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Undecidability Results for Low Complexity Degree Structures
, 1997
"... We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and manyone reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras whichwas introduced in #7#. The method can be applied in fact to all hyperpolynomial time clas ..."
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We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and manyone reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras whichwas introduced in #7#. The method can be applied in fact to all hyperpolynomial time classes. 1 Introduction If h is a time constructible function which dominates all polynomials, then, by the methods of the deterministic time hierarchy theorem, DTIME#h# properly contains P . Therefore, a polynomial time reducibility like polynomial time many#one or Turing reducibility induces a nontrivial degree structure on DTIME#h#, which is an uppersemilattice with least element 0. By the methods of Ladner ##6#, also see #4#, Chapter I.7#, this degree structure is dense. This was so far the only fact known to hold in general for all such structures. Here we prove that all those degree structures are # Partially supported by the New Zealand Marsden Fund for Basic Science under grant VIC50...
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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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...
The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
 Memoirs of the American Mathematical Society
"... Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In ..."
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Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible ordinal
Differences between Resource Bounded Degree Structures
"... We exhibit a structural difference between the truthtable degrees of the sets which are truthtable above 0 # and the PTIMETuring degrees of all sets. ..."
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We exhibit a structural difference between the truthtable degrees of the sets which are truthtable above 0 # and the PTIMETuring degrees of all sets.