## Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability

##### Date

##### Authors

Ziegler, Martin

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

##### Description

It is folklore particularly in numerical and computer sciences that, instead
of solving some general problem f:A->B, additional structural information about
the input x in A (that is any kind of promise that x belongs to a certain
subset A' of A) should be taken advantage of. Some examples from real number
computation show that such discrete advice can even make the difference between
computability and uncomputability. We turn this into a both topological and
combinatorial complexity theory of information, investigating for several
practical problems how much advice is necessary and sufficient to render them
computable.
Specifically, finding a nontrivial solution to a homogeneous linear equation
A*x=0 for a given singular real NxN-matrix A is possible when knowing
rank(A)=0,1,...,N-1; and we show this to be best possible. Similarly,
diagonalizing (i.e. finding a BASIS of eigenvectors of) a given real symmetric
NxN-matrix is possible when knowing the number of distinct eigenvalues: an
integer between 1 and N (the latter corresponding to the nondegenerate case).
And again we show that N-fold (i.e. roughly log N bits of) additional
information is indeed necessary in order to render this problem (continuous
and) computable; whereas for finding SOME SINGLE eigenvector of A, providing
the truncated binary logarithm of the least-dimensional eigenspace of A--i.e.
Theta(log N)-fold advice--is sufficient and optimal.

Comment: added Sections 5.1 and 5.2

Comment: added Sections 5.1 and 5.2

##### Keywords

Computer Science - Computational Complexity, Mathematics - Logic, F.1.1, F.2.1, F.4.1, H.1.1, G.1.3