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Decidability Of The TwoQuantifier Theory Of The Recursively Enumerable Weak TruthTable Degrees And Other Distributive Upper SemiLattices
 Journal of Symbolic Logic
, 1996
"... . We give a decision procedure for the 89theory of the weak truthtable (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wttdegrees by a map which preserves the least and greatest e ..."
Abstract

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. We give a decision procedure for the 89theory of the weak truthtable (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wttdegrees by a map which preserves the least and greatest elements: A finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semilattice has a decidable twoquantifier theory. These criteria are applied not only to the weak truthtable degrees of the recursively enumerable sets but also to various substructures of the polynomial manyone (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexitytheoretic results. The fact that the pmdegrees of the recursive sets have a decidable twoquantifier theor...