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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Undecidable and creative theories

Recursively enumerable sets and degrees

Preface- Acknowledgments- Introduction- I Background- Preliminaries- Computability Theory- Kolmogorov Complexity of Finite Strings- Relating Plain and Prefix-Free Complexity- Effective Reals- II Randomness of Sets- Martin-Lof Randomness- Other Notions of Effective Randomness- Algorithmic Randomness and Turing Reducibility- III Relative Randomness- Measures of Relative Randomness- The Quantity of K- and Other Degrees- Randomness-Theoretic Weakness- Lowness for Other Randomness Notions- Effective Hausdorff Dimension- IV Further Topics- Omega as an Operator- Complexity of CE Sets- References- Index

Algorithmic Randomness and Complexity

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

High recursively enumerable degrees and the anti-cupping property

Publisher Summary This chapter discusses some reasons for generalizing recursion theory (g.r.t.). The chapter corrects two common errors: the first is to suppose that there is just one use of g.r.t., the other is to suppose that there are so many (depending on unspecified purposes). Existing results on various generalizations are referred. The formulation and analysis of the aims of g.r.t. themselves are focussed. The metarecursion theory is discussed. The chapter presents the purposes of g.r.t. that include (a) advancing other parts of logic and mathematics; (b) understanding the mathematical character of ordinary recursion theory; and (c) analysis of a general concept of computation. Work on admissible sets constitutes a refinement because it concerns consequences of only special instances of the replacement scheme. An application of classical recursion theory is that this object is well-suited to the study of infinitary languages.

Undecidable and creative theories

Citations

Recursively enumerable sets and degrees

Algorithmic Randomness and Complexity

Degrees of Unsolvability

High recursively enumerable degrees and the anti-cupping property

Some Reasons for Generalizing Recursion Theory

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