Type 1 Computability Type 2 Computability
For any representation δ there is a derived numbering vδ, defined by νδ(i):=δφi, of the δ-computable elements. In this chapter the relation between δ-computability and νδ-computability is investigated. It is easy to show that the restriction of a (δ,δ,)-computable function to the computable elements is (νδ,νδ’)-computable. The converse does not seem to be true in general. Only for two special cases a positive answer is known: for “effective” metric spaces (special cases have been proved independently by Ceitin, by Kreisel, Lacombe, and Shoenfield, and by Moschovakis) and for “computable” cpo’s (a special case has been proved by Myhill and Sheperdson). As a corollary we obtain a theorem which characterizes the νδ-r.e. subsets of the δ-computable elements for “computable” cpo’s (Rice /Shapiro theorem). The proofs of the main theorems are rather sophisticated.
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