Recursion Theoretic Characterizations of Complexity Classes of Counting Functions

There has been a great eeort in giving machine independent, algebraic characterizations of complexity classes, especially of functions. Astonishingly, no satisfactory characterization of the prominent class # P has been known up to now. Here, we characterize # P as the closure of a set of simple arithmetical functions under summation and weak product. Building on that result, the hierarchy of counting functions, which is the closure of # P under substitution, is characterized; remarkably without using the operator of substitution, since we can show that in the context of this hierarchy the operation of modiied subtraction is as powerful as substitution. This leads us to a number of consequences concerning closure of # P under certain arith-metical operations. Analogous results are achieved for the class Gap-P which is the closure of # P under subtraction .