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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Probabilities of first-order sentences about unary functions


Author: James F. Lynch
Journal: Trans. Amer. Math. Soc. 287 (1985), 543-568
MSC: Primary 03C13; Secondary 03B25, 03B48
MathSciNet review: 768725
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Abstract: Let $ f$ be any fixed positive integer and $ \sigma $ a sentence in the first-order predicate calculus of $ f$ unary functions. For positive integers $ n$, an $ n$-structure is a model with universe $ \{ 0,1, \ldots ,n - 1\} $ and $ f$ unary functions, and $ \mu (n,\sigma )$ is the ratio of the number of $ n$-structures satisfying $ \sigma $ to $ {n^{nf}}$, the number of $ n$-structures. We show that $ {\lim _{n \to \infty }}\mu (n,\sigma )$ exists for all such $ \sigma $, and its value is given by an expression consisting of integer constants and the operators $ + , - , \cdot ,/$, and $ {e^x}$.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768725-2
Article copyright: © Copyright 1985 American Mathematical Society