Invertibility of linear finite automata over a ring

  • Renji Tao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)


This paper deals with the invertibility problem of linear finite automata over a finite commutative ring with 1. It is shown that for any linear finite automaton M over a finite commutative ring with 1 the following statements are equivalent: 1. M is weakly invertible, 2. the transfer function matrix of M has a left inverse matrix, 3. there is a weak inverse linear finite automaton of M. And for linear finite automata over the ring of integers modulo q, this paper gives a decision procedure for an invertible (a weakly invertible, resp.) linear finite automaton with delay t and a construction method of its linear inverse (weak inverse, resp.) with delay t.


Finite Field Finite Automaton Zero Divisor Elementary Transformation Invertible Matrice 
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  1. 1.
    D. A. Huffman, Canonical forms for information-lossless finite-state logical machines, IRE Trans. Cir. Theory, 6(1959),special supplement, 41–59.Google Scholar
  2. 2.
    J. L. Massey and M. K. Sain, Inverse of linear sequential circuits, IEEE Trans. Comput., 17(1968),330–337.Google Scholar
  3. 3.
    G. D. Forney, Convolution codes I:algebraic structure, IEEE Trans. Infor. Theory, 16(1970),720–738.Google Scholar
  4. 4.
    H. J. Clinton, Finite Automata Admitting Inverses with Some Application to Cryptography, North Canolina State University at Raleigh, 1970.Google Scholar
  5. 5.
    Tao Jen-chi, invertible linear finite automata, Scientia Sinica, 16(1973),565–581.Google Scholar
  6. 6.
    A. A. Kyrmit, Information Lossless Automata of Finite Order, New York: Wiley, 1974.Google Scholar
  7. 7.
    Tao Renji, Invertibility of Finite Automata, Beijing: Science Press, 1979.(Chinese)Google Scholar
  8. 8.
    Chen Shihua, On the structure of weak inverses of a weakly invertible linear finite automaton, Chinese J. of Computer, 4(1981),409–419. (in Chinese)Google Scholar
  9. 9.
    Tao Renji, Relationship between bounded error propagation and feedforward invertibility, Kexue Tongbao,27(1982),680–682.Google Scholar
  10. 10.
    Tao Renji, Some results on the structure of feedforward inverses, Scientia Sinica, 27(1984),157–162.Google Scholar
  11. 11.
    Chen Shihua, On the structure of finite automata of which M′ is an(weak) inverse with delay t, J. of Comput. Sci. & Tech., 1(1986),no.2,54–59.Google Scholar
  12. 12.
    Chen Shihua, On the structure of (weak) inverses of an (weakly) invertible finite automaton. J. of Comput. & Tech., 1(1986),no.3,92–100.Google Scholar
  13. 13.
    Qi Yulu,Chen Shihua and Tao Renji, A finite automaton cryptosystem and its software implementation, in International Conference on Computer and Communications Proceedings 1986,550–557.Google Scholar
  14. 14.
    Tao Renji and Chen Shihua, A finite automaton public key cryptosystem and digital signatures, Chinese J. of Computer,8(1985),401–409. (in Chinese)Google Scholar
  15. 15.
    Tao Renji and Chen Shihua, Two varieties of finite automaton public key cryptosystem and digital signatures, J. of Comput. Sci. & Tech., 1(1986),no.1,9–18.Google Scholar
  16. 16.
    B. R. McDonald, Finite rings with identity, New York: Marcel Dekker, 1974.Google Scholar
  17. 17.
    B. R. McDonard, Linear algebra over commutative rings, New York: Marcel Dekker, 1984.Google Scholar
  18. 18.
    P. Camion, L. S. Levy and H. B. Mann, Linear equations over commutative ring, J. of Algebra, 18(1971),432–446.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Renji Tao
    • 1
  1. 1.Institute of SoftwareAcademia SinicaBeijingChina

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