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Parameter Definability in the Recursively Enumerable Degrees
"... The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly defin ..."
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Cited by 34 (13 self)
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The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...
Intervals of the Lattice of Computably Enumerable Sets and Effective Boolean Algebras
, 1997
"... We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree s ..."
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Cited by 6 (3 self)
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We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [6] by giving an example of a nondefinable subclass of E which has an arithmetical index set and is invariant under automorphisms. 1 Introduction Intervals play an important role in the study of the lattice E of computably enumerable (c.e.) sets under inclusion. Several interesting properties of a c.e. set can be given alternative definitions in terms of the structure of L(A), the lattice of c.e. supersets of A. For instance, hyperhypersimplicity of a coinfinite c.e. set A is equivalent to L(A) being a boolean algebra, and A is rmaximal if and only if L(A) has no nontrivial complemented elements. A further typ...
Effectively dense Boolean algebras and their applications
"... A computably enumerable boolean algebra B is effectively dense if for each x 2 B we can effectively determine an F (x) x such that x 6= 0 implies 0 ! F (x) ! x. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a boolean algebra. A ..."
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Cited by 4 (3 self)
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A computably enumerable boolean algebra B is effectively dense if for each x 2 B we can effectively determine an F (x) x such that x 6= 0 implies 0 ! F (x) ! x. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of E (the lattice of computably enumerable sets under inclusion) which are not boolean algebras. We derive a similar result for theories of certain initial segments "low down" of subrecursive degree structures. 1 Introduction We describe a uniform method to interpret Th(N; +; \Theta) in the theories of a wide variety of seemingly wellbehaved structures. These structures stem from formal logic, complexity theory and computability theory. In many cases, they are closely related to dense distributive lattices. The results can be summarized by saying that, in spite of the structure's apparen...
CONTENTS
"... 1 Introduction to computability theory 1 1.1 The basic concepts 2 1.1.1 Partial computable functions 2 1.1.2 Computably enumerable sets 5 ..."
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1 Introduction to computability theory 1 1.1 The basic concepts 2 1.1.1 Partial computable functions 2 1.1.2 Computably enumerable sets 5
Interpreting N in the computably enumerable weak truth table degrees
"... We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an im ..."
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We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an important further problem is the question of the actual complexity of the theory. If the structure is arithmetical, then its theory can be interpreted in true arithmetic, i.e. Th(N; +; \Theta). Thus an upper bound is ; (!) , the complexity of Th(N; +; \Theta). Here an interpretation of theories is a manyone reduction based on a computable map defined on sentences in some natural way. An example of an arithmetical structure is D T ( ; 0 ), the Turing degrees of \Delta 0 2 sets. Shore [16] proved that true arithmetic can be interpreted in Th(D T ( ; 0 )). A stronger result is interpretability without parameters of a copy of (N; +; \Theta) in the structure (interpretability of struc...
Differences of Computably Enumerable Sets
, 1999
"... We consider the lower semilattice D of differences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a definable subclass. 1 Introduction A persistent open problem about the lattice E of computably enumerable (c.e.) sets under inclusi ..."
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We consider the lower semilattice D of differences of c.e. sets under inclusion. It is shown that D is not distributive as a semilattice, and that the c.e. sets form a definable subclass. 1 Introduction A persistent open problem about the lattice E of computably enumerable (c.e.) sets under inclusion is to determine the least number k such that the \Sigma k theory is undecidable. Lachlan [6] proved that the \Sigma 2 theory in the language of lattices is decidable, while one of the various known proofs of undecidability for Th(E), in that case due to Harrington, shows that in fact the \Sigma 8 theory in the language of lattices is undecidable (see [10], p. 381 for a sketch of that proof). Thus a very unsatisfying gap of 6 quantifier alternations remains. The reason why the undecidability proofs are so "bad" is that the coding used is very indirect. For instance, first one codes the class of finite symmetric graphs (which has an hereditarily undecidable \Sigma 2 theory) in the cl...