Calculus of Cost Functions

Abstract

Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive strength. We show that the class of additive cost functions describes the K-trivial sets. We prove a cost function basis theorem, and give a general construction for building computably enumerable sets that are close to being Turing complete.