On the Structure of Valiant's Complexity Classes

One of the most important developments in theoretical computer science is the concept of NP-completeness. Recently, initiated by a paper by Blum, Shub, and Smale [6] (BSS-model), there has been a growing interest in investigating such concepts over general algebraic structures, with the purpose of classifying the complexity of continous problems. But already ten years earlier, Valiant [26, 28] had developed a convincing analogue of the theory of NP-completeness in an entirely algebraic framework, in connection with his famous hardness result for the permanent [27]. In fact, the generating functions of many NP-complete graph problems turn out to be complete in Valiant’s sense (cf. [7]). The major differences between the BSS-model and Valiant’s model are the absence of uniformity conditions in the latter, and the fact that only straight-line computations are considered (no branching). Both structured models are adapted to the framework of polynomial computations, and we believe that they will be useful for classifing the intrinsic