Summary: Combinatory Models and Symbolic Computation
Karl Aberer 1
We introduce an algebraic model of computation which is especially useful for the
description of computations in analysis. On one level the model allows the representation
of algebraic computation and on an other level approximate computation is represented.
Furthermore programs are themselves algebraic expressions. Therefore it is possible to
algebraically manipulate programs of symbolic and numerical computation, thus provid-
ing symbolic computation with a rm semantic foundation and giving a natural model
for mixed symbolic-numerical computation. We illustrate these facts with examples.
A substantial part of computer algebra deals with problems arising from analysis, see for
example Buchberger et al., 1983, Davenport et al., 1988]. This computational approach to
analysis makes use of algebraic representations of analytic structures, as for example di eren-
tial elds Kaplansky, 1957]. The objects used for these algebraic computations in analysis are
represented exactly. The computational approach to analysis which exploits the set-theoretic
properties of structures de ned over the real numbers, leads to computational methods, which
are usually related to numerical computation. In this case usually approximative representa-
tions for objects are used.
Computational models for analysis which origin from recursion theory, like, e.g., recursive