Abstract
The first-order logical theory Th\(({\mathbb{N}},x + 1,F(x))\) is proved to be complete for the class ATIME-ALT\((2^{O(n)},O(n))\) when \(F(x) = 2^{x}\), and the same result holds for \(F(x) = c^{x}, x^{c} (c \in {\mathbb{N}}, c \ge 2)\), and F(x) = tower of x powers of two. The difficult part is the upper bound, which is obtained by using a bounded Ehrenfeucht–Fraïssé game.
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Michel, P. Computational complexity of logical theories of one successor and another unary function. Arch. Math. Logic 46, 123–148 (2007). https://doi.org/10.1007/s00153-006-0031-1
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DOI: https://doi.org/10.1007/s00153-006-0031-1