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Showing papers on "Computability published in 1977"


BookDOI
01 Jan 1977
TL;DR: Provability - introduction to formal languages truth and deducibility the continuum problem and forcing the continuum Problem and constructible sets computability - recursive functions and Church's thesis diophantine sets and algorithmic undecidability provability and computability Goedel's incompleteness theorem recursive groups.
Abstract: Provability - introduction to formal languages truth and deducibility the continuum problem and forcing the continuum problem and constructible sets computability - recursive functions and Church's thesis diophantine sets and algorithmic undecidability provability and computability Goedel's incompleteness theorem recursive groups.

185 citations


Journal ArticleDOI
TL;DR: It is proved that a Turing computable functionf is computable in binary Horn clauses, which are a subset of first order logic and do not need more than one function symbol.
Abstract: It is proved that a Turing computable functionf is computable in binary Horn clauses, which are a subset of first order logic. Moreover, it is proved that the binary Horn clauses do not need more than one function symbol. The proofs comprise computable relations that can be run efficiently as logic programs on a computer.

101 citations


Book
01 Jan 1977

51 citations


Journal ArticleDOI
Dana Scott1
TL;DR: Logic has been long interested in whether answers to certain questions are computable in principle, since the outcome puts bounds on the possibilities of formalization; however, much remains to be done in showing how abstract conceptualizations can (or cannot) be actualized before the authors can say they have a unified theory.
Abstract: Logic has been long interested in whether answers to certain questions are computable in principle, since the outcome puts bounds on the possibilities of formalization. More recently, precise comparisons in the efficiency of decision methods have become available through the developments in complexity theory. These, however, are applications to logic, and a big question is whether methods of logic have significance in the other direction for the more applied parts of computability theory.Programming languages offer an obvious opportunity as their syntactic formalization is well advanced; however, the semantical theory can hardly be said to be complete. Though we have many examples, we have still to give wide-ranging mathematical answers to these queries: What is a machine? What is a computable process? How (or how well) does a machine simulate a process? Programs naturally enter in giving descriptions of processes. The definition of the precise meaning of a program then requires us to explain what are the objects of computation (in a way, the statics of the problem) and how they are to be transformed (the dynamics).So far the theories of automata and of nets, though most interesting for dynamics, have formalized only a portion of the field, and there has been perhaps too much concentration on the finite-state and algebraic aspects. It would seem that the understanding of higher-level program features involves us with infinite objects and forces us to pass through several levels of explanation to go from the conceptual ideas to the final simulation on a real machine. These levels can be made mathematically exact if we can find the right abstractions to represent the necessary structures.The experience of many independent workers with the method of data types as lattices (or partial orderings) under an information content ordering, and with their continuous mappings, has demonstrated the flexibility of this approach in providing definitions and proofs, which are clean and without undue dependence on implementations. Nevertheless much remains to be done in showing how abstract conceptualizations can (or cannot) be actualized before we can say we have a unified theory.

50 citations



01 Jan 1977
TL;DR: This report presents an algorithm for the five-color problem which has an asymptotic computing time which is proportional to the square of the number of regions, at worst and can be modified to produce six-color maps in linear time.
Abstract: Aside from the mathematical question as to whether any planar map can be colored with four or five colors so that no adjacent regions are assigned the same color, there is a computational problem of finding a coloration for a given map. This report presents an algorithm for the five-color problem which has an asymptotic computing time which is proportional to the square of the number of regions, at worst. The algorithm can be modified to produce six-color maps in linear time. The algorithm has been implemented for practical use in an image processing system and can be used in practical cases to produce four-color maps.

