Abstract
A procedure for comparing language operations is formulated and used to study some general algebraic properties of such operations as well as the interrelationships of some specific well known operations. The comparison scheme is used to partition the family of all language operations into a collection of equivalence classes that forms an upper semilattice, When full semi-AFL operations are used as a basis of comparison, the Min and Max operations are shown to be in the same class as relative complementation and reside above the class containing Kleene closure. This gives the result that any complementation closed full semi-AFL is a full AFL closed under Min and Max.
Similar content being viewed by others
References
B. S. Baker andR. V. Book, Reversal-Bounded Multi-Pushdown Machines,J. Comput. System Sci.,8, June 1974, pp. 315–332.
Y. Bar-Hillel,Language and Information, Addison-Wesley, Reading, Massachusetts, 1964.
G. Birkhoff,Lattice Theory, 3rd ed., American Mathematical Society, Providence, Rhode Island, 1967.
C. C. Elgot andJ. E. Mezei, On Relations Defined by Generalized Finite Automata,IBM Journal of Research and Dev.,9, January 1965, pp. 47–68.
S. Ginsburg,The Mathematical Theory of Context Free Languages, McGraw-Hill, New York, 1966.
S. Ginsburg andE. H. Spanier, Mappings of Languages by Two-Tape Devices,J. ACM,12, July 1965, pp. 423–434.
S. Ginsburg andE. H. Spanier, Finite-Turn Pushdown Automata,J. SIAM Control,4, August 1966, pp. 429–453.
S. Ginsburg andS. A. Greibach, Deterministic Context Free Language,Information and Control,9, December 1966, pp. 620–648.
S. Ginsburg, S. A. Greibach andJ. Hopcroft, Studies in Abstract Families of Languages,Memoirs, Amer. Math. Soc.,87, 1969.
J. Goldstine. Substitution and Bounded Languages,J. Comput. System Sci.,6, February 1972, pp. 9–29.
S. A. Greibach, The Unsolvability of the Recognition of Linear Context-Free Languages.J. ACM,13, October 1966, pp. 582–587.
S. A. Greibach, An Infinite Hierarchy of Context-Free Languages,J. ACM,16, January 1969, pp. 91–106.
S. A. Greibach, Checking Automata and One-Way Stack Languages.J. Comput. System Sci.,3, May 1969, pp. 196–217.
S. A. Greibach, Chains of Full AFL's.Math. Systems Theory,4, September 1970. pp. 231–242.
S. A. Greibach, Characteristic and Ultrarealtime Languages.Information and Control,18, February 1971. pp. 65–98.
S. A. Greibach, Syntactic Operators on Full SemiAFLs.J. Comput. System Sci..6, February 1972, pp. 30–76.
G. H. Hardy, andE. M. Wright,An Introduction to the Theory of Numbers, 4th ed., Oxford University Press, London, 1960.
I. N. Herstein.Topics in Algebra, Blaisdell Publishing Company. Waltham. Massachusetts. 1964.
J. E. Hopcroft andJ. D. Ullman.Formal Languages and Their Relation to Automata Addison-Wesley. Reading. Massachusetts. 1969.
S. Y. Itoga. “A Scheme for Comparing Language Operations.” Ph. D. Dissertation. University of California at Los Angeles. Los Angeles. CA, 1973.
O. Ore,Theory of Graphs, American Mathematical Society. Providence. Rhode Island. 1962.
S. Scheinberg, Note on the Boolean Properties of Context-Free Languages.Information and Control,3, December 1960. pp. 372–375.
Author information
Authors and Affiliations
Additional information
This paper was supported in part by the National Science Foundation under grant GJ-803 and in part by the U.S. Army Research Office.
Rights and permissions
About this article
Cite this article
Itoga, S.Y. Comparing language operations. Math. Systems Theory 10, 305–321 (1976). https://doi.org/10.1007/BF01683281
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01683281