The isomorphism problem for tree-automatic ordinals with addition

This paper studies tree-automatic ordinals (or equivalently, well-founded linearly ordered sets) together with the ordinal addition operation +. Informally, these are ordinals such that their elements are coded by finite trees for which the linear order relation of the ordinal and the ordinal addition operation can be determined by tree automata. We describe an algorithm that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic

The model-theoretic complexity of automatic linear orders

Automatic structures are—possibly infinite—structures which are finitely presentable by means of finite automata on strings or trees. Largely motivated by the fact that their first-order theories are uniformly decidable, automatic structures gained a lot of attention in the "logic in computer science" community during the last fifteen years. This thesis studies the model-theoretic complexity of automatic linear orders in terms of two complexity measures: the finite-condensation rank and the Ramsey degree. The former is an ordinal which indicates how far a linear order is away from being dense. The corresponding main results establish optimal upper bounds on this rank with respect to several notions of automaticity. The Ramsey degree measures the model-theoretic complexity of well-orders by means of the partition relations studied in combinatorial set theory. This concept is investigated in a purely set-theoretic setting as well as in the context of automatic structures.Auch im Buchhandel erhältlich: The model-theoretic complexity of automatic linear orders / Martin Huschenbett Ilmenau : Univ.-Verl. Ilmenau, 2016. - xiii, 228 Seiten ISBN 978-3-86360-127-0 Preis (Druckausgabe): 16,50

The Isomorphism Problem for ω-Automatic Trees

The main result of this paper is that the isomorphism problem for ω-automatic trees of finite height is at least as hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, Montalbán, and Nies [9] showing that the isomorphism problem for ω-automatic structures is not ∑_2^1. Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for ω-automatic trees of finite height is recursively equivalent with second-order arithmetic. On the way to our main results, we show lower and upper bounds for the isomorphism problem for ω-automatic trees of every finite height: (i) It is decidable (∏_1^0-complete, resp.) for height 1 (2, resp.), (ii) (∏_1^1-hard and in (∏_2^1 for height 3, and (iii) (∏_(n-3)^1- and (∏_(n-3)^1-hard and in (∏_(2n-4)^1(assuming CH) for all n ≥ 4. All proofs are elementary and do not rely on theorems from set theory. Complete proofs can be found in [18]

The Isomorphism Problem for ω-Automatic Trees (Extended Abstract)

The main result of this paper is that the isomorphism for ω-automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, Montalbán, and Nies [9] showing that the isomorphism problem for ω-automatic structures is not Σ 1 2. Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for ω-automatic trees of finite height is recursively equivalent with second-order arithmetic. On the way to our main results, we show lower and upper bounds for the isomorphism problem for ω-automatic trees of every finite height: (i) It is decidable (Π 0 1-complete, resp,) for height 1 (2, resp.), (ii) Π 1 1-hard and in Π 1 2 for height 3, and (iii) Π 1 n−3- and Σ 1 n−3-hard and in Π 1 2n−4 (assuming CH) for all n ≥ 4. All proofs are elementary and do not rely on theorems from set theory