Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Deterministic representation of probabilistic systems by ergodic machines

  • 38 Accesses

Abstract

Homomorphic simplification is one way of defining valid simplication of deterministic models. To allow the use of similar techniques for probabilistic systems, we introduce the concept of an ergodic machine. An ergodic machine is a specification for a deterministic system and it represents and associated probabilistic system when viewed properly. The probabilistic features arise by using an ergodic transformation on a “hidden” partial state space. An ergodic transformation can be thought of as an ideal random number generator. In this paper, we develop the concepts of ergodic machines necessary for faithfully representing probabilistic systems and make connections to markov chains and pseudo random number generators.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    S. Aggarwal, Ergodic Machines—Probabilistic and Approximate Homomorphic Simplification,University of Michigan, Tech. Report LCG-184, August 1975.

  2. [2]

    P. Billingsley,Ergodic Theory and Information, John Wiley and Sons, Inc., 1965.

  3. [3]

    L. Breiman,Probability, Addison-Wesley, 1968.

  4. [4]

    G. C. Corynen,A. Mathematical Theory of Modeling and Simulation, Ph.D. Thesis, Computer, Information, and Control Engineering, The University of Michigan, 1974.

  5. [5]

    W. Feller,An Introduction to Probability Theory, John Wiley and Sons, 1968.

  6. [6]

    N. Y. Foo, Homomorphic Simplification of Systems,University of Michigan, Technical Report LCG-156, July 1974.

  7. [7]

    P. R. Halmos,Lectures on Ergodic Theory, Tokyo: The Mathematical Society of Japan (1956).

  8. [8]

    J. Hartmanis andR. E. Stearns,Algebraic Structure Theory of Sequential Machines, Prentice-Hall, 1969.

  9. [9]

    D. E. Knuth,The Art of Computer Programming, Vol. 2, Addison-Wesley, 1971.

  10. [10]

    C. V. Page, Equivalences Between Probabilistic and Deterministic Sequential Machines,Inform. and Control (1966), 469–520.

  11. [11]

    A. Paz, Homomorphisms Between Stochastic Sequential Machines and Related Problems,Math. System Theory, Vol.2, (1968), 223–245.

  12. [12]

    A. Paz,Introduction to Probabilistic Automata, Academic Press (1971).

  13. [13]

    B. P. Zeigler, On the Formulation of Problems in Modelling and Simulation within the Framework of Mathematical Systems Theory,Proc. Sixth International Congr. on Cybernetics, Namur, Belgium, 1972.

  14. [14]

    B. P. Zeigler,Theory of Modelling and Simulation, John Wiley and Sons. 1976.

Download references

Author information

Additional information

This research represents a portion of the author's Ph.D. dissertation at the University of Michigan under the supervision of Prof. Bernard Zeigler. The work was partially supported by NSF Grant DCR71-01997 and the U.S. Energy Research and Development Administration, contract No. W-7405-Eng-48.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aggarwal, S. Deterministic representation of probabilistic systems by ergodic machines. Math. Systems Theory 10, 345–361 (1976). https://doi.org/10.1007/BF01683284

Download citation

Keywords

  • State Space
  • Random Number
  • Computational Mathematic
  • Number Generator
  • Probabilistic System