sentences:  formula_59)
  
  We will define first-order with iteration, formula_60; here formula_61 is a (class of) functions from integers to integers, and for different classes of functions formula_61 we will obtain different complexity classes formula_60.In this section we will write formula_64 to mean formula_65 and formula_66 to mean formula_67.We first need to define quantifier blocks (QB), a quantifier block is a list formula_68 where the formula_69s are quantifier-free -formulae and formula_70s are either formula_13 or formula_14.