Next-bit Test - Completeness For Yao's Test

Completeness For Yao's Test

The next-bit test is a particular case of Yao's test for random sequences, and passing it is therefore a necessary condition for passing Yao's test. However, it has also been shown a sufficient condition by Yao.

We prove it now in the case of probabilistic Turing machine, since Adleman has already done the work of replacing randomization with non-uniformity in his theorem. The case of boolean circuits cannot be derived from this case (since it involves deciding potentially undecidable problems), but the proof of Adleman's theorem can be easily adapted to the case of non-uniform boolean circuits families.

Let a distringuer for the probabilistic version of Yao's test, i.e. a probabilistic Turing machine, running in polynomial time, such that there is a polynomial such that for infinitely many

Let . We have : and . Then, we notice that . Therefore, at least one of the should be no smaller than .

Next, we consider probability distributions and on . Distribution is the probability distribution of choosing the first bits in with probability given by, and the remaining bits uniformly at random. We have thus :

We thus have (a simple calculus trick shows this), thus distributions and can be distinguished by . Without loss of generality, we can assume that, with a polynomial.

This gives us a possible construction of a Turing machine solving the next-bit test : upon receiving the first bits of a sequence, pads this input with a guess of bit and then random bits, chosen with uniform probability. Then it runs, and outputs if the result is, and else.