Mathematical systems theory

, Volume 6, Issue 4, pp 359–374

Realization is universal

  • J. A. Goguen
Article

Abstract

Behavior is left adjoint to minimal realization, as functors between certain categories of machines and behaviors. This gives a succinct characterization of minimal realization valid for discrete as well as linear machines, with or without finiteness condition. Realization theory is therefore expressed in contemporary algebra in a way which reveals its inner structure and suggests generalizations. An adjunction between regular sets and finite state acceptors follows as a corollary.

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References

  1. [1]
    M. A. Arbib,Theories of Abstract Automata, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.Google Scholar
  2. [2]
    M. A. Arbib andH. P. Zeiger, On the relevance of abstract algebra to control theory,Automatica 5 (1969), 589–606.Google Scholar
  3. [3]
    Y. Give'on andY. Zalcstein, Algebraic structures in linear systems theory,J. Computer and System Sciences 4 (1970), 539–556.Google Scholar
  4. [4]
    J. A. Goguen, “Mathematical Representation of Hierarchically Organized Systems”, inGlobal Systems Dynamics (E. Attinger, editor), S. Karger, Basel, 1970.Google Scholar
  5. [5]
    J. A. Goguen, “Systems and Minimal Realization”, inProc. 1971 IEEE Conference on Decision and Control, Miami, Florida, 1972.Google Scholar
  6. [6]
    J. A. Goguen, Minimal realization of machines in closed categories, to appear inBull. Amer. Math. Soc. Google Scholar
  7. [7]
    R. E. Kalman, “Introduction to the Algebraic Theory of Linear Dynamical Systems”,Lecture Notes in Operations Research and Mathematical Economics, Vol. 11, Springer Verlag, Berlin, 1969.Google Scholar
  8. [8]
    S. MacLane,Categories for the Working Mathematician, Springer Verlag, Berlin, 1972.Google Scholar
  9. [9]
    S. MacLane andG. Birkhoff,Algebra, MacMillan, New York, 1967.Google Scholar
  10. [10]
    M. B. Mesarović, “Foundations for a General Systems Theory”, inViews on General Systems, Proceedings of the Second Systems Symposium at Case Institute of Technology, J. Wiley, New York, 1964.Google Scholar
  11. [11]
    B. Mitchell,Theory of Categories, Academic Press, New York, 1965.Google Scholar
  12. [12]
    A. Nerode, Linear automaton transformations,Proc. Amer. Mach. Soc. 9 (1958), 541–544.Google Scholar
  13. [13]
    Y. Rouchaleau, Finite-dimensional, constant, discrete-time, linear dynamical systems over some classes of commutative rings, Ph.D. Dissertation, Operations Research Dept., Stanford University, 1972.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1973

Authors and Affiliations

  • J. A. Goguen
    • 1
  1. 1.Mathematical Sciences DepartmentIBM T. J. Watson Research CenterYorktown HeightsUSA

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