Turing Machines with One-sided Advice and Acceptance of the co-RE Languages

We resolve an old problem, namely to design a ‘natural’ machine model for accepting the complements of recursively enumerable languages. The model we present is based on Turing machines with ‘onesided’ advice, which are Turing machines with advice with a restricted scenario for the use of their advice during computations. We show that Turing machines with one-sided advice accept precisely the co-RE languages, even though the advices can be arbitrary. We prove that Turing machines with one-sided advice are not less powerful than their unrestricted version: for all languages L ∈ co-RE, L is accepted by an ordinary Turing machine with advice f if and only L is accepted by a Turing machine with one-sided advice f . We show that co-RE filters into an infinite proper hierarchy, based on the complexity (‘length’) of the advice f(n) a one-sided Turing machine needs for accepting a language. The model and the results dualise to the ordinary RE-languages.