Randomness and Counting 6.1 Probabilistic Polynomial-time Construction 6.4 (the Reduction): on Input a Natural Number N > 2 Do

I owe this almost atrocious variety to an institution which other republics do not know or which operates in them in an imperfect and secret manner: the lottery. So far, our approach to computing devices was somewhat conservative: we thought of them as executing a deterministic rule. A more liberal and quite realistic approach , which is pursued in this chapter, considers computing devices that use a probabilistic rule. This relaxation has an immediate impact on the notion of ee-cient computation, which is consequently associated with probabilistic polynomial-time computations rather than with deterministic (polynomial-time) ones. We stress that the association of eecient computation with probabilistic polynomial-time computation makes sense provided that the failure probability of the latter is negligible (which means that it may be safely ignored). The quantitative nature of the failure probability of probabilistic algorithm provides one connection between probabilistic algorithms and counting problems. The latter are indeed a new type of computational problems, and our focus is on counting eeciently recognizable objects (e.g., NP-witnesses for a given instance of set in NP). Randomized procedures turn out to play an important role in the study of such counting problems. Summary: Focusing on probabilistic polynomial-time algorithms, we consider various types of probabilistic failure of such algorithms (e.g., actual error versus failure to produce output). This leads to the formulation of complexity classes such as BPP, RP, and ZPP. The results presented include the existence of (non-uniform) families of polynomial-size circuits that emulate probabilistic polynomial-time algorithms (i.e., BPP P=poly) and the fact that BPP resides in the (second level of the) Polynomial-time Hierarchy (i.e., BPP 2). We then turn to counting problems; speciically, counting the number of solutions for an instance of a search problem in PC (or, equivalently, 203 204 CHAPTER 6. RANDOMNESS AND COUNTING counting the number of NP-witnesses for an instance of a decision problem in NP). We distinguish between exact counting and approximate counting (in the sense of relative approximation). In particular, while any problem in PH is reducible to the exact counting class #P, approximate counting (for #P) is (probabilisticly) reducible to NP. In general, counting problems exhibit a \richer structure" than the corresponding search (and decision) problems, even when considering only natural problems. For example, some counting problems are hard in the exact version (e.g., are #P-complete) but easy to approximate, while others are NP-hard to approximate. In some cases #P-completeness is due to …