech. Report, available as ftp://boole.stanford.edu/pub/gupthes.ps.Z.C.A.R. Hoare. Communicating sequential processes.

*Communications of the ACM*, 21(8):666â€“672, August 1978.P.T. Johnstone.

*Stone Spaces*. Cambridge University Press, 1982.G. Kahn. The semantics of a simple language for parallel programming. In

*Proc. IFIP Congress 74*North-Holland, Amsterdam, 1974.G.M. Kelly.

*Basic Concepts of Enriched Category Theory*, London Math. Soc. Lecture Notes. 64. Cambridge University Press, 1982.Y. Lafont and T. Streicher. Games semantics for linear logic. In

*Proc. 6th Annual IEEE Symp. on Logic in Computer Science*, pages 43â€“49, Amsterdam, July 1991.R. Milner.

*Communication and Concurrency*. Prentice-Hall, 1989.R. Milner. Action calculi, or syntactic action structures. In

*MFCS'93*, Proceedings, volume 711 of*Lecture Notes in Computer Science*, pages 105â€“121, Springer-Verlag, 1993.R. Milner, J. Parrow, and D Walker. A calculus of mobile processes.

*Information and Control*, 100:1â€“77, 1992.M. Nielsen, G. Plotkin, and G. Winskel. Petri nets, event structures, and domains, part I.

*Theoretical Computer Science*, 13:85â€“108, 1981.C.A. Petri. Fundamentals of a theory of asynchronous information flow. In

*Proc. IFIP Congress 62*, pages 386â€“390, 1962. North-Holland, Amsterdam.V.R. Pratt. Some constructions for order-theoretic models of concurrency. In

*Proc. Conf. on Logics of Programs*, volume 193 of*Lecture Notes in Computer Science*, pages 269â€“283, Springer-Verlag, 1985.V.R. Pratt. Modeling concurrency with partial orders.

*Int. J. of Parallel Programming*, 15(1):33â€“71, February 1986.V.R. Pratt. The duality of time and information. In

*Proc. of CONCUR'92*, volume 630 of*Lecture Notes in Computer Science*, pages 237â€“253, Springer-Verlag, 1992.V.R. Pratt. The second calculus of binary relations. In

*MFCS'93*, Proceedings, volume 711 of*Lecture Notes in Computer Science*, pages 142â€“155, Springer-Verlag, 1993.V.R. Pratt. Chu spaces: Automata with quantum aspects. In

*Proc. Workshop on Physics and Computation (PhysCornp'94)*, Dallas, 1994. IEEE.V.R. Pratt. Time and information in sequential and concurrent computation. In

*Proc. Theory and Practice of Parallel Programming (TPPP'94)*, Sendai, Japan, November 1994.V.R. Pratt. Rational mechanics and natural mathematics. In

*TAPSOFT'95*, volume 915 of*Lecture Notes in Computer Science*, pages 108â€“122, Springer-Verlag, 1995.V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In

*Logic in Computer Science*, pages 444â€“454. IEEE Computer Society, June 1995.H.A. Priestley. Representation of distributive lattices.

*Bull. London Math. Soc*, 2:186â€“190, 1970.K.I. Rosenthal.

*Quantales and their applications*. Longman Scientific and Technical, 1990.D. Scott. Data types as lattices.

*SIAM Journal on Computing*, 5(3):522â€“587, 1976.R.A.G Seely. Linear logic, *-autonomous categories and cofree algebras. In

*Categories in Computer Science and Logic*, volume 92 of*Contemporary Mathematics*, pages 371â€“382, held June 1987, Boulder, Colorado, 1989.M. Stone. The theory of representations for Boolean algebras.

*Trans. Amer. Math. Soc*, 40:37â€“111, 1936.B. Trakhtenbrot. Origins and metamorphoses of the trinity: logic, nets, automata. In D. Kozen, editor,

*Logic in Computer Science*, pages 506â€“507. IEEE Computer Society, June 1995.R. Van Glabbeek and G. Plotkin. Configuration structures. In

*Logic in Computer Science*, pages 199â€“209. IEEE Computer Society, June 1995.

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Pratt, V. (1995). Chu spaces and their interpretation as concurrent objects. In: van Leeuwen, J. (eds) Computer Science Today. Lecture Notes in Computer Science, vol 1000. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015256

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