Dot operators

Well-known examples of dot operators are the existential, the counting, and the BP-operator. We will generalize this notion of a dot operator so that every language A will determine an operator A·. In fact, we will introduce the more general notion of promise dot operators for which the BP-operator is an example. Dot operators are a re5nement of the leaf language concept because the class determined by a leaf language A equals A · P. Moreover, we are able to represent not only classes but reducibilities, in fact most of the known polynomial-time reducibilities can be represented by dot operators. We show that two languages determine the same dot operator if and only if they are reducible to each other by polylog-time computable monotone projections. c © 2001 Elsevier Science B.V. All rights reserved.