We introduce Computational Depth, a measure for the amount of "nonrandom" or "useful" information in a string by considering the difference of various Kolmogorov complexity measures. We investigate three instantiations of Computational Depth:• Basic Computational Depth, a clean notion capturing the spirit of Bennett's Logical Depth. We show that a Turing machine M runs in time polynomial on average over the time-bounded universal distribution if and only if for all inputs x, M uses time exponential in the basic computational depth of x. • Sublinear-time Computational Depth and the resulting concept of Shallow Sets, a generalization of sparse and random sets based on low depth properties of their characteristic sequences. We show that every computable set that is reducible to a shallow set has polynomial-size circuits. • Distinguishing Computational Depth, measuring when strings are easier to recognize than to produce. We show that if a Boolean formula has a nonnegligible fraction of its satisfying assignments with low depth, then we can find a satisfying assignment efficiently.Torturing an uninformed witness cannot give information about the crime. Leonid Levin [16] ଁ Preliminary versions of different parts of this paper appeared as [1] and [2]. Much of the research for this paper occurred at the NEC Research Institute.

The effective fractal dimensions at the polynomial-space level and above can all be equivalently defined as the C-entropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspace-dimension is equivalent to the PSPACE-entropy rate.At lower levels of complexity the equivalence proofs break down. In the polynomial-time case, the P-entropy rate is a lower bound on the p-dimension. Equality seems unlikely, but separating the P-entropy rate from p-dimension would require proving P = NP.We show that at the finite-state level, the opposite of the polynomial-time case happens: the REG-entropy rate is an upper bound on the finite-state dimension. We also use the finite-state genericity of Ambos-Spies and Busse [Automatic forcing and genericity: On the diagonalization strength of finit automata, in: Proc. fourth Int. Conf. on Discrete Mathematics and Theoretical Computer Science, 2003, Springer, Berlin, pp. 97-108] to separate finite-state dimension from the REG-entropy rate. However, we point out that a block-entropy rate characterization of finite-state dimension follows from the work of Ziv and Lempel [Compression of individual sequences via variable rate coding, IEEE Trans. Inform. Theory 24 (1978) 530-536] on finitestate compressibility and the compressibility characterization of finite-state dimension by Dai et al. [Finite-state dimension, Theoret. Comput. Sci. 310(1-3) (2004) 1-33].As applications of the REG-entropy rate upper bound and the block-entropy rate characterization, we prove that every regular language has finite-state dimension 0 and that normality is equivalent to finite-state dimension 1.

We make progress in understanding the complexity of the graph reachability problem in the context of unambiguous logarithmic space computation; a restricted form of nondeterminism. As our main result, we show a new upper bound on the directed planar reachability problem by showing that it can be decided in the class unambiguous logarithmic space (UL). We explore the possibility of showing the same upper bound for the general graph reachability problem. We give a simple reduction showing that the reachability problem for directed graphs with thickness two is complete for the class nondeterministic logarithmic space (NL). Hence an extension of our results to directed graphs with thickness two will unconditionally collapse NL to UL.

The multiple-instance learning (MIL) model has been very successful in application areas such as drug discovery and content-based imageretrieval. Recently, a generalization of this model and an algorithm for this generalization were introduced, showing significant advantages over the conventional MIL model in certain application areas. Unfortunately, this algorithm is inherently inefficient, preventing scaling to high dimensions. We reformulate this algorithm using a kernel for a support vector machine, reducing its time complexity from exponential to polynomial. Computing the kernel is equivalent to counting the number of axis-parallel boxes in a discrete, bounded space that contain at least one point from each of two multisets P and Q. We show that this problem is #P-complete, but then give a fully polynomial randomized approximation scheme (FPRAS) for it. Finally, we empirically evaluate our kernel.

We prove that AM (and hence Graph Nonisomorphism) is in NP if for some ǫ > 0, some language in NE ∩ coNE requires nondeterministic circuits of size 2 ǫn . This improves results of Arvind and Köbler and of Klivans and van Melkebeek who proved the same conclusion, but under stronger hardness assumptions. The previous results on derandomizing AM were based on pseudorandom generators. In contrast, our approach is based on a strengthening of Andreev, Clementi and Rolim's hitting set approach to derandomization. As a spin-off, we show that this approach is strong enough to give an easy proof of the following implication: for some ǫ > 0, if there is a language in E which requires nondeterministic circuits of size 2 ǫn , then P = BPP. This differs from Impagliazzo and Wigderson's theorem "only" by replacing deterministic circuits with nondeterministic ones.