Mathematical systems theory

, Volume 8, Issue 2, pp 105–126 | Cite as

Semantic preserving translations

  • David B. Benson


Let X1, X2 be derivation systems (freex-categories) generated by context free grammars. Let X0 be a translation category withx-functorsf i :X0→X i ,i=1, 2. Let T be an Ω*-theory, a generalization of algebraic theories. LetI i :X i →T be algebraic interpretations of the derivations systems, giving the semantics of derivation systems. The translation category X0 is shown to preserve the common semantics through the translation if there is a natural transformation from the functorf2ºI2 to the functorf1ºI1. This is used to show that certain elementary conditions on well-behaved generalized2 sequential machine maps (g2sm maps) result in semantics preservation by the g2sm maps.


Computational Mathematic Natural Transformation Elementary Condition Algebraic Theory Context Free Grammar 
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  1. [1]
    A. V. Aho andJ. D. Ullman, Syntax directed translations and the pushdown assembler,J. Comp. System Sci. 3 (1969), 37–56.Google Scholar
  2. [2]
    A. V. Aho andJ. D. Ullman, Properties of syntax directed translations,J. Comp. System Sci. 3 (1969), 319–334.Google Scholar
  3. [3]
    D. B. Benson, Syntax and semantics: a categorical view,Information and Control 17 (1970), 145–160.Google Scholar
  4. [4]
    R. M. Burstall andP. J. Landin, Programs and their proofs: an algebraic approach,Machine Intelligence 4, American Elsevier, 1969.Google Scholar
  5. [5]
    P. M. Cohn,Universal Algebra, Harper and Row, New York, 1965.Google Scholar
  6. [6]
    K. Culik, Well-translatable grammars andAlgol-like languages, inFormal Language Description Languages (T. B. Steel, Ed.), North-Holland, Amsterdam, 1966.Google Scholar
  7. [7]
    S. Eilenberg andJ. B. Wright, Automata in general algebras,Information and Control 11 (1967), 452–470.Google Scholar
  8. [8]
    T. V. Griffiths, Some remarks on derivations in general rewriting systems,Information and Control 12 (1968), 27–54.Google Scholar
  9. [9]
    G. Hotz, Eindeutigkeit und Mehrdeutigkeit formaler Sprachen,Elektron. Informationsarbeit. Kybernetik 2 (1966), 235–247.Google Scholar
  10. [10]
    E. T. Irons, A syntax directed compiler for ALGOL 60,Comm. ACM 4 (1961), 51–55.Google Scholar
  11. [11]
    D. E. Knuth, Semantics of context-free languages,Math. Systems Theory 2 (1968), 127–145.Google Scholar
  12. [12]
    F. W. Lawvere, Functorial semantics of algebraic theories,Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869–872.Google Scholar
  13. [13]
    P. M. Lewis andR. E. Stearns, Syntax directed transduction,J. Assoc. Comput. Mach. 15 (1968), 465–488.Google Scholar
  14. [14]
    S. MacLane andG. Birkhoff,Algebra, MacMillan, New York, 1967.Google Scholar
  15. [15]
    B. Pareigis,Categories and Functors, Academic Press, New York, 1970.Google Scholar
  16. [16]
    L. Petrone, Syntactic mappings of context-free languages,Proc. IFIP Congress 2 (1965), 590–591.Google Scholar
  17. [17]
    C. D. Shepard, Languages in general algebras,Conf. Record ACM Symp. Theory Comp., ACM, New York, 1969, 155–163.Google Scholar
  18. [18]
    C. P. Schnorr, Transformational classes of grammars,Information and Control 14 (1969), 252–277.Google Scholar
  19. [19]
    C. P. Schnorr andH. Walter, Pullbackkonstruktionen bei Semi-Thuesystemen,Elektron. Informationsarbeit. Kybernetik 5 (1969), 27–36.Google Scholar
  20. [20]
    J. W. Thatcher, Characterizing derivation trees of context-free grammars through a generalization of finite automata theory,J. Comp. System Sci. 1 (1967), 317–322.Google Scholar
  21. [21]
    J. W. Thatcher, Generalized2 sequential machine maps,J. Comp. System Sci. 4 (1970), 339–367.Google Scholar
  22. [22]
    F. B. Thompson, English for the computer,AFIPS Conf. Proc. 29, FJCC, Spartan Books, Washington D.C., 1966, 349–356.Google Scholar
  23. [23]
    H. Walter, Verallgemeinerte Pullbackkonstruktionen bei Semi-Thuesystemen und Grammatiken,Elektron. Informationsarbeit. Kybernetik 6 (1970), 239–254.Google Scholar
  24. [24]
    N. Wirth andH. Weber, EULER: A generalization of ALGOL and its formal definition: Part I,Comm. ACM 9 (1966), 13–23.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1974

Authors and Affiliations

  • David B. Benson
    • 1
  1. 1.Computer Science DepartmentWashington State UniversityPullmanUSA

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