ON NON-CLASSICAL THEORY OF COMPUTABILITY

Definition of arithmetical functions with indeterminate values of arguments is given. Notions of computability, strong computability and λ-definability for such functions are introduced. Monotonicity and computability of every λ-definable arithmetical function with indeterminate values of arguments is proved. It is proved that every computable, naturally extended arithmetical function with indeterminate values of arguments is λ-definable. It is also proved that there exist strong computable, monotonic arithmetical functions with indeterminate values of arguments, which are not λ-definable. The δ-redex problem for strong computable, monotonic arithmetical functions with indeterminate values of arguments is defined. It is proved that there exist strong computable, λ-definable arithmetical functions with indeterminate values of arguments, for which the δ-redex problem is unsolvable.