2 citations


Journal ArticleDOI
Leif Larsen1
TL;DR: In this article, it was shown that CPo(U, W) can be computed for most finitely generated (henceforth f.g.) Fuchsian groups, which are the discrete subgroups of the group of all 2 • 2 real matrices with determinant + 1.
Abstract: where \" ~ a \" denotes the conjugaey relation in G. In this paper we show how these sets CPo(U, W) can be effectively computed for most finitely generated (henceforth f.g.) Fuchsian groups. The Fuchsian groups are the discrete subgroups of the group of all 2 • 2 real matrices with determinant + 1. By a result of Poinear$ [11] (see also [8]), the class of f.g. Fuchsian groups consists of free products of cyclic groups, together with the groups

1 citations


Journal ArticleDOI
TL;DR: This monograph is suitable as a supplement to student with a strong background in mathematics who wants to existing texts for a course in information theory or a course to find out what coding theory is all about before tackling a broader distortion theory.
Abstract: recent progress in the field, it is also suitable as a supplement to student with a strong background in mathematics who wants to existing texts for a course in information theory or a course in rate find out what coding theory is all about before tackling a broader distortion theory. treatise in this field. This monograph contains a series of lectures given by the author in July, 1973 at the Summer School on Data Transmission at the \" International Centre for Mechanical Sciences, \" Udine, Italy. Apart from omitting the fact that only binary block codes are discussed, the title accurately describes the contents of the monograph. The first chapter gives the definition and examples of binary block codes, followed by a brief discussion of the use of codes in communication systems. This chapter ends with Shannon's coding theorem for the binary symmetric channel and a mention of existing upper and lower bounds on the minimum distance of block codes. Chapter 2 is a short introduction to linear block codes and their associated generator and parity-check matrices. The encoding procedure and decoding technique using the standard array for the code are discussed briefly. In this chapter, a number of theorems are stated without proof. Chapter 3 is a thorough discussion of the Golay code. Several properties of this code are given, including those associated with t-designs. Two decoding algorithms for the Golay code are presented , one due to Berlekamp and another, a threshold decoding algorithm, due to Goethals. In this chapter, the author provides proofs for some, but not all, of the stated theorems. Chapter 4 is devoted to the proof of the MacWilliams's identities and of Gleason's theorem which relates the weight enum-erators of self-dual codes to the weight enumerators of the extended Hamming and Golay codes. The last chapter is devoted to cyclic codes. After presenting some of the general theory, the author gives a brief description of Hamming, Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon, and Justesen codes. The classical Peterson decoding technique for binary BCH codes is presented, and reference is made to Berlekamp's iterative decoding algorithm. In each chapter, the author provides the reader with numerous examples which well illustrate the theory presented. At the end of each chapter he gives suggestions for further reading. There are 113 references in this monograph of 78 pages which makes it a very good source of references. But what is more …

1 citations


Proceedings Article
01 Jan 1977
TL;DR: The effective numberings of M with respect to F are defined in two equivalent ways, firstly as the class of numberings equivalent to the term numbering, and secondly as the minimal numbering in theclass of those numberings for which the functions from F become computable.
Abstract: Let M := [F;G], i.e. the smallest set including G which is closed under the functions from F, where F and G are finite. Any numbering v of M implies v-computability on M. The effective numberings of M with respect to F are defined in two equivalent ways, firstly as the class of numberings equivalent to the term numbering, and secondly as the minimal numbering (with respect to many-one reducibility) in the class of those numberings for which the functions from F become computable.

Journal ArticleDOI
TL;DR: The theory of ( Z, Q )-machines contains as special cases the theory of some discrete, continuous, as well as hybrid computers, and is proved that the paper contains one general mathematical model for a uniform theory of computability, closely related to existing computer practice.
Abstract: In this paper we define the notion of an abstract ( Z, Q )-machine, which is a mathematical model for a device uniquely extending the functions of a real variable on the set Z to the set Q , where Z, Q are some subsets of the set of all nonnegative real numbers and Z ⊊ Q . Every such extension is called a computation of the machine. Any function which is a computation of some ( Z, Q )machine is called ( Z, Q )-computable. Similarly, a set of functions is called ( Z, Q )-computable if it is the set of all computations of some ( Z, Q )-machine. We examine the basic properties of these notions. It is proved that the theory of ( Z, Q )-machines contains as special cases the theory of some discrete, continuous, as well as hybrid computers. Consequently, the paper contains one general mathematical model for a uniform theory of computability, closely related to existing computer practice.