Recursively enumerable sets and degrees
01 Mar 1987-Bulletin of the American Mathematical Society (American Mathematical Society)-Vol. 84, Iss: 6, pp 1149-1181
TL;DR: In this paper, the relation of the structure of an R set to its degree is discussed, and the infinite injury priority method is proposed to solve the problem of scaling and splitting R sets.
Abstract: 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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Book•
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TL;DR: This book provides the first comprehensive introduction to Dynamic Logic, a system of remarkable unity that is theoretically rich as well as of practical value.
Abstract: From the Publisher:
Among the many approaches to formal reasoning about programs, Dynamic Logic enjoys the singular advantage of being strongly related to classical logic. Its variants constitute natural generalizations and extensions of classical formalisms. For example, Propositional Dynamic Logic (PDL) can be described as a blend of three complementary classical ingredients: propositional calculus, modal logic, and the algebra of regular events. In First-Order Dynamic Logic (DL), the propositional calculus is replaced by classical first-order predicate calculus. Dynamic Logic is a system of remarkable unity that is theoretically rich as well as of practical value. It can be used for formalizing correctness specifications and proving rigorously that those specifications are met by a particular program. Other uses include determining the equivalence of programs, comparing the expressive power of various programming constructs, and synthesizing programs from specifications.
This book provides the first comprehensive introduction to Dynamic Logic. It is divided into three parts. The first part reviews the appropriate fundamental concepts of logic and computability theory and can stand alone as an introduction to these topics. The second part discusses PDL and its variants, and the third part discusses DL and its variants. Examples are provided throughout, and exercises and a short historical section are included at the end of each chapter.
1,631 citations
Cites methods from "Recursively enumerable sets and deg..."
...35 is from Urzyczyn (1987). There is a di erent proof of this result, using Adian structures, which appears in Stolboushkin (1989)....
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Book•
01 Jan 1989TL;DR: This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning and is sufficiently detailed to provide an introduction to research in this area.
Abstract: This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning. Among the topics dealt with are: classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The book is self-contained, and yet sufficiently detailed to provide an introduction to research in this area.
871 citations
Book•
01 Jan 2009
TL;DR: This book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.
Abstract: The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. Covering the basics as well as recent research results, this book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.
638 citations
Cites background from "Recursively enumerable sets and deg..."
...Things left out here can be found in Soare (1987) or Odifreddi (1989)....
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...Harrington and Soare (1991) studied a further Post property which is even first-order definable in E , the lattice of c....
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...The following extension, also from Jockusch and Soare (1972), shows that there is Y ∈ P avoiding the cone above a noncomputable set B....
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...The uses γs(x) are pictured as movable markers (Soare, 1987)....
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...For details on these deep theorems, see Oddifreddi (1989, 1999) or Soare (1987)....
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TL;DR: Abramsky as discussed by the authors introduced a notion of universes of discourse for various computational situations, and a standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms.
496 citations
Proceedings Article•
19 Apr 1994
TL;DR: An intensional model for the programming language PCF is described, in which the types of PCF are interpreted by games, and the terms by certain "history-free" strategies are interpreted.
469 citations
Cites background or methods from "Recursively enumerable sets and deg..."
...rsal terms will work by simulating the evaluation tree corresponding to σ. 42 Firstly, we recall some notations from recursion theory. We fix an acceptable numbering of the partial recursive functions [Soa87] such that φn is the n’th partial recursive function and Wn is the n’th r.e. set. We also fix a recursive pairing function h−,−i: ω×ω →ω and a recursive coding of finite sequences. A recursive strategy ...
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...5.1 RecursiveStrategies We shall develop effective versions of Gand E. Our treatment will be very sketchy, as the details are lengthy and tedious, but quite routine. We refer to standard texts such as [Soa87] for background. We say that a game A is effectively given if there is a surjective map eA : ω →MA with respect to which λA (with some coding of {P,O,Q,A}) and the characteristic functions of PA and ≈A...
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References
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TL;DR: This chapter discusses the application of the diagonal process of the universal computing machine, which automates the calculation of circle and circle-free numbers.
Abstract: 1. Computing machines. 2. Definitions. Automatic machines. Computing machines. Circle and circle-free numbers. Computable sequences and numbers. 3. Examples of computing machines. 4. Abbreviated tables Further examples. 5. Enumeration of computable sequences. 6. The universal computing machine. 7. Detailed description of the universal machine. 8. Application of the diagonal process. Pagina 1 di 38 On computable numbers, with an application to the Entscheidungsproblem A. M. ...
7,642 citations
TL;DR: Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular and generalizations of recursion theory.
3,665 citations
2,459 citations
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Abstract: Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org.
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