NP-hardness of circuit minimization for multi-output functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions

Derandomizing from Random Strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings

Stochastic Description of Agglomeration and Growth Processes in Glasses

We show how growth by agglomeration can be described by means of algebraic or differential equations which determine the evolution of probabilities of various local configurations. The minimal fluctuation condition is used to define vitrification. Our methods have been successfully used for the description of glass formation.Comment: 9 pages, 1 figure, LaTeX 2e, uses ws-ijmpb.cls ; submitted to International Journal of Modern Physics

Our Dream and our Nightmare: Anomie and Violence in Historical Periods

This paper explores outbreaks of violence in two American historical periods through the examination of Durkheim \u27s theories on anomie. Core values behind the American Dream, individualism, and a frontier mentality are also examined. Both the American western frontier and the Post-Reconstruction South experienced periods of violent unrest among the general population. Both of these historical periods were under conditions of anomie associated with the lack of regulation from social institutions, anxiety over the removal (or sudden presentation of) existing opportunities, and the forces of individualism and the penchant for violence provided by American ethics