Decidability of the Two-Quantifier Theory of the Recursively Enumerable Weak Truth-Table Degrees and Other Distributive Upper Semi-Lattices

We give a decision procedure for the theory of the weak truth table wtt degrees of the recursively enumerable sets The key to this decision procedure is a characterization of the nite lattices which can be embedded into the r e wtt degrees by a map which preserves the least and greatest elements A nite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the lter generated by its cuppable elements are disjoint We formulate general criteria that allow one to conclude that a distributive upper semi lattice has a decidable two quanti er theory These criteria are applied not only to the weak truth table degrees of the recursively enumerable sets but also to various substructures of the polynomial many one pm degrees of the recursive sets These applications to the pm degrees require no new complexity theoretic results The fact that the pm degrees of the recursive sets have a decidable two quanti er theory answers a question raised by Shore and Slaman in