Abstract
The class of all quantifier-free formulas constructed from atomic formulas of the form (x+y= z),(x=1), and (x¦y) is considered, where the predicate symbol “¦” is interpreted as the divisibility relation on nonnegative integers. The decidability isproved of the set of all formulas of this form which are true for at least one choice of values for the variables. This result is equivalent to the decidability of the universal theory of natural numbers with addition and divisibility.
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Literature cited
Yu. V. Matiyasevich, “Recursively enumerable sets are Diophantine,” Dokl. Akad. Nauk SSSR,191, No. 2, 279–282 (1970).
N. K. Kosovskii, “On decidable systems consisting simultaneously of word equations and inequations in lengths of words,” Zap. Nauch. Sem. Leningr. Otd. Mat. Aka. Nauk SSSR,40, 24–49 (1974).
A. P. Bel'tyukov, “Representing enumerable sets with sums and divisibility,” Report Fourth Student Scientific Conference, Abstr., LGU, 4–5 (1975).
L. Lipshitz, “The Diophantine problem for +, ¦,” Preprint (1975).
Yu. L. Ershov, “The elementary theory of maximal normed fields,” Dokl. Akad. Nauk SSSR,165, No. 1, 21–23 (1965).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Mathematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 15–28, 1976. Results presented September 26, 1974.
The author wishes to thank his advisor N. K. Kosovskii, who suggested the topic of this paper, for his aid in checking and formulating these results.
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Bel'tyukov, A.P. Decidability of the universal theory of natural numbers with addition and divisibility. J Math Sci 14, 1436–1444 (1980). https://doi.org/10.1007/BF01693974
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DOI: https://doi.org/10.1007/BF01693974