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Pp (Complexity)In complexity theory, PP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to Probabilistic, Polynomial time. The complexity class was defined by Gill in 1977. If a decision problem is in PP, then there is an algorithm for it that is allowed to flip coins and make random decisions.It is guaranteed to run in polynomial time. If the answer is YES, the algorithm will answer YES with probability more than 1/2. If the answer is NO, the algorithm will answer YES with probability at most 1/2. PP vs BPP The distinction from BPP is in the error probability that is allowed (1/3 vs. 1/2). BPP can be viewed as a notion of efficient probabilistic algorithm. If a problem is in BPP, then there is an algorithm which distinguishes YES and NO instances with a small probability of failure. In contrast, PP is a complexity class that describes inefficient algorithms. For a PP algorithm and a YES instance, the probability of the algorithm outputting YES can be just slightly higher than 1/2, for example, 1/2+1/2n. If this is the case, we cannot distinguish this YES instance from a NO instance on which algorithm outputs YES with probability exactly 1/2, unless we repeat the algorithm an exponential number of times. PP compared to other complexity classes PP contains BPP, since probabilistic algorithms described in the definition of BPP form a subset of those in the definition of PP. PP also contains NP. To prove that, we show that satisfiability belongs to PP. Consider a probabilistic algorithm that, given a formula F(x1, x2, ..., xn) chooses an assignment x1, x2, ..., xn uniformly at random. Then, the algorithm checks if the assignment makes the formula F true. If yes, it outputs YES. Otherwise, it outputs YES with probability 1/2 and NO with probability 1/2. If the formula is unsatisfiable, the algorithm will always output YES with probability 1/2. If there exists a satisfying assignment, it will output YES with probability more than 1/2 (exactly 1/2 if it picked an unsatisfying assignement and 1 if it picked a satisfying assignment, averaging to some number greater than 1/2). Thus, this algorithm puts satisfiability in PP. A similar argument works for any other problem in NP. PP also contains BQP, the class of decision problems solvable by efficient polynomial time quantum computers. A polynomial time Turing machine with a PP oracle is even more powerful (if P ≠ NP). Namely, PH is contained in PPP. This result was shown by Seinosuke Toda in 1989 and is known as Toda's theorem. PP is contained in PSPACE. Complete problems and other properties Unlike BPP, PP is a syntantic, rather than semantic class. Any probabilistic machine recognizes some language in PP. In contrast, given a description of a probabilistic machine, it is hard to determine if it recognizes a language in BPP. PP has natural complete problems, for example, MAJSAT. MAJSAT is a decision problem in which one is given a Boolean formula F. The answer must be YES if more than half of all assignments x1, x2, ..., xn make F true and NO otherwise. PP is closed under union, intersection and complement. References - C. Papadimitriou. Computational Complexity, chapter 11. Addison-Wesley, 1994.
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