Towards a computational theory of statistical tests

The authors initiate a computational theory of statistical tests. Loosely speaking, an algorithm is a statistical test if it rejects a 'negligible' fraction of strings. A statistical test is universal for a class of algorithms if it rejects all (but finitely many) of the strings rejected by each algorithm in the class. They consider the existence and efficiency of universal statistical tests for various classes of statistical tests. They also consider the relation between ensembles passing statistical tests of particular complexity and ensembles which are indistinguishable from uniform by algorithms of the same complexity. Some results refer to relatively simple statistical tests (e.g. those implemented by counter machines).>