Calculus of Cost Functions

Andre Nies

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arXiv · 1703.01577

Calculus of Cost Functions

Andre Nies

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

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Andre Nies. 2017-03-05. Calculus of Cost Functions. https://arxiv.org/abs/1703.01577

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