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A mathematical modeling of pure, recursive algorithms
 Logic at Botik ’89
, 1989
"... This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are ..."
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Cited by 9 (6 self)
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This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are the soundness of the reduction calculus of FLR (which models faithful, algorithm–preserving compilation) for this semantics, and the robustness of the class of algorithms assigned to a structure under algorithm adjunction. This is the second in a sequence of papers begun with [16] in which we develop a foundation for the theory of computation based on a mathematical modeling of recursive algorithms. The general features, aims and methodological assumptions of this program were discussed and illustrated by examples in the preliminary, expository report [15]. In [16] we studied the formal language of recursion FLR which is the main technical tool for this work, we developed several alternative denotational semantics for it and we established a key unique termination theorem for a reduction calculus which models faithful (algorithm–preserving) compilation. Here we will define the intensional semantics of FLR for structures with given (pure) recursors, the set–theoretic objects we use to model pure (side–effect–free) algorithms: the intension of a term t on each structure A is a recursor which models the algorithm expressed by t on A. Basic results of the paper During the preparation of this paper the author was partially supported by an NSF
Characteristic Properties of MajorantComputability over the Reals
 Proc. of CSL'98, LNCS, 1584
, 1999
"... ..."
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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Cited by 2 (2 self)
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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 ...
IOS Press
"... Computability in type2 objects with wellbehaved type1 oracles is pnormal ..."
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Computability in type2 objects with wellbehaved type1 oracles is pnormal
This document in subdirectoryRS/02/26/ Fixed Points on Abstract Structures without the Equality Test
, 909
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
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...
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...
A Logical Approach to Specification of Hybrid Systems
"... . The main subject of our investigation is behaviour of the continuous components of hybrid systems. By a hybrid system we mean a network of digital and analog devices interacting at discrete times. A firstorder logical formalisation of hybrid systems is proposed in which the trajectories of th ..."
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. The main subject of our investigation is behaviour of the continuous components of hybrid systems. By a hybrid system we mean a network of digital and analog devices interacting at discrete times. A firstorder logical formalisation of hybrid systems is proposed in which the trajectories of the continuous components are presented by majorantcomputable functionals. 1 Introduction In the recent time, attention to the problems of exact mathematical formalisation of complex systems such as hybrid systems is constantly raised. By a hybrid system we mean a network of digital and analog devices interacting at discrete times. An important characteristic of hybrid systems is that they incorporate both continuous components, usually called plants, as well as digital components, i.e. digital computers, sensors and atuators controlled by programs. These programs are designed to select, control, and supervise the behaviours of the continuous components. Modelling, design, and investigation...
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,...