On the Complexity of the Successivity Relation in Computable Linear Orderings
نویسنده
چکیده
In this paper, we solve a long-standing open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing ∆3isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.
منابع مشابه
Ordered Structures and Computability Ordered Structures and Computability Dedication
We consider three questions surrounding computable structures and orderings. First, we make progress toward answering a question of Downey, Hirschfeldt, and Goncharov by showing that for a large class of countable linear orderings, the Turing degree spectrum of the successor relation is closed upward in the c.e. degrees. Next we consider computable partial orders (specifically, finitely branchi...
متن کاملDegree spectra of the successor relation of computable linear orderings
We establish that for every computably enumerable (c.e.) Turing degree b, the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings, the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.
متن کامل2 Antonio Montalbán In
My area of research centers around Computability Theory, a branch of Mathematical Logic. Inside computability theory, I have worked in various different areas. I have been particularly interested in the programs of Computable Mathematics, Reverse Mathematics and Turing Degree Theory. The former one studies the computability aspects of mathematical theorems and structures. The second one analyze...
متن کاملOn computable self-embeddings of computable linear orderings
Wemake progress toward solving a long-standing open problem in the area of computable linear orderings by showing that every computable η-like linear ordering without an infinite strongly η-like interval has a computable copy without nontrivial computable self-embedding. The precise characterization of those computable linear orderings which have computable copies without nontrivial computable ...
متن کاملPrime Models of Theories of Computable Linear Orderings
We answer a long-standing question of Rosenstein by exhibiting a complete theory of linear orderings with both a computable model and a prime model, but no computable prime model. The proof uses the relativized version of the concept of limitwise monotonic function. A linear ordering is computable if both its domain and its order relation are computable; it is computably presentable if it is is...
متن کامل