Peter A. Fejer

Degrees of Unsolvability

Modern computability theory began with Turing [Turing, 1936], where he introduced the notion of a function computable by a Turing machine. Soon after, it was shown that this definition was equivalent to several others that had been proposed previously and the Church-Turing thesis that Turing computability captured precisely the informal notion of computability was commonly accepted. This isolation of the concept of computable function was one of the greatest advances of twentieth century mathematics and gave rise to the field of computability theory. Among the first results in computability theory was Church and Turing’s work on the unsolvability of the decision problem for first-order logic. Computability theory to a great extent deals with noncomputable problems. Relativized computation, which also originated with Turing, in [Turing, 1939], allows the comparison of the complexity of unsolvable problems. Turing formalized relative computation with oracle Turing machines. If a set A is computable relative to a set B, we say that A is Turing reducible to B. By identifying sets that are reducible to each other, we are led to the notion of degree of unsolvability first introduced by Post in [Post, 1944]. The degrees form a partially ordered set whose study is called degree theory. Most of the unsolvable problems that have arisen outside of computability theory are computably enumerable (c.e.). The c.e. sets can intuitively be viewed as unbounded search problems, a typical example being those formulas provable in some effectively given formal system. Reducibility allows us to isolate the most difficult c.e. problems, the complete problems. The standard method for showing that a c.e. problem is undecidable is to show that it is complete. Post [Post, 1944] asked if this technique always works, i.e., whether there is a noncomputable, incomplete c.e. set. This problem came to be known as Post’s Problem and it was origin of degree theory. Degree theory became one of the core areas of computability theory and attracted some of the most brilliant logicians of the second half of the twentieth century. The fascination with the field stems from the quite sophisticated techniques needed to solve the problems that arose, many of which are quite easy to state. The hallmark of the field is the priority method introduced by

Decidability of the Two-Quantifier Theory of the Recursively Enumerable Weak Truth-Table Degrees and Other Distributive Upper Semi-Lattices

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 nite lattices which can be embedded into the r e wtt degrees by a map which preserves the least and greatest elements A nite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the lter 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 quanti er 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 quanti er theory answers a question raised by Shore and Slaman in

Embeddings of N5 and the contiguous degrees

Abstract Downey and Lempp (J. Symbolic Logic 62 (1997) 1215–1240) have shown that the contiguous computably enumerable (c.e.) degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5-element lattice N 5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local nondistributivity coincides with local nonmodularity in the computably enumerable degrees.

Embedding Lattices with Top Preserved Below Non‐GL2 Degrees

A minimal degree not realizing least possible jump. A recursively enumerable degree that is not the top of a diamond in the Turing degrees. 18 6] Peter A. Fejer and Richard A. Shore. Embeddings and extensions of embeddings in the r.e. tt-and wtt-degrees. 17 Corollary 8 If a is not in GL 2 , then any nontrivial recursively presented lattice can be embedded (as a lattice) into D (a) preserving least and greatest element, if they exist. Corollary 9 If a is not in GL 2 , then any nite nontrivial partial order can be embedded into D (a), preserving least and greatest elements, if they exist, and all joins and infs that exist in the partial order. Corollary 10 If a is not in GL 2 , then the 9 theory of the structure D (a) in the language L = (; _; ^; 0; 1) is decidable. Proof: This proof is essentially as in Fejer-Shore 6], Corollary to Theorem 3. That is, given an 9 sentence = 9x 1 : : : 9x n in L with quantiier free (where _ is taken to be a three place relation symbol), we claim that is valid in D (a) if and only if is valid in some nontrivial bounded partial order with at most 2(n + 2) 3 + (n + 2) elements. We refer the reader to 6] for details. must be some t t 0 such that g t has at least n zeros, and hence g has at least n zeros. Now take s 1 s 0 such that g s 1 has at least n zeros and let s s 1 be such that h(s) F(s). (This is possible since F does not dominate h.) Then by the construction, either there is an attack made on R n before stage s + 1 which is not canceled by the end of stage s and is not discredited at stage s + 1 or else an attack is made on R n at stage s + 1. Let t + 1 s + 1 be the stage at which the attack on R n which exists at the end of stage s + 1 was made. If, during the attack at stage t + 1 on R n , a search was made for a string or strings of length h(t) and no such string(s) were …

Embedding Distributive Lattices Preserving 1 below a Nonzero Recursively Enumerable Turing Degree

One way to try to gain an understanding of the various degree-theoretic structures which recursion theorists study is to see what lattices can be embedded into them. Lattice embeddings have been used to show that such structures have an undecidable theory (via embeddings as initial segments) and to show that the theory of such structures is decidable up to a certain quantifier level. Many results in recursion theory can be stated as results about lattice embeddings even if they were not originally phrased that way.

Every incomplete computably enumerable truth-table degree is branching

Abstract. If r is a reducibility between sets of numbers, a natural question to ask about the structure ?r of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching (i.e., the meet of two elements strictly above it). For the commonly studied reducibilities, the answer to this question is known except for the case of truth-table (tt) reducibility. In this paper, we answer the question in the tt case by showing that every tt-incomplete computably enumerable truth-table degree a is branching in ?tt. The fact that every Turing-incomplete computably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as noticed by Nies, also follows from a relativization of a result of Degtev. We give a proof here using the Ambos-Spies technique because it has not yet appeared in the literature. The proof uses an infinite injury argument. Our main result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a finite injury proof.

LATTICE REPRESENTATIONS FOR COMPUTABILITY THEORY

Abstract Lattice representations are an important tool for computability theorists when they embed nondistributive lattices into degree-theoretic structures. In this expository paper, we present the basic definitions and results about lattice representations needed by computability theorists. We define lattice representations both from the lattice-theoretic and computability-theoretic points of view, give examples and show the connection between the two types of representations, discuss some of the known theorems on the existence of lattice representations that are of interest to computability theorists, and give a simple example of the use of lattice representations in an embedding result.