Mathematical systems theory

, Volume 3, Issue 3, pp 193–221 | Cite as

A general theory of translation

  • A. V. Aho
  • J. E. Hopcroft
  • J. D. Ullman
Article

Abstract

The concept of a translation is fundamental to any theory of compiling. Formally, atranslation is any set of pairs of words. Classes of finitely describable translations are considered in general, from the point of view of balloon automata [17, 18, 19].

A translation can be defined by atransducer, a device with an input tape and an output terminal. If, with inputx, the stringy appears at the output terminal, then (x, y) is in the translation defined by the transducer. One can also define a translation by a two input taperecognizer. Ifx andy are placed on the two tapes, the recognizer tells if (x, y) is in the defined translation.

One can define closed classes of transducers and recognizers by:
  1. (1)

    restricting the way in which infinite storage may be used (pushdown structure, stack structure, etc.),

     
  2. (2)

    allowing the finite control to be nondeterministic or deterministic,

     
  3. (3)

    allowing one way or two way motion on the input tapes.

     
We have some results on classes of translations which can be categorized roughly into three types.
  1. (a)

    Translations defined by certain classes of transducers and recognizers are equivalent.

     
  2. (b)

    Translations of a given class are sometimes closed under composition and decomposition with a finite memory translation (gsm mapping).

     
  3. (c)

    A nondeterministically defined translation can be expressed as the composition of a finitely defined translation and a related deterministically defined translation in many cases.

     

In addition, ifC is a class of translations, then one can write a compiler-compiler to render any translationT inC and only if the following question is solvable: For any translationT inC and stringx, does there exist ay such that (x, y) is inT? We shall show that, in general, the decidability of this question is equivalent to the decidability of one or more questions from automata theory, depending upon the type of devices defining the classC.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Irons, A syntax directed compiler forAlgol 60,Comm. ACM. 4 (1961), 51–55.Google Scholar
  2. [2]
    A. V. Aho, Indexed grammars—An extension of context free grammars,J. Assoc. Comput. Mach. 15 (1968), 647–671.Google Scholar
  3. [3]
    D. J. Rosenkrantz, Programmed grammars and classes of formal languages,J. Assoc. Comput. Mach. 16 (1969), 107–131.Google Scholar
  4. [4]
    R. W. Floyd, On the nonexistence of a phrase structure grammar forAlgol 60,Comm. ACM. 5 (1962), 483–484.Google Scholar
  5. [5]
    P. M. Lewis II andR. E. Stearns, Syntax Directed Transduction,J. Assoc. Comput. Mach. 15 (1968), 464–488.Google Scholar
  6. [6]
    K. Culik, Well-translatable languages andAlgol-like languages, IFIP Working Conference onFormal Language Description Languages pp. 76–85, N. Holland Press, Amsterdam, 1966.Google Scholar
  7. [7]
    M. Paull, “Bilateral Descriptions of Syntactic Mappings”,First Annual Princeton Conference on Information Sciences and Systems, 1967.Google Scholar
  8. [8]
    S. Ginsburg, Examples of abstract machines,IRE Trans. EC-11 (1962), 132–135.Google Scholar
  9. [9]
    C. C. Elgot andJ. E. Mezei, On relations defined by generalized finite automata,IBM J. Res. Develop. 9 (1965), 47–68.Google Scholar
  10. [10]
    S. Ginsburg andG. Rose, Operations which preserve definability in languages,J. Assoc. Comput. Mach. 10 (1963), 175–195.Google Scholar
  11. [11]
    S. Ginsburg andG. Rose, A characterization of machine mappings,Canad. J. Math. 18 (1966), 381–388:Google Scholar
  12. [12]
    J. N. Gray andM. A. Harrison, The theory of sequential relations,Information and Control,9 (1966), 435–468.Google Scholar
  13. [13]
    S. Ginsburg andG. Rose, Preservation of languages by transducers,Information and Control,9 (1966), 153–176.Google Scholar
  14. [14]
    S. Ginsburg andS. A. Greibach, Abstract families of languages,IEEE Conference Record of 1967 8th Annual Symposium on Switching and Automata Theory, pp. 128–139.Google Scholar
  15. [15]
    M. O. Rabin andD. Scott, Finite automata and their decision problems,IBM J. Res. Develop. 3 (1959), 114–125.Google Scholar
  16. [16]
    A. L. Rosenberg, Onn-tape Finite State Acceptors,IEEE Conference Record of 5th Annual Symposium on Switching Circuit Theory and Logical Design, pp. 76–81, 1964.Google Scholar
  17. [17]
    J. E. Hopcroft andJ. D. Ullman, An approach to a unified theory of automata,Bell System Tech. J. 46 (1967), 1793–1829.Google Scholar
  18. [18]
    J. E. Hopcroft andJ. D. Ullman, Decidable and undecidable questions about automata,J. Assoc. Comput. Mach. 15 (1968), 317–324.Google Scholar
  19. [19]
    J. E. Hopcroft andJ. D. Ullman, “An Approach to a Unified Theory of Automata” (extended abstract),IEEE Conference Record of 1967 8th Annual Symposium on Switching and Automata Theory, Oct. 1967, 140–147.Google Scholar
  20. [20]
    S. Ginsburg, S. A. Greibach andM. A. Harrison, Stack automata and compiling,J. Assoc. Comput. Mach. 14 (1967), 172–201.Google Scholar
  21. [21]
    S. Ginsburg,The Mathematical Theory of Context-Free Languages, McGraw-Hill, New York, N.Y., 1966.Google Scholar
  22. [22]
    J. N. Gray, M. A. Harrison andO. Ibarra, Two-way pushdown automata,Information and Control 11 (1967), 30–70.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1969

Authors and Affiliations

  • A. V. Aho
    • 1
  • J. E. Hopcroft
    • 1
  • J. D. Ullman
    • 1
  1. 1.Bell Telephone Laboratories, IncorporatedMurray HillUSA

Personalised recommendations