Multivalued Functions: Turing Characterizations and Complexity Results Domenico Sacc`a and Francesco Scarcello DEIS, University of Calabria 87030 Rende (CS), Italy {sacca,scarcello}@deis.unical.it Abstract This paper presents a nice Turing characterization of im- portant classes of multivalued functions (used to model search prob- lems) which elucidates properties of such classes as well as relationships among them. The results are based on a Turing transducer, called write- nondeterministic (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 intro- duced 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 func- tions, i.e., those functions such that (any)one of their results can be computed using a polynomial-time single-valued function. Keywords: Computational complexity, multivalued functions, Turing machines. 1 Introduction and Overview of Results Classic complexity theory focuses on studying the properties of classes of decision problems (alias, languages) in the place of studying classes of search problems (alias, functions), the underlying assumption being that properties of classes of functions can be obtained fairly easily by extending corresponding results on languages. However, as shown by recent papers, this is not completely true: in moving from language complexity to function complexity, new ad hoc formal analysis tools are needed to deal with non-determinism — indeed functions are often multivalued (MV), for search problems in general admit multiple solutions. Many classes of MV functions are known in the literature. In particular, the class NPMV is defined as the set of all MV functions f such that both (i) f is polynomially balanced (i.e., for each x, the size of each result in f (x) is polynomially bounded in the size of x) and (ii) graph(f ) is in NP. By analogy, the classes NPMV g and coNPMV are defined as the classes of all polynomially- balanced multivalued functions f for which graph(f ) is respectively in P and in coNP . These classes have no characterization in terms of Turing machines.