Multivalued Functions: Turing Characterizations and Complexity Results

This paper presents a nice Turing characterization of important classes of multivalued functions (used to model search problems) which elucidates properties of such classes as well as relationships among them. The results are based on a Turing transducer, called writenondeterministic (WND) transducer, which extends deterministic Turing machines with a simple non-deterministic construct which leaves track of each guess on the output tape. Moreover, the paper shows that oracle WND-transducers characterize a hierarchy of function classes corresponding to the polynomial hierarchy PH. A suitable notion of reduction for MV function-classes is also introduced to overcome some drawbacks of previously-proposed reductions. Finally, a simple restriction on the amount of nondeterminism of WND transducers gives a Turing characterization of the “tractable” MV functions, i.e., those functions such that (any)one of their results can be computed using a polynomial-time single-valued function.