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

Abstract: 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.