  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.
  This works dates from 2010 and was submitted in 2013 to the long-delayed
volume "The Incomputable" arising from the 2012 Cambridge Turing year.
