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

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 ∈ A (that is any kind of promise that x belongs to a certain subset A ⊆ 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 n × n-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 n × n-matrix A is possible when knowing the...

• Type: Conference
• Year: 2009
• Where: CCA
• Authors: Martin Ziegler
• Keywords: Additional Structural Information, CCA 2009, Combinatorial Complexity Theory, Real Number Computation, Theoretical Computer Science