The WHILE hierarchy of program schemes is infinite

  • Can Adam Albayrak
  • Thomas Noll
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1378)


We exhibit a sequence S n (n ≥ 0) of while program schemes, i. e., while programs without interpretation, with the property that the while nesting depth of S n is n, and prove that any while program scheme which is scheme equivalent to S n , i. e., equivalent for all interpretations over arbitrary domains, has while nesting depth at least n. This shows that the while nesting depth imposes a strict hierarchy (the while hierarchy) when programs are compared with respect to scheme equivalence and contrasts with Kleene's classical result that every program is equivalent to a program of while nesting depth 1 (when interpreted over a fixed domain with arithmetic on non-negative integers). Our proof is based on results from formal language theory; in particular, we make use of the notion of star height of regular languages.


  1. 1.
    Corrado Böhm and Giuseppe Jacopini. Flow diagrams, Turing machines and languages with only two formation rules. Communications of the ACM, 9(5):366–371, 1966.CrossRefGoogle Scholar
  2. 2.
    Steven Brown, David Gries, and Thomas Szymanski. Program schemes with pushdown stores. SIAM Journal on Computing, 1:242–268, 1972.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Rina S. Cohen and Janusz A. Brzozowski. General properties of star heigt of regular events. Journal of Computer and System Sciences, 4:260–280, 1970.MathSciNetGoogle Scholar
  4. 4.
    F. Dejean and M. P. Schützenberger. On a question of Eggan. Information and Control, 9:23–25, 1966.CrossRefMathSciNetGoogle Scholar
  5. 5.
    L. C. Eggan. Transition graphs and the star-height of regular events. The Michigan Mathematical Journal, 10:385–397, 1963.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Iu. I. Ianov. On matrix program schemes. Communications of the ACM, 12(1):3–6, 1958.CrossRefGoogle Scholar
  7. 7.
    Iu. I. Ianov. On the equivalence and transformation of program schemes. Communications of the ACM, 10(1):8–12, 1958.CrossRefGoogle Scholar
  8. 8.
    Iu. I. Ianov. The logical schemes of algorithms. Problems of Cybernetics, 1:82–140, 1960.Google Scholar
  9. 9.
    Klaus Indermark. On a class of schematic languages. In R. Aguilar, editor, Formal Languages and Programming, Proceedings of a Seminar Organized by UAM-IBM Scientific Center, pages 1–13, 1975.Google Scholar
  10. 10.
    S. C. Kleene. General recursive functions of natural numbers. Mathematische Annalen, 112:727–742, 1936.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    D. C. Luckham, D. M. R. Park, and M. S. Paterson. On formalised computer programs. Journal of Computer and System Sciences, 4(3):220–249, 1970.MathSciNetGoogle Scholar
  12. 12.
    Michael S. Paterson and Carl E. Hewitt. Comparative schematology. Technical Report AI memo 201, MIT AI Lab, Publications Office, 545 Technology Sq. Cambridge, MA 02139, 1970.Google Scholar
  13. 13.
    Joseph D. Rutledge. On Ianov's program schemata. Journal of the ACM, 11(1):1–9, 1964.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Arto Salomaa. Jewels of formal language theory. Computer Science Press, 1981.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Can Adam Albayrak
    • 1
  • Thomas Noll
    • 1
  1. 1.RWTH AachenAachenGermany

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