Loop-programs and Polynomially Computable Functions

By slightly modifying the original definition of loop-programs by Meyer and Ritchie, a modified hierarchy of loop(n) programs, Ƥn, is obtained, with the following characteristics. Let ℒ'n be the class of functions defined by programs in Ƥ'n and ℒ'n be the Grzegorzcyk hierarchy. Then ℒ'n= for n>1, where in particular E2 has a natural counterpart in loop programs. It also turns out that E2 can be characterized as the class of all recursive functions that are computable with a polynomial number of steps by modified programs.