Results 1 
6 of
6
Embedding finite lattices into the computably enumerable degrees  a status survey
 In Proceedings of Logic Colloquium
, 2002
"... Abstract. We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by t ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the socalled joinsemidistributive lattices (or lattices without socalled “critical triple”), the collection of embeddable lattices forms a Π0 2set. This paper surveys recent joint work by the authors, concentrating on restricting the number of meets by considering “quasilattices”, i.e., finite upper semilattices in which only some meets of incomparable elements are specified. In particular, we note that all finite quasilattices with one meet specified are embeddable; and that the class of embeddable finite quasilattices with two meets specified, while nontrivial, forms a computable set. On the other hand, more sophisticated techniques may be necessary for finite quasilattices with three meets specified. 1.
The ∀∃theory of R(≤, ∨, ∧) is undecidable
 TRANS. AMER. MATH. SOC
, 2004
"... 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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
The ∀∃ theory of D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.
ON THE EXISTENCE OF A STRONG MINIMAL PAIR
"... Abstract. We show that there is a strong minimal pair in the computably enumerable Turing degrees, i.e., a pair of nonzero c.e. degrees a and b such that a ∩ b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a. 1. ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We show that there is a strong minimal pair in the computably enumerable Turing degrees, i.e., a pair of nonzero c.e. degrees a and b such that a ∩ b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a. 1.
EMBEDDINGS INTO THE COMPUTABLY ENUMERABLE DEGREES
"... Abstract. We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directio ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directions which have been pursued to try to solve the problem, and present some related open questions. 1.