COMPUTABILITY OVER MODELS OF DECIDABLE THEORIES

sigms-definability in hereditarily finite superstructures over algebraic systems is studied. We prove the sigma-definability criterion, which is then used as a basis for establishing the reduction theorem for regular theories and for obtaining a characterization of simple theories. The idea of a nonstandard recursion theory is developed using subfields of the field of reals as an example. partial algebraic description is given for a distributive upper semilattice of m sigma-degrees in hereditarily finite superstructures over models of simple theories.