On Turing Machines Deciding According to the Shortest Computations ^†

In this paper we propose and analyse from the computational complexity point of view several new variants of nondeterministic Turing machines. In the first such variant, a machine accepts a given input word if and only if one of its shortest possible computations on that word is accepting; on the other hand, the machine rejects the input word when all the shortest computations performed by the machine on that word are rejecting. We are able to show that the class of languages decided in polynomial time by such machines is ${\mathbf{P}}^{\mathbf{NP}[log]}$. When we consider machines that decide a word according to the decision taken by the lexicographically first shortest computation, we obtain a new characterization of ${\mathbf{P}}^{\mathbf{NP}}$. A series of other ways of deciding a language with respect to the shortest computations of a Turing machine are also discussed.