Computable Linear Ordering Research Articles

The Block Relation in Computable Linear Orders

The block relation B(x, y) in a linear order is satisfied by elements that are finitely far apart; a block is an equivalence class under this relation. We show that every computable linear order with dense condensation-type (i.e., a dense collection of blocks) but no infinite, strongly η-like interval (i.e., with all blocks of size less than some fixed, finite k) has a computable copy with the non-block relation ¬B(x, y) computably enumerable. This implies that every computable linear order has a computable copy with a computable non-trivial self-embedding, and that the long-standing conjecture characterizing those computable linear orders every computable copy of which has a computable non-trivial self-embedding (as precisely those that contain an infinite, strongly η-like interval) holds for all linear orders with dense condensation-type.

Presentations of the successor relation of computable linear ordering

We prove that a nontrivial degree spectrum of the successor relation of either strongly η-like or non-η-like computable linear orderings is closed upwards in the class of all computably enumerable degrees. We also show that the degree spectrum contains 0 if and only if either it is trivial or it contains all computably enumerable degrees.

ON THE COMPLEXITY OF THE SUCCESSIVITY RELATION IN COMPUTABLE LINEAR ORDERINGS

In this paper, we solve a long-standing open question (see, e.g. Downey [6, Sec. 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 [Formula: see text] 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 [Formula: see text]-isomorphisms, 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.

On computable self-embeddings of computable linear orderings

AbstractWe solve a longstanding question of Rosenstein, and make 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 self-embedding remains open.

Initial segments of computable linear orders with additional computable predicates

We study computable linear orders with computable neighborhood and block predicates. In particular, it is proved that there exists a computable linear order with a computable neighborhood predicate, having a Π 1 0 -initial segment which is isomorphic to no computable order with a computable neighborhood predicate. On the other hand, every Σ 1 0 -initial segment of such an order has a computable copy enjoying a computable neighborhood predicate. Similar results are stated for computable linear orders with a computable block predicate replacing a neighborhood relation. Moreover, using the results obtained, we give a simpler proof for the Coles–Downey–Khoussainov theorem on the existence of a computable linear order with Π 2 0 -initial segment, not having a computable copy.

Degree 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.

Π10 classes and strong degree spectra of relations

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

On self-embeddings of computable linear orderings

The Dushnik–Miller Theorem states that every infinite countable linear ordering has a nontrivial self-embedding. We examine computability-theoretical aspects of this classical theorem.

Computable choice functions for computable linear orderings

AbstractA choice set for a computable linear ordering is a set which contains one element from each maximal block of the ordering. We obtain a partial characterization of the computable linear order‐types for which each computable model has a computable choice set, and a full characterization in the relativized case; Every model of the linear order‐type α of degree ≤ d has a choice set of degree ≤ d iff α can written as a finite sum of order‐types, each of which either has finitely many blocks, or has order‐type n · η for some integer n.

Prime 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 isomorphic to a computable linear ordering. (There are natural generalizations of these notions to other kinds of structures; for instance, see [10] for details.) There is a large body of research on computable linear orderings ([4] gives an extensive overview). Much of this work has been focused on the relationship between classical and effective order types, but it is also interesting to take an approach inspired by classical model theory and study the relationship between effective order types and theories of linear orderings, asking, for instance, what kinds of computable linear orderings exist within the models of a given theory of linear orderings. Taking this approach, Rosenstein ended his book Linear Orderings [12] by asking whether a complete theory of linear orderings with a computable model and a prime model must have a computable prime model. This question was repeated in the Problem Sessions section of [11] (Problem 7.20), and it has recently been included in [3] (Question 3.18). In this paper, we give a negative answer to Rosenstein's question. We assume familiarity with basic notions and results from computability theory and model theory (standard references are [13] and [5], respectively). For structures A and B in the same language, we will write A ok B to mean that player 3 has a winning strategy in the Ehrenfeucht-Fraisse game EFk [A, B] of length k. Recall that if A -k B for all k e w then A B. See Section 3.3 of [5] for details. We will use the following relativized version of a notion due originally to Khisamiev [6]. 1. Definition. Let a be a Turing degree. A function f is a-limitwise monotonic if there exists an a-computable binary function g such that, for all n, s E w, g(n, s) ? Received by the editors December 29, 1999 and, in revised form, February 25, 2000. 2000 Mathematics Subject Classification. Primary 03D45; Secondary 06A05, 03C57, 03C50, 03C65. The author's research was supported by the Marsden Fund of New Zealand. The author thanks the anonymous referee for valuable historical comments. ?)2001 American Mathematical Society

Degree Spectra of Relations on Computable Structures

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. (A valuable reference is the handbook [7]. In particular, the introduction and the articles by Ershov and Goncharov and by Harizanov give useful overviews, while the articles by Ash and by Goncharov cover material related to the topic of this communication.)Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable (c. e.) degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough.

On Initial Segments of Computable Linear Orders

We show there is a computable linear order with a \(\prod {_2^0 } \) initial segment that is not isomorphic to any computable linear order.