Abstract

We give a decision procedure for the $\forall\exists$-theory of the weak truth-table $$ degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. $wtt$-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one $$ 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-quantifier theory answers a question raised by Shore and Slaman in [21].