TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.

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Recursively enumerable sets and degrees

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

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Degrees of Unsolvability

A class of recursively enumerable sets may be classified either as an object in itself -the range of a two-place function in the obvious way or by means of the corresponding set of indices. The latter approach is not only more precise but also, as we show below, provides an alternative method for solving certain problems on recursively enumerable sets and their degrees of unsolvability. The main result of the present paper is the computation, for every recursively enumerable degree a, of the degree (in fact, isomorphism-type) of the index-set corresponding to the recursively enumerable sets of degree a: its degree is a(. It follows from a theorem of Sacks [10] that the degrees of such index-sets are exactly those which are > 0(3) and recursively enumerable in 0(3). In particular, this proves Rogers' conjecture [9] that the index-set corresponding to Ol) is of degree 0(4); partial results on this problem have been obtained by Rogers [9] and by Lacombe (unpublished). The most interesting immediate consequence of our result is a different proof of Sacks' theorem [11] that the recursively enumerable degrees are dense. We refer the reader to Kleene [5] and Sacks [10] for our basic terminology and notation. A useful summary of many results which connect degrees with the arithmetical hierarchy is presented in [9], which is a good background to the present paper since without it the latter would not exist. For an assortment of results on classes of recursively enumerable sets the reader is referred to [2]. If e is a number and A is a set, then we define the partial function OA by setting:

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On the degrees of index sets. II

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Recursive Properties of Abstract Complexity Classes

Cellular automaton A (one-dimensional) cellular automaton is a linear array of cells which are connected to both of their nearest neighbors. The total number of cells in the array is determined by the input data. They are exactly in one of a finite number of states, which is changed according to local rules depending on the current state of a cell itself and the current states of its neighbors. The state changes take place simultaneously at discrete time steps. The input mode for cellular automata is called parallel. One can suppose that all cells fetch their input symbol during a pre-initial step. Iterative array Basically, iterative arrays are cellular automata whose leftmost cell is distinguished. This socalled communication cell is connected to the input supply and fetches the input sequentially. The cells are initially empty, that is, in a special quiescent state. Formal language The data on which the devices operate are strings built from input symbols of a finite set or alphabet. A subset of strings over a given alphabet is a formal language. Signal Signals are used to transmit and encode information in cellular automata. If a cell changes to the state of its neighbor after some k time steps, and if subsequently its neighbors and their neighbors do the same, then the basic signal moves with speed k through the array. With the help of auxiliary signals, rather complex signals can be established. Closure property Closure properties of families of formal languages indicate their robustness under certain operations. A family of formal languages is closed under some operation, if any application of the operation on languages from the family yields again a language from the family. Turing machine A Turing machine is the simplest form of a universal computer. It captures the idea of an effective procedure or algorithm. At any time the machine is in any one of a finite number of states. It is equipped with an infinite tape divided into cells and a read-write head scanning a single cell. Each cell may contain a symbol from a finite set or alphabet. Initially, the finite input is written in successive cells. All other cells are empty. Dependent on a list of instructions, which serve as the program for the machine, the action is determined completely by the current state and the symbol currently scanned by the head. The action comprises the symbol to be written on the current cell, the new state of the machine, and the information of whether the head should move left or right. Decidability A formal problem with two alternatives is decidable, if there is an algorithm or a Turing machine that solves it and halts on all inputs. That is, given an encoding of some instance of the problem, the algorithm or Turing machine returns the correct answer yes or no. The problem is semidecidable, if the algorithm halts on all instances for which the answer is yes.

Cellular Automata and Language Theory.

(2) to start a classification of recursively enumerable classes which parallels the classification of recursively enumerable sets initiated by Post in [15]. A mapping of numbers (or ordered n-tuples of numbers) onto numbers is called a function. Numbers and functions are denoted by small Latin letters, sets by small Greek letters and classes by capital Latin letters. 'a-,B' stands for the set of all numbers which belong to ax, but not to ,B, while 'A -B' stands for the class of all sets which belong to A, but not to B. We denote the range of the function f(x) by 'pf(x)' or 'pf'. If a is the range of the everywhere defined function f(x), we say that a can be generated by f(x). Let g(n, x) be defined for every ordered pair of numbers and let A be the class of all sets which occur at least once in the sequence pg(O, x), pg(l, x), We say that A can be generated by g(n, x). A set is recursively enumerable (r.e.) if it is empty or it can be generated by a recursive function of one variable. Similarly, a class of r.e. sets is recursively enumerable, if it is empty or consists only of the empty set, or the class of its nonempty members can be generated by a recursive function of two variables. Let 'F' stand for the class of all r.e. sets and 'Q' for the class of all finite sets. Then Q is a proper subclass of F and it is easily seen that both F and Q are r.e. classes [2, T2.2 ]. A set is called decidable or recursive if there exists a recursive procedure which enables us to test membership in the set, i.e., if its characteristic function is recursive. A decidable set is said to have a solvable decision problem, and an undecidable set an unsolvable decision problem. The class of all recursive sets is denoted by 'E'; this class properly includes Q and is properly

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J. C. E. Dekker

Papers

Some theorems on classes of recursively enumerable sets