About and around Computing over the Reals

Computation Procedures • Abstract Computation Procedures (ACPs), (Feferman 1992); should have been called Abstract Recursion Procedures. • Here structures are specified by (possibly) manysorted domains, individual constants, partial functions, and partial monotonic functionals of type level 2. ACP Computable Functions and Functionals • The ACP schemata are given by Explicit Definition in type levels 1 and 2, and LFP Recursion in type 2. • ACP(A) = the set of partial functions over A generated by the ACP schemata. • ACP*(A) = ACP(A, N, A*) Relations to the Other Approaches • While(A) = ACP(A) and While*(A) = ACP*(A) by Xu and Zucker 2005. • So BSS finite and infinite dim. computable fns. on R are subsumed under the ACP approach. • The type 2 functionals generated in ACP(N) are just the partial recursive functionals, so the Seff.approx. approach is also subsumed under the ACP approach. Extensional/Intensional Aspects • The foregoing theories are all extensional. • ACP(N) can also be given an intensional interpretation by replacing the partial functions and functionals by Gödel numbers. • Each type 2 functional in this interpretation of ACP(N) is an effective operator in the MyhillShepherdson sense. • Actual computers can actually compute on codes.