Journal of Symbolic Logic

Automatic structures of bounded degree revisited

Dietrich Kuske and Markus Lohrey

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The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic structure of bounded degree is decidable in doubly exponential space (for injective automatic presentations, this holds even uniformly). This result is shown to be optimal since we also present a string automatic structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We prove similar results also for tree automatic structures. These findings close the gaps left open in [28] by improving both the lower and the upper bounds.

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J. Symbolic Logic Volume 76, Issue 4 (2011), 1352-1380.

First available in Project Euclid: 11 October 2011

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Kuske, Dietrich; Lohrey, Markus. Automatic structures of bounded degree revisited. J. Symbolic Logic 76 (2011), no. 4, 1352--1380. doi:10.2178/jsl/1318338854.

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