Descriptive complexity of #P functions: A new perspective

Abstract We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that and appear as classes of this hierarchy. In this way, we unconditionally place properly in a strict hierarchy of arithmetic classes within . Furthermore, we show that some of our classes admit efficient approximation in the sense of FPRAS. We compare our classes with a hierarchy within defined in a model-theoretic way by Saluja et al and argue that our approach is better suited to study arithmetic circuit classes such as which can be descriptively characterized as a class in our framework.

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