Recursion theoretic characterizations of complexity classes of counting functions

Heribert Vollmer, Klaus W. Wagner
1996 Theoretical Computer Science  
There has been a great effort in giving machine-independent, algebraic characterizations of complexity classes, especially of functions. Astonishingly, no satisfactory characterization of the prominent class #P is known up to now. Here, we characterize #P as the closure of a set of simple arithmetical functions under summation and weak product. Based on that result, the hierarchy of counting functions, which is the closure of #P under substitution, is characterized, remarkably without using the
more » ... operator of substitution, since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution. This leads us to a number of consequences concerning closure of #P under certain arithmetical operations.
doi:10.1016/0304-3975(95)00237-5 fatcat:3esk7336bbh3rpznmclz6enr7a