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Decidability of the Two-Quantifier Theory of the Recursively Enumerable Weak Truth-Table degrees and other Distributive Upper Semi-Lattices
Journal of Symbolic Logic (1996)
  • Klaus Ambos-Spies
  • Peter A Fejer, University of Massachusetts Boston
  • Steffen Lempp, University of Wisconsin - Madison
  • Manuel Lerman, University of Connecticut - Storrs
Abstract

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 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 (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-quantifier theory answers a question raised by Shore and Slaman in [21].

Keywords
  • recursively enumerable weak truth-table degree,
  • recursive polynomial many-one degree,
  • decidable fragment
Disciplines
Publication Date
1996
Publisher Statement

Published by the Journal for Symbolic Logic.

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1183745082

Citation Information
Klaus Ambos-Spies, Peter A Fejer, Steffen Lempp and Manuel Lerman. "Decidability of the Two-Quantifier Theory of the Recursively Enumerable Weak Truth-Table degrees and other Distributive Upper Semi-Lattices" Journal of Symbolic Logic Vol. 61 Iss. 3 (1996)
Available at: http://works.bepress.com/peter_fejer/2/