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ηREPRESENTATION OF SETS AND DEGREES
"... Let A = {a0 < a1 < a2 <...} which does not include 0 or 1 and η the order type of the rationals. When A can be represented by a computable linear order of the ordertype ..."
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Let A = {a0 < a1 < a2 <...} which does not include 0 or 1 and η the order type of the rationals. When A can be represented by a computable linear order of the ordertype
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
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In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set
Characterizing the Computable Structures: Boolean Algebras and Linear Orders
, 2007
"... A countable structure (with finite signature) is computable if its universe can be identified with ω in such a way as to make the relations and operations computable functions. In this thesis, I study which Boolean algebras and linear orders are computable. Making use of Ketonen invariants, I study ..."
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A countable structure (with finite signature) is computable if its universe can be identified with ω in such a way as to make the relations and operations computable functions. In this thesis, I study which Boolean algebras and linear orders are computable. Making use of Ketonen invariants, I study the Boolean algebras of low Ketonen depth, both classically and effectively. Classically, I give an explicit characterization of the depth zero Boolean algebras; provide continuum many examples of depth one, rank ω Boolean algebras with range ω + 1; and provide continuum many examples of depth ω, rank one Boolean algebras. Effectively, I show for sets S ⊆ ω + 1 with greatest element, the depth zero Boolean algebras Bu(S) and Bv(S) are computable if and only if S \{ω} is Σ 0 n↦→2n+3 in the Feiner Σhierarchy. Making use of the existing notion of limitwise monotonic functions and the new notion of limit infimum functions, I characterize which shuffle sums of ordinals below ω + 1 have computable copies. Additionally, I show that the notions of limitwise monotonic functions relative to 0 ′ and limit infimum functions coincide.
On limitwise monotonicity and maximal block functions
, 2014
"... Link to publication record in Explore Bristol Research ..."
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Link to publication record in Explore Bristol Research
THE BLOCK RELATION IN COMPUTABLE LINEAR ORDERS
, 904
"... A block in a linear order is an equivalence class when factored by the block relation B(x, y), satisfied by elements that are finitely far apart. We show that every computable linear order with dense condensationtype (i.e. a dense collection of blocks) but no infinite, strongly ηlike interval (i.e ..."
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A block in a linear order is an equivalence class when factored by the block relation B(x, y), satisfied by elements that are finitely far apart. We show that every computable linear order with dense condensationtype (i.e. a dense collection of blocks) but no infinite, strongly ηlike interval (i.e. with all blocks of size less than some fixed, finite k) has a computable copy with the nonblock relation ¬B(x, y) computably enumerable. This implies that every computable linear order has a computable copy with a computable nontrivial selfembedding, and that the longstanding conjecture characterizing those computable linear orders every computable copy of which has a computable nontrivial selfembedding (as precisely those that contain an infinite, strongly ηlike interval) holds for all linear orders with dense condensationtype. 2
Prime Models of Theories of Computable Linear Orderings
"... Abstract We answer a longstanding 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 com ..."
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Abstract We answer a longstanding 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; see for instance [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.
LIMITWISE MONOTONIC FUNCTIONS AND THEIR APPLICATIONS
"... Abstract. We survey what is known about limitwise monotonic functions and sets and discuss their applications in effective algebra and computable model theory. Additionally, we characterize the computably enumerable degrees that are totally limitwise monotonic, show the support strictly increasing 0 ..."
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Abstract. We survey what is known about limitwise monotonic functions and sets and discuss their applications in effective algebra and computable model theory. Additionally, we characterize the computably enumerable degrees that are totally limitwise monotonic, show the support strictly increasing 0 ′limitwise monotonic sets on Q do not capture the sets with computable strong ηrepresentations, and study the limitwise monotonic spectra of a set. 1.