Elementary Bounded Languages

The POL be the class of polynomials having nonnegative integer coefficients and EXP the class of exponential functions. We call the closure of POL ∪ EXP under superposition, primitive recursion, and exponentiation, the class of elementary functions ( EF ). We have obtained that every elementary bounded language (i.e., language in the form { w 1 f 1 ( n ) ⋯ w t f t ( n | n ࢠ N k , w i words f i ࢠ EF }) is context-sensitive. A concept for the computability of the functions usinggrammars is given, and it is shown that every function from EF is computable inthis manner, using context-sensitive grammars. By considering a new unaryoperation with respect to languages, called polynomial iteration (which is ageneralization of star closure) we prove that the class of context-sensitivelanguages is closed with respect to this operation but neither the class of contextfreenor the class of regular languages is closed.