Climbing up the Elementary Complexity Classes with Theories of Automatic Structures

Automatic structures are structures that admit a finite presentation via automata. Their most prominent feature is that their theories are decidable. In the literature, one finds automatic structures with non-elementary theory (e.g., the complete binary tree with equal-level predicate) and automatic structures whose theories are at most 3-fold exponential (e.g., Presburger arithmetic or infinite automatic graphs of bounded degree). This observation led Durand-Gasselin to the question whether there are automatic structures of arbitrary high elementary complexity. We give a positive answer to this question. Namely, we show that for every h ≥ 0 the forest of (infinitely many copies of) all finite trees of height at most h+ 2 is automatic and it’s theory is complete for STA(∗, exph(n, poly(n)), poly(n)), an alternating complexity class between h-fold exponential time and space. This exact determination of the complexity of the theory of these forests might be of independent interest. 2012 ACM Subject Classification Theory of computation → Complexity theory and logic

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