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Primitive recursive algebraic theories and program schemes

Published online by Cambridge University Press:  17 April 2009

W. Kühnel ,
J. Meseguer ,
M. Pfender  and
I. Sols
W. Kühnel
Affiliation:
Fachbereich Mathematik, Technische Universität Berlin, Berlin, Germany;
J. Meseguer
Affiliation:
Departamento de Algebra y Fundamentos, Facultad de Ciencias, Universidad, Santiago de Compostela, Spain;
M. Pfender
Affiliation:
Fachbereich Mathematik, Technische Universität Berlin, Berlin, Germany;
I. Sols
Affiliation:
Departamento de Matematica, Universidad de Zaragoza, Zaragoza, Spain.
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We introduce primitive recursion as a generation process for arrows of algebraic theories in the sense of Lawvere and carry over important results on algebraic theories and functorial semantics to the enriched setting of “primitive recursive algebra”: existence of free primitive recursive theories and of theories presented by operations and equations on primitive recursive functions; existence of free models presented by generators and equations. Finally semantical correctness of translations is reduced to correctness for the basic operations. There is a connection to the theory of program schemes: program schemes involving primitive recursion correspond to arrows of a primitive recursive theory freely generated over a graph of basic operations. This theory T can be viewed as a programming language with “arithmetics” given by the basic operations and with DO-loops. A machine loaded with a compiler for T can be interpreted as a T-model in Lawvere's sense, preserving primitive recursion.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 17 , Issue 2 , October 1977 , pp. 207 - 233
Copyright
Copyright © Australian Mathematical Society 1977

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