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Decidability of the two-quantifier theory of the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices

Published online by Cambridge University Press:  12 March 2014

Klaus Ambos-Spies ,
Peter A. Fejer ,
Steffen Lempp  and
Manuel Lerman
Klaus Ambos-Spies
Affiliation:
Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany, E-mail: ambos@math.uni-heidelberg.de
Peter A. Fejer
Affiliation:
Department of Mathematics, and Computer Science, University of Massachusetts at Boston, Boston, MA 02125-3393, USAfejer@cs.umb.edu
Steffen Lempp
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA, E-mail: lempp@math.wisc.edu
Manuel Lerman
Affiliation:
Department of Mathematics, University of Connecticut, U-9, Storrs, CT 06269-3009, USAmlerman@math.uconn.edu
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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 degreerecursive polynomial many-one degreedecidable fragment
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 3 , September 1996 , pp. 880 - 905
Copyright
Copyright © Association for Symbolic Logic 1996

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References

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