Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0811.3782

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

Martin Ziegler
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CCmath.LO

Keep this discovery

BibTeXRIS

Martin Ziegler. 2009-09-02. Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability. https://arxiv.org/abs/0811.3782

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