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THE SLAMANWEHNER THEOREM IN HIGHER RECURSION THEORY
"... A central concern of computable model theory is the restriction that algebraic structure imposes on the information content of an object of study. One asks about a countable object, what information is coded intrinsically into this object, which cannot be avoided by passing to an isomorphic copy of ..."
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A central concern of computable model theory is the restriction that algebraic structure imposes on the information content of an object of study. One asks about a countable object, what information is coded intrinsically into this object, which cannot be avoided by passing to an isomorphic copy of the object? Given a
SPECTRA OF THEORIES AND STRUCTURES
"... Abstract. We introduce the notion of a degree spectrum of a complete theory to be the set of Turing degrees that contain a copy of some model of the theory. We generate examples showing that not all degree spectra of theories are degree spectra of structures and viceversa. To this end, we give a ne ..."
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Abstract. We introduce the notion of a degree spectrum of a complete theory to be the set of Turing degrees that contain a copy of some model of the theory. We generate examples showing that not all degree spectra of theories are degree spectra of structures and viceversa. To this end, we give a new necessary condition on the degree spectrum of a structure, specifically showing that the set of PA degrees and the upward closure of the set of 1random degrees are not degree spectra of structures but are degree spectra of theories. 1.
NOTES ON THE JUMP OF A STRUCTURES
, 2009
"... We introduce the notions of a complete set of computably infinitary Π 0 n relations on a structure, of the jump of a structure, and of admitting nth jump inversion. ..."
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We introduce the notions of a complete set of computably infinitary Π 0 n relations on a structure, of the jump of a structure, and of admitting nth jump inversion.
Separating the Degree . . .
, 2009
"... In computable model theory, mathematical structures are studied on the basis of their computability or computational complexity. The degree spectrum DgSp(A) of a countable structure A is one way to measure the computability of the structure. Given various classes of countable structures, such as lin ..."
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In computable model theory, mathematical structures are studied on the basis of their computability or computational complexity. The degree spectrum DgSp(A) of a countable structure A is one way to measure the computability of the structure. Given various classes of countable structures, such as linear orders, groups, and graphs, we separate two classes K1 and K2 in the following way: we say that K1 is distinguished from K2 with respect to degree spectrum if there is an A ∈ K1 such that for all B ∈ K2, DgSp(A) ̸ = DgSp(B). In the dissertation, we will investigate this separation idea. We look at specific choices for K1 and K2—for example, we show that linear orders are distinguished from finitecomponents graphs, equivalence structures, rank1 torsionfree abelian groups, and daisy graphs with respect to degree spectrum. Out of these proofs, there comes a general pattern for the kinds of structures from which linear orders are distinguished with respect to degree spectrum. In the future, we may also replace linear orders with possibly
THE COMPLEXITY OF COMPUTABLE CATEGORICITY
"... We show that the index set complexity of the computably categorical structures is Π11complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably ..."
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We show that the index set complexity of the computably categorical structures is Π11complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably categorical but not relatively ∆0αcategorical.