Minimization and NP multifunctions

Abstract

The implicit characterizations of the polynomial-time computable functions FP given by Bellantoni–Cook and Leivant suggest that this class is the complexity-theoretic analog of the primitive recursive functions. Hence, it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's definition of FP by bounded recursion and for Bellantoni–Cook's safe recursion and prove that the resulting classes capture exactly NPMV , the nondeterministic polynomial-time computable partial multifunctions. We also consider the relationship between our schemes and a notion of nondeterministic recursion defined by Leivant and show that the latter characterizes the total functions of NPMV . We view these results as giving evidence that NPMV is the appropriate analog of partial recursive. This view is reinforced by earlier results of Spreen and Stahl who show that for many of the relationships between partial recursive functions and r.e. sets, analogous relationships hold between NPMV and NP sets. Furthermore, since NPMV is obtained from FP in the same way as the recursive functions are obtained from the primitive recursive functions (when defined via function schemes), this also gives further evidence that FP is properly seen as playing the role of primitive recursion.