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

Real-time recognition of substring repetition and reversal


Real-time multitape Turing machine algorithms are presented for recognizing the languages {wxyxz∈Σ*:|w|=r|x,|z| =t|x|} and {wxyx R z ∈Σ*:|w|=r|x,|z| =t|x|} for fixedr, s, andt and for string-matching with “forced mismatches.”

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


  1. 1.

    A. J. Atrubin, A one-dimensional real-time iterative multiplier,IEEE Transactions on Electronic Computers EC-14 (1965), 394–399.

  2. 2.

    S. N. Cole, Real-time computation by iterative arrays of finite-state machines. Doctoral Thesis, Harvard University, Cambridge, Massachusetts, August 1964.

  3. 3.

    S. N. Cole, Real-time computation byn-dimensional iterative arrays of finite-state machines,IEEE Transactions on Computers C-13 (1969), 349–365.

  4. 4.

    S. A. Cook andS. O. Aanderaa, On the minimum computation time of functions,Transactions of the American Mathematical Society 142 (1969), 291–314.

  5. 5.

    N. J. Fine andH. S. Wilf, Uniqueness theorems for periodic functions,Proceedings of the American Mathematical Society 16 (1965), 109–114.

  6. 6.

    N. J. Fine and M. S. Paterson, String-matching and other products, inComplexity of Computation (SIAM-AMS Proceedings 7), R. M. Karp, ed., American Mathematical Society, Providence, Rhode Island (1974), 113–125.

  7. 7.

    P. C. Fischer, A. R. Meyer, andA. L. Rosenberg, Real-time simulation of multihead tape units,Journal of the Association for Computing Machinery 19 (1972), 590–607.

  8. 8.

    Z. Galil, String-matching in real time, submitted for publication.

  9. 9.

    Z. Galil, Palindrome recognition in real time by a multitape Turing machine,Journal of Computer and System Sciences, to appear.

  10. 10.

    D. E. Knuth, J. H. Morris, Jr., andV. R. Partt, Fast pattern matching in strings,SIAM Journal on Computing 6 (1977), 323–350.

  11. 11.

    R. C. Lyndon andM. P. Schützenberger, The equationa M=N c P in a free group,Michigan Mathematical Journal 9 (1962), 289–298.

  12. 12.

    G. Manacher, A new linear-time “on-line” algorithm for finding the smallest initial palindrome of a string,Journal of the Association for Computing Machinery 22 (1975), 346–351.

  13. 13.

    J. I. Seiferas, Iterative arrays with direct central control,Acta Informatica 8 (1977), 177–192.

  14. 14.

    A. O. Slisenko, Recognition of palindromes by multihead Turing machines (in Russian), inProblems in the Constructive Trend in Mathematics. VI (Proceedings of the Steklov Institute of Mathematics, No. 129), V. P. Orevkov and N. A. Sanin, ed., Academy of Sciences of the USSR (1973), 30–202; English translation by R. H. Silverman, American Mathematical Society, Providence, Rhode Island (1976), 25–208.

  15. 15.

    A. O. Slisenko, private communication, July 1976.

  16. 16.

    A. O. Slisenko, A simplified proof of real-time recognizability of palindromes on Turing machines (in Russian), inTheoretical Applications of Methods of Mathematical Logic. II (Proceedings of the Scientific Seminars of the Leningrad Branch of the Steklov Institute of Mathematics,68, G. E. Mintz and V. P. Orevkov, ed., Academy of Sciences of the USSR (1977), 123–139.

Download references

Author information

Additional information

This research was supported in part by the National Science Foundation under Grant MCS77-06613 (first author) and the Bat-Sheva Fund (second author). Part of the work was conducted while the second author was at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York. A preliminary report was included in a paper the authors presented at the Seventeenth Annual IEEE Symposium on Foundations of Computer Science, Houston, Texas, October 1976.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Seiferas, J., Galil, Z. Real-time recognition of substring repetition and reversal. Math. Systems Theory 11, 111–146 (1977).

Download citation

Key words and phrases

  • palindrome
  • real-time
  • Turing machine
  • language recognition
  • repetition
  • reversal
  • string-matching
  • pattern-matching
  • periodicity
  • predictability condition