Let A be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to A. With probability 1, each of the relativized classes LOGSPACEA, pA, NpA, ppA, and PSPACE A is properly contained in the nexi. Also, NP A co-NP a with probability 1. By contrast, with probability 1 the class pA coincides with the class BPP A of languages recognized by probabilistic oracle machines with error probability uniformly bounded below 1/2. NP A is shown, with probability 1, to contain a pA-immune set, i.e., a set having no infinite subset in pA. The relationship of pA-immunity to p-sparseness and NpA-completeness is briefly discussed: pA-immune sets in NP A can be sparse or moderately dense, but not co-sparse. Relativization with respect to a random length-preserving permutation 7r, instead of a random oracle A, yields analogous results and in addition the proper containment, with probability 1, of P in NP f3 co-NP , which we have been unable to decide for a simple random oracle. Most of these results are shown by straightforward counting arguments, applied to oracle-dependent languages designed not to be recognizable without a large number of oracle calls. It is conjectured that all pA-invariant statements that are true with probability of subrecursive language classes uniformly relativized to a random oracle are also true in the unrelativized case.
In distributed storage systems built using commodity hardware, it is necessary to have data redundancy in order to ensure system reliability. In such systems, it is also often desirable to be able to quickly repair storage nodes that fail. We consider a scheme-introduced by El Rouayheb and Ramchandran-which uses combinatorial block design in order to design storage systems that enable efficient (and exact) node repair. In this work, we investigate systems where node sizes may be much larger than replication degrees, and explicitly provide algorithms for constructing these storage designs. Our designs, which are related to projective geometries, are based on the construction of bipartite cage graphs (with girth 6) and the concept of mutually-orthogonal Latin squares. Via these constructions, we can guarantee that the resulting designs require the fewest number of storage nodes for the given parameters, and can further show that these systems can be easily expanded without need for frequent reconfiguration.
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