Abstract:
We consider a class, denoted APP, of real-valued functions f:{0,1}^n\rightarrow [0,1] such that f can be approximated, to within any epsilon>0, by a probabilistic Turing machine running in time poly(n,1/epsilon). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a natural complete problem: computing the acceptance probability of a given Boolean circuit; in contrast, no complete problems are known for BPP. We observe that all known complexity-theoretic assumptions under which BPP is easy (i.e., can be efficiently derandomized) imply that APP is easy; on the other hand, we show that BPP may be easy while APP is not, by constructing an appropriate oracle.