Showing papers on "Computability published in 1969"

Theory of Recursive Functions and Effective Computability

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Abstract first order computability. II

Abstract: The recent development of recursion theory has turned in part toward studying notions of computability on domains other than the natural numbers. Without any attempt for completeness, we mention the theories of Takeuti [23] (and others), Machover [15], Kripke [10], Kreisel-Sacks [9] and Platek [20] on classes or sets of ordinals, Kleene's theory of recursive functionals on objects of arbitrary finite type over the integers [6], [7] (and others) and the theories of computability on arbitrary structures of Fraisse [1], Lacombe [11], [12], Kreisel [8] (and others) and more recently Platek [20], Montague [16] and Lambert [13]. (Levy's development of a hierarchy of set-theoretic predicates in [14] is also relevant.) Some of these theories attempt to abstract the computational (or combinatorial) aspects of recursion theory while others (notably Kreisel's and Montague's) conceive of recursion theory as a branch of definability theory. Here we study computability theory on abstract (unordered) domains primarily as a tool for hierarchy theory. The term "first order" in the title indicates that we restrict ourselves to computabilities relative to given functions (i.e. objects of type-i)

Uniformly reflexive structures: On the nature of gödelizations and relative computability

Abstract: In this paper we present an axiomatic theory within which much of the theory of computability can be developed in an abstract manner. The paper is based on the axiomatically defined concept of a Uniformly Reflexive Structure (U.R.S.). The axioms are chosen so as to capture what we view to be the essential properties of a "godelization" of a set of functions on arbitrary infinite domain. It can be shown that (with a "standard g6delization") both the partial recursive functions and the meta-recursive functions satisfy the axioms of U.R.S. In the first part of this paper, we define U.R.S. and develop the basic working theorems of the subject (e.g., analogues of the Kleene recursion theorems). The greater part of the paper is concerned with applying these basic results to (1) investigating the properties of godelizations, and (2) developing an intrinsic theory of relative com- putability. The notion of relative computability which we develop is equivalent to Turing reducibility when applied to the partial recursive functions. Applied to appropriate U.R.S. on arbitrary domains, it provides an upper-semi-lattice ordering on the set of all functions (both total and partial) on that domain. 0. Introduction. In this paper we present the axioms of an abstract theory of computability (the theory of Uniformly Reflexive Structures) and employ it to establish a number of general results on godelizations and to develop an intrinsic approach to the study of relative computability. To a first approximation, a Uniformly Reflexive Structure (a U.R.S.) is a set of functions on an arbitrary infinite domain together with a special indexing of the functions by elements of the domain. We call these indexings g6delizations; this terminology is apt in that the partial recursive functions with a "standard g6deli- zation" (such as given by Davis (1)) form a U.R.S. For each element u of the domain, the indexing gives us a 1-ary function denoted

Enumerability · Decidability Computability: An Introduction to the Theory of Recursive Functions

Abstract: Reading a book is also kind of better solution when you have no enough money or time to get your own adventure. This is one of the reasons we show the enumerability decidability computability an introduction to the theory of recursive functions as your friend in spending the time. For more representative collections, this book not only offers it's strategically book resource. It can be a good friend, really good friend with much knowledge.

Computability over arbitrary fields

Abstract: In most attempts to make precise the concept of a computable function, or decidable predicate, over a field F, it is considered necessary that the elements of F should be in some sense effectively describable, and hence that F itself should be countable. This is the attitude taken in the study of computable fields (see Rabin1). Our proposed definition of computability over arbitrary fields is based on the Shepherdson - Sturgis2 concept of an unlimited register machine.

Axioms for time dependence of resource consumption in computing recursive functions

Abstract: Any formal definition of effective computability includes the idea of graduality of computation: whether it is a matter of Markov's algorithms (ANM) or of Turing Machines (TM) or of any other equivalent mathematical system, the elaboration progresses by discrete and successive instants.