Abstract
Rational relations (finite transductions) which are equivalence relations are discussed. After establishing a containment hierarchy, the complexity of canonical function computation and a number of class membership decision problems are studied. The following classes are considered:
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(1)
Rational Equivalence Relations,
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(2)
Equivalence Kernels of Rational Functions,
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(3)
Deterministic Rational Equivalence Relations,
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(4)
Equivalence Kernels of Subsequential Functions,
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(5)
Recognizable Equivalence Relations,
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(6)
Length-bounded Rational Equivalence Relations, and
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(7)
Finite Equivalence Relations.
Except for one open case ((1) = (2)?), Hasse diagrams are given to show the relative containments in the general and one-letter-alphabet cases. Canonical function application for an input of length n is shown to be O(n 2) time and space for (1), O(n) time and space for (2), (3), and (6), and O(n) time and constant space for the others. It is shown that transitivity, symmetry, reflexivity, and membership in any of (1) through (5) are undecidable properties for rational relations whereas membership in (6) or (7) is decidable.
Work supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A0237
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References
Jean Berstel. Transductions and Context-Free Languages. B. G. Teubner, Stuttgart, Germany, 1979.
Christian Choffrut. Une caractérisation des fonctions séquentielles et des fonctions sous-séquentielles en tant que relations rationnelles. Theoretical Computer Science, 5:325–338, 1977.
Leon Davidson. Retrieval of misspelled names in an airlines passenger record system. Communications of the ACM, 5(3):169–171, 1962.
Samuel Eilenberg. Automata, Languages, and Machines, vol. A. Academic Press, New York, 1974.
C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. IBM Journal of Research, 9:47–65, 1965.
Patrick C. Fischer and Arnold L. Rosenberg. Multitape one-way nonwriting automata. Journal of Computer and System Sciences, 2:88–101, 1968.
Ivan P. Fellegi and Alan B. Sunter. A theory of record linkage. Journal of the Americal Statistical Association, 64:1183–1210, 1969.
Seymour Ginsburg. The Mathematical Theory of Context-Free Languages. McGraw-Hill, New York, 1966.
J. Howard Johnson. Formal Models for String Similarity. PhD thesis, University of Waterloo, 1983. Available as University of Waterloo Research Report CS-83-32.
J. Howard Johnson. Do rational equivalence relations have regular cross-sections? In Proceedings of the 12th International Conference on Automata, Languages, and Programming, pages 300–309, Springer-Verlag LNCS 194, 1985.
Donald E. Knuth. Sorting and Searching. Addison-Wesley, Reading, Mass., 1973.
Gwendolyn B. Moore, John L. Kuhns, Jeffrey L. Trefftzs, and Christine A. Montgomery. Accessing Individual Records from Personal Data Files using Non-unique Identifiers. Technical Report NBS Special Publication 500-2, U.S. Dept. of Commerce—National Bureau of Standards, 1977. Available from the National Technical Information Service.
H. B. Newcombe and J. M. Kennedy. Record linkage: making maximum use of the discriminating power of identifying information. Communications of the ACM, 5(11):563–566, 1962.
M. P. Schützenberger. A remark on finite transducers. Information and Control, 4:185–196, 1961.
M. P. Schützenberger. Sur les relations rationelles. In Automata theory and formal languages: 2nd GI Conference, pages 209–213, 1975.
Gio Wiederhold. Database Design. McGraw-Hill, New York, 1977.
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© 1986 Springer-Verlag Berlin Heidelberg
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Johnson, J.H. (1986). Rational equivalence relations. In: Kott, L. (eds) Automata, Languages and Programming. ICALP 1986. Lecture Notes in Computer Science, vol 226. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16761-7_66
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DOI: https://doi.org/10.1007/3-540-16761-7_66
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