The Algorithmization of Physics: Math Between Science and Engineering - Artificial Intelligence and Symbolic Computation - page 4

view of these facts, Buchberger [7, 3, 5] has defined symbolic computation (in par-
ticular: computer algebra) as that part of algorithmic mathematics (in particular:
algorithmic algebra) which solves problems stated in non-algorithmic domains
by translating them into isomorphic algorithmic representations.
Let us look at three immediate examples:

1. The traditional definition of polynomials introduces them as certain (infi-
nite!) congruence classes over some term algebra in the variety of unital com-
mutative rings [13]. The modern definition starts from a monoid ring [12],
which turns out to be an isomorphic translation of the former, basically
encoding the canonical representatives of the congruence classes.
2. As another example, consider the cardinal question of ideal membership in
algebraic geometry: As infinite sets, ideals cannot directly be treated by
algorithmic methods; representing them via Grobner bases allows a finitary
description and a solution of the ideal membership problem [1, 2, 6].
3. Finally, let us consider an example from a traditional domain of symbolic
computation that does not belong to computer algebra, namely automated
theorem proving . It is based on translating the non-algorithmic (semantic)
concept of consequence into the algorithmic (syntactic) concept of deducibil-
ity, the isomorphism being guaranteed by Godel's Completeness Theorem.

Returning to the example of boundary value problems introduced above, we
should also note that there are actually two famous approaches for algorith-
mization: symbolic computation takes the path through algebraization, whereas
numerical computation goes through approximation. Simplifying matters a bit,
we could say that symbolic computation hunts down the algebraic structure
of the continuum, numerical computation its topological structure 1 . But while
industrial mathematics is virtually flooded with numerical solvers (mostly of
finite-element type), it is strange to notice that computer algebra systems like
Maple or Mathematica do not provide any command for attacking those BVPs
that have a symbolic solution. My own research on symbolic functional analysis
can be seen as an endeavor to change this situation.

4 Symbolic Functional Analysis:

Conquering a New Territory for Computing

I will now sketch my own contribution to the exciting process of conquering more
and more territory through algebraization and algorithmization. As mentioned
before, it deals with certain boundary value problems .Moreprecisely,weare

1 The ideal algorithmic approach to problem solving would be to combine the best
parts of both symbolic and numerical computation, which is the overall objective
of a 10-year special research project (SFB) at Johannes Kepler University. My own
research there takes place in the subproject [17] on symbolic functional analysis; for
some details, see the next section.