We survey recent developments related to the Minimum Circuit Size Problem and time-bounded Kolmogorov Complexity.

This dissertation presents the results of my research in two areas: parallel algorithms/circuit complexity, and algorithmic motion planning. The chapters on circuit complexity examine the parallel complexity of several fundamental problems (such as integer division) in the model of small depth circuits. In the later chapters on motion planning, we turn to the computationally intensive problem o...

The thesis consists of three parts. The first part investigates the AC complexity of subgraph isomorphism problem that is, detecting whether an n-vertex input graph contains a P -subgraph for some fixed P . For the average-case complexity, where the input is a distribution on random graphs at the threshold and small error is allowed, we are able to explicitly characterize the minimal AC size up...

We investigate in a method for proving separation results for abstract classes of languages. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so called equations, which are capable of describing arbitrary Boolean algebras of languages....

The main result of this paper is a (n 1=4) lower bound on the size of a sigmoidal circuit computing a speciic AC 0 2 function. This is the rst lower bound for the computation model of sigmoidal circuits with unbounded weights. We also give upper and lower bounds for the same function in a few other computation models: circuits of AND/OR gates, threshold circuits, and circuits of piecewise-ratio...