Computable Functionals of Finite Type I

Publisher Summary This chapter introduces the ordinary recursion theory (ORT). The theory has two salient features: (a) although partial recursive functionals are constructed, the arguments of these functionals are required to be totally defined and (b) the arguments are considered solely as extensions. The chapter describes several notational preliminaries and various classes of functional, which must certainly be counted as computable. A description of computable functionals is developed in the chapter, which is based on indexed schemata. The schemata constitute, in effect, an inductive definition of the value of a computation; there is a tacit assumption that a computation only gets a value if it does so as a consequence of them. The chapter examines the form of this inductive definition in greater detail. The manipulations that are required to prove that various processes are computable are also presented in the chapter. The specification of a computation tree contains a lot of detail that—though it may be relevant for ORT—is not relevant to the specific problems raised by the introduction of arguments of higher type.