Model-theoretic complexity of automatic structures

Abstract

a b s t r a c tWe study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: (1) The ordinal height of any automatic well-founded partial order is bounded by ω ω . (2) The ordinal heights of automatic well-founded relations are unbounded below ω CK 1 , the first non-computable ordinal. (3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ω CK 1 , ω CK 1 +1. (4) For any computable ordinal α, there is an automatic successor tree of Cantor-Bendixson rank α.

Automatic Structures and Groups

“…In [20] it is proven that if X is regular and X admits an ordinal α then α < ω ω . Generally, Khoussainov and Minnes [15] showed that if X is FA recognizable and X admits a well-founded partial order A then the height of A is below ω ω . Another nice example is a recent result by Tsankov [29] showing that no regular language admits the structure (Q; +), the additive group of rational numbers.…”

Tree-automatic scattered linear orders

“…In [24] it is proven that if X is regular and X admits an ordinal α then α < ω ω . Generally, Delhommé [6] showed that if X is regular and X admits a well-founded partial order A then the height of A is below ω ω (see also [13]). Also, as mentioned above, Delhommé [6] shows that no regular tree language admits ordinals greater or equal to ω ω ω .…”

Hamiltonicity of automatic graphs

“…In contrast to these positive results, Khoussainov, Nies, and Rubin have shown that the isomorphism problem for automatic graphs is Σ 1 1 -complete [17]. Results on the model theoretic complexity of automatic structures can be found in [15].…”

“…The proof idea is to transform a recursive successor tree into an automatic one by adding the computation (i.e., sequence of transitions) that verifies the edge (u, v) as a path between the nodes u and v; a similar idea was used in [20,15].…”

“…Using a proof technique from [20,15], we prove that the fundamental Σ 1 1 -complete problem in recursion theory, namely the existence of an infinite path in a recursive tree remains Σ 1 1 -complete if the input tree is automatic. For this result it is crucial that the tree is a successor tree, which means that it is an acyclic graph, where every node is reachable from a root node and every node except the root has exactly one incoming edge.…”