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Characteristic Properties of MajorantComputability over the Reals
 Proc. of CSL'98, LNCS, 1584
"... . Characteristic properties of majorantcomputable realvalued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorantcomputability proposed in [13 ..."
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Cited by 5 (5 self)
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. Characteristic properties of majorantcomputable realvalued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorantcomputability proposed in [13]. A modeltheoretical characterization of majorantcomputability realvalued functions and their domains is investigated. A theorem which connects the graph of a majorantcomputable function with validity of a finite formula on the set of hereditarily finite sets on IR, HF( IR) (where IR is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the definition of majorantcomputability and the notions of computability earlier proposed by Blum et al., Edalat, Sunderhauf, PourEl and Richards, StoltenbergHansen and Tucker is given. Examples of majorantcomputable realvalued functions are presented. 1 Introduction In the recent time, attention to the prob...
Projections Of Semicomputable Relations On Abstract Data Types
, 1991
"... DATA TYPES J.V. TUCKER Department of Mathematics and Computer Science, University College of Swansea, Swansea SA2 8PP, Wales and J.I. ZUCKER Department of Computer Science and Systems, McMaster University, Hamilton, Ontario L8S 4K1, Canada ABSTRACT We consider projections of semicomputable rela ..."
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DATA TYPES J.V. TUCKER Department of Mathematics and Computer Science, University College of Swansea, Swansea SA2 8PP, Wales and J.I. ZUCKER Department of Computer Science and Systems, McMaster University, Hamilton, Ontario L8S 4K1, Canada ABSTRACT We consider projections of semicomputable relations on abstract structures We show that they arise in several contexts, including nondeterministic extensions of ` while' programs with arrays by means of arbitrary initializations and random assignments. They form a basis for an investigation of the concept of algorithmic specifications of relations with nondeterministic search. An important technique in this investigation is the study of computation trees for imperative programs, in order to prove characterization theorems for semicomputable sets, of a form first developed by E. Engeler. Keywords: abstract data types, computability, manysorted algebras 0. Introduction The mathematical theory of computable functions and relations over ...
A New Approach to Computability on Real Numbers
, 1997
"... We introduce majorant computability of functions on reals. A structural theorem is proved, which connects the graph of a majorantcomputable function with validity of a finite formula on the set of the hereditarily finite set HF( ¯ R) (where ¯ R is an elementary proper extension of standard real nu ..."
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We introduce majorant computability of functions on reals. A structural theorem is proved, which connects the graph of a majorantcomputable function with validity of a finite formula on the set of the hereditarily finite set HF( ¯ R) (where ¯ R is an elementary proper extension of standard real numbers). The class of majorantcomputable functions in our approach include an interesting class of real total functions having meromorphic extension on C. This class in particular contains functions which are solutions of known differential equations. The notion of a majorantcomputable functional on the set of total majorantcomputable real functions is defined. As an example of a majorantcomputable functional the Riemann integral is proposed. 1 Section 1. Introduction Section 1 Introduction The semiring of natural numbers is the classical structure for which the concept of computability has been defined and studied rather well. Though several definitions were independently proposed,...
MajorantComputability And Definability Over The Reals
"... The concept of majorantcomputability over the reals which integrates methods and ideas of continuous mathematics and the modern mathematical logic is introduced. Properties of majorantcomputable realvalued functions are studied. Definability of majorantcomputable realvalued functions and majora ..."
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The concept of majorantcomputability over the reals which integrates methods and ideas of continuous mathematics and the modern mathematical logic is introduced. Properties of majorantcomputable realvalued functions are studied. Definability of majorantcomputable realvalued functions and majorantenumerable subsets of the reals is investigated in this framework. A theorem which connects the graph of a majorantcomputable function with validity of a finite formula on the set of hereditarily finite sets on R, HF( R) (where R is an elementary proper extension of the standard reals) is proved. The Mandelbrot set and Julia sets are given as examples od sets that are majorantenumerable. Keywords: majorantcomputabile function, majorantenumerable set, computational process, approximation, definability. 1 Introduction We introduce the concept of majorantcomputability over the reals which integrates methods and ideas of continuous mathematics and the modern mathematical logic, in pa...
Some Properties of MajorantComputability
"... We introduce the concept of majorantcomputability over the reals which integrates methods and ideas of the modern mathematical logic in particular definability theory and continuous mathematics. We study properties of majorantcomputable real and realvalued functions and functionals. In this frame ..."
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We introduce the concept of majorantcomputability over the reals which integrates methods and ideas of the modern mathematical logic in particular definability theory and continuous mathematics. We study properties of majorantcomputable real and realvalued functions and functionals. In this framework we investigate definability of majorantcomputable real functions and functionals. A structural theorem is proved which connects the graph of a majorantcomputable function with validity of a finite formula on the set of hereditarily finite sets on R HF( R) (where R is an elementary proper extension of the standard real numbers). We also find interesting examples of real functions, a functional and a set which are majorantcomputable. Keywords: majorantcomputability, computational process, approximations, definability. 1 Introduction The semiring of natural numbers is a classical structure for which the concept of computability has been defined and studied rather well. Although seve...