Descriptive complexity of Hash P functions

A logic-based framework for defining counting problems is given, and it is shown that it exactly captures the problems in Valiant's counting class Hash P. The expressive power of the framework is studied under natural syntactic restrictions, and it is shown that some of the subclasses obtained in this way contain problems in Hash P with interesting computational properties. In particular, using syntactic conditions, a class of polynomial-time-computable Hash P problems is isolated, as well as a class in which every problem is approximable by a polynomial-time randomized algorithm. These results set the foundation for further study of the descriptive complexity of the class Hash P. In contrast, it is shown, under reasonable complexity theoretic assumptions, that it is an undecidable problem to tell if a counting problem expressed in the framework is polynomial-time computable or is approximable by a randomized polynomial-time algorithm. Some open problems are discussed.<
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