### Probabilistic Type-2 Operators and "Almost"-Classes

R.V. Book, H. Vollmer, K.W. Wagner
1998 Computational Complexity
We define and examine several probabilistic operators ranging over sets (i.e., operators of type 2), among them the formerly studied ALMOST-operator. We compare their power and prove that they all coincide for a wide variety of classes. As a consequence, we characterize the ALMOST-operator which ranges over infinite objects (sets) by a bounded-error probabilistic operator which ranges over strings, i.e. finite objects. This leads to a number of consequences about complexity classes of current
more » ... terest. As applications, we obtain (a) a criterion for measure 1 inclusions of complexity classes, (b) a criterion for inclusions of complexity classes relative to a random oracle, (c) a new upper time bound for ALMOST-PSPACE, and (d) a characterization of ALMOST-PSPACE in terms of checking stack automata. Finally, a connection between the power of ALMOST-PSPACE and that of probabilistic £ ¥ ¤ § ¦ circuits is given. (for bounded error probabilistic polynomial time) which are regarded as natural probabilistic counterparts of the deterministic class¨; and moreover is felt to be the class of "tractable" problems (since the error bound can be made arbitrarily small). A preliminary abstract of this paper appeared in the proceedings of the 23rd International Colloquium on Automata, Languages, and Programming. But how do define probabilistic analogues for other (possibly not deterministic) classes? The "traditional" way is to consider operators in an abstract way as we will do in this paper. This kind of randomness can best be visualized as allowing Turing machines access to a random tape, or equivalently supplying them together with their regular input with an input sequence of random bits. Thus, here the random bits may be multiply accessed. (This should be contrasted with the machines with built-in probabilism described above: If those machines want to re-use their random bits later, they have to store them on their worktape-which might make a difference for space-bounded computations. Therefore the aforementioned built-in probabilism is also called one-way access to randomness, see  .) Well known examples for operators as just described are Wagner's counting operator ¢ ¡  , and the corresponding bounded error operator BP