Efficient Simulations by a Biased Coin

A coin of bias p, denoted by C p , is a coin such that an outcome of the coin-tossing is H with probability p, and T with probability 1−p. Let S p be the set of biases efficiently simulated by a probabilistic Turing machine with C p . The main results in this paper are the following: (1) The set S 1/2 is equal to the set of the biases in finite figures on the binary system, (2) The set S 1/3 ie equal to the set of the biases in finite figures on the ternary system, (3) The set S b/a in reduced form with a>3, is properly contained in the set of the biases in finite figures on the a-ary system.