The Field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic

  title={The Field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic},
  author={Alexander Kartzow},
Recently, Schlicht and Stephan lifted the notion of automatic-structures to the notion of (finite-word) ordinal-automatic structures. These are structures whose domain and relations can be represented by automata reading finite words whose shape is some fixed ordinal $\alpha$. We lift Delhomm\'e's relative-growth-technique from the automatic and tree-automatic setting to the ordinal-automatic setting. This result implies that the random graph is not ordinal-automatic and infinite integral… Expand
Algorithmic Solutions via Model Theoretic Interpretations
Model theoretic interpretations are an important tool in algorithmic model theory. Their applications range from reductions between logical theories to the construction of algorithms for problems,Expand
Technical Report Column
Simultaneous Approximation of Constraint Satisfaction Problems, Amey Bhangale, Swastik Kopparty, Sushant Sachdeva, TR14-098. The Complexity of DNF of Parities, Gil Cohen, Igor Shinkar, TR14-099. TheExpand


Structures without Scattered-Automatic Presentation
This paper proves the following limitations on the class of \(\mathfrak{L}\)-automatic structures for a fixed \(\ mathfrak {L}\) of finite condensation rank 1 + α. Expand
The Rank of Tree-Automatic Linear Orderings
It is proved that the FC-rank of every tree-automatic linear ordering is below omega^omega, and an analogue for tree- automatic linear orderings where the branching complexity of the trees involved is bounded is shown. Expand
A hierarchy of tree-automatic structures
It is obtained that there exist infinitely many ωn- automatic, hence also ω-tree-automatic, atomless boolean algebras, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorph under an alternate axiom AT, strengthening a result of [14]. Expand
Pumping for ordinal-automatic structures
A pumping lemma for alpha-automata (processing finite alpha-words, i.e., words of length alpha that have one fixed letter at all but finitely many positions) is developed and a sharp bound on the height of the finite word alpha-automatic well-founded order forests is provided. Expand
Model-theoretic complexity of automatic structures
The following results are proved: The ordinal height of any automatic well-founded partial order is bounded by ωω, and the ordinal heights of automaticWell-founded relations are unbounded below (ω1CK). Expand
Automatic linear orders and trees
It is shown that every infinite path in an automatic tree with countably many infinite paths is a regular language. Expand
Automata on ordinals and automaticity of linear orders
A method for proving non-automaticity is described and this is applied to determine the optimal bounds for the ranks of linear orders recognized by finite state automata. Expand
Tree-Automatic Well-Founded Trees
It is shown that the isomorphism problem for tree-automatic well-founded trees is complete for level $\Delta^0_{\omega^ \omega}$ of the hyperarithmetical hierarchy (under Turing-reductions). Expand
Transfinite Automata Recursions and Weak Second Order Theory of Ordinals
We identify the ordinal α with the set of all ordinals x < α. The weak second order theory of [α, <] is the interpreted formalism WST [α, <] which makes use of: (a) the propositional connectives withExpand
Automatic structures: richness and limitations
It is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations, and the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete. Expand