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Mathematical systems theory

, Volume 6, Issue 4, pp 359–374 | Cite as

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.

Keywords

Computational Mathematic Realization Theory State Acceptor Finiteness Condition Minimal Realization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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