Computations in Higher Types
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574 Johan Moldestad
Computations in Higher Types
Springer-Verlag Berlin.Heidelberg. New York 1977
Author Johan M o l d e s t a d M a t e m a t i s k Institutt Postboks 1053 Blindern, O s l o 3 / N o r w a y
Library of Congress Cataloging in Publication Data
Moldestad~ Joham, 1946Computations in higher types. (Lecture notes in mathematics ; 574) Bibliography: p. Includes index. i. Recursive functions. I. Title. II. Series: Lecture motes in mathematics (Berlin) ~ 574° QA3.L28 no. 574 [QA248.5] 510'.8s [511'.3] 77-1375
AMS Subject Classifications (1970): 02F27, 02F29
ISBN 3-540-08132-1 ISBN 0-387-08132-1
Springer-Verlag Berlin • Heidelberg • New York Springer-Verlag New York • Heidelberg - Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 197"7 Printed in Germany Printing and binding: Beltz, Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface §§ i - 9
and § q3
have previously appeared in the Preprint Series
of the Institute of Mathematics,
University of Oslo (No. 2
tation ~heories on two Types by Joham Holdestad). changes have been made. myself.
In § 40
§§ 4 0 - q4
is
Compu-
A few corrections and
a joint work of Dag Normann and
the notion of a countable recursion structure is due to
him~and independently to Jan Bergstra both of us.
4976
Part B
[23].
Part A
is written by me, and part C
in § 4 q
is due to
by Dag Normann.
§ 42
is written by me in collaboration with professor Jens Erik Fenstad. Jens Erik Fenstad inspired me to write this book. valuable tions,
comments concerning selection of topics,
etc.
improvements,
correc-
I owe him great debt for this and for the interest he has
shown in the work. cussions,
He has given
I am also grateful to Dag Normann for helpful dis-
and to Randi M@ller and Sigrid Cordtzen £or the typewriting.
Oslo, q976 Johan Moldestad
Contents Page q
Abstract
44
§ q
The computation
§ 2
Recursion
§ 3
Connection with Kleene
§ ~
Recursion
§ 5
Kleene recursion
§ 6
Computation
§ ~
Abstract Kleene theories
§ 8
Normal
§ 9
More about Hahloness
§ 40
Calculation
§ lq
Gaps
§ q2
0n Platek:
§ q3
A final comment
n+2,
on
domain
g~
47 recursion
in higher types
in normal lists on
n>O
34
in normal objects
of type
theores on
computation
26
5O 54
6O
theories
64
on
99
of the lengths of some computations
q09 q20
"Foundations
of Recursion
concerning
Theory"
the two types
q50 ~93
References
q95
Index of notations
197
Subject index
2OO
Abstract R e c u r s i o n in h i g h e r types was first studied by Kleene. fundamental paper I, Ii"
In
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