Results 1 
5 of
5
Degree structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
"... ..."
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

Cited by 4 (0 self)
 Add to MetaCart
. 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...
2004], The 89theory of R( ; _; ^) is undecidable
 Trans. Am. Math. Soc
"... Abstract The three quantifier theory of (R; ^T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman [1998]. The two quantifier theory includes the lattice embedding problem and its decidability is a long standing open question. A negative s ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract The three quantifier theory of (R; ^T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman [1998]. The two quantifier theory includes the lattice embedding problem and its decidability is a long standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ^T but includes function symbols.
The ∀∃theory of R(≤, ∨, ∧) is undecidable
 Trans. Amer. Math. Soc
, 2004
"... Abstract. The three quantifier theory of (R, ≤T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a longstanding open question. A negative ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. The three quantifier theory of (R, ≤T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a longstanding open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ≤T but includes function symbols. Theorem. The two quantifier theory of (R, ≤, ∨, ∧), the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on R) is undecidable. The same result holds for various lattices of ideals of R which are natural extensions of R preserving join and infimum when it exits. 1.