Editing PP (complexity)

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{{short description|Class of problems in computer science}}
[[File:Randomised Complexity Classes 2.svg|alt=Diagram of randomised complexity classes|thumb|upright=1.25|PP in relation to other probabilistic complexity classes ([[ZPP (complexity)|ZPP]], [[RP (complexity)|RP]], co-RP, [[BPP (complexity)|BPP]], [[BQP]]), which generalise [[P (complexity)|P]] within [[PSPACE]]. It is unknown if any of these inclusions are strict.]]

In [[Computational complexity theory|complexity theory]], '''PP''', or '''PPT''' is the class of [[decision problem]]s 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.<ref name=gill>{{Cite journal | doi=10.1137/0206049|title = Computational Complexity of Probabilistic Turing Machines| journal=SIAM Journal on Computing| volume=6| issue=4| pages=675–695|year = 1977|last1 = Gill|first1 = John}}</ref>

If a decision problem is in '''PP''', then there is an algorithm running in polynomial time that is allowed to make random decisions, such that it returns the correct answer with chance higher than 1/2. In more practical terms, it is the class of problems that can be solved to any fixed degree of accuracy by running a randomized, polynomial-time algorithm a sufficient (but bounded) number of times.

Turing machines that are polynomially-bound and probabilistic are characterized as '''PPT''', which stands for probabilistic polynomial-time machines.<ref>{{cite book|last1=Lindell|first1=Yehuda |last2=Katz|first2=Jonathan|date=2015|title=Introduction to Modern Cryptography|publisher=Chapman and Hall/CRC|isbn=978-1-4665-7027-6|language=en|edition=2|page=46}}</ref> This characterization of Turing machines does not require a bounded error probability. Hence, '''PP''' is the complexity class containing all problems solvable by a PPT machine with an error probability of less than 1/2.

An alternative characterization of '''PP''' is the set of problems that can be solved by a [[nondeterministic Turing machine]] in polynomial time where the acceptance condition is that a majority (more than half) of computation paths accept. Because of this some authors have suggested the alternative name ''Majority-P''.<ref>Lance Fortnow. Computational Complexity: Wednesday, September 4, 2002: Complexity Class of the Week: PP. http://weblog.fortnow.com/2002/09/complexity-class-of-week-pp.html</ref>

==Definition==
A language ''L'' is in '''PP''' if and only if there exists a probabilistic Turing machine ''M'', such that
* ''M'' runs for polynomial time on all inputs
* For all ''x'' in ''L'', ''M'' outputs 1 with probability no less than 1/2
* For all ''x'' not in ''L'', ''M'' outputs 1 with probability strictly less than 1/2.

Alternatively, '''PP''' can be defined using only deterministic Turing machines. A language ''L'' is in '''PP''' if and only if there exists a polynomial ''p'' and deterministic Turing machine ''M'', such that
* ''M'' runs for polynomial time on all inputs
* For all ''x'' in ''L'', the fraction of strings ''y'' of length ''p''(|''x''|) which satisfy ''M(x,y)'' = 1 is greater than 1/2
* For all ''x'' not in ''L'', the fraction of strings ''y'' of length ''p''(|''x''|) which satisfy ''M(x,y)'' = 1 is less than 1/2.

In this definition, the string ''y'' corresponds to the output of the random coin flips that the probabilistic Turing machine would have made.

In both definitions, "less than" can be changed to "less than or equal to" (see below), and the threshold 1/2 can be replaced by any fixed rational number in (0,1), without changing the class.

==PP vs BPP==
'''[[Bounded-error probabilistic polynomial|BPP]]''' is a subset of '''PP'''; it can be seen as the subset for which there are efficient probabilistic algorithms. The distinction is in the error probability that is allowed: in '''BPP''', an algorithm must give correct answer (YES or NO) with probability exceeding some fixed constant c > 1/2, such as 2/3 or 501/1000. If this is the case, then we can run the algorithm a number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the [[Chernoff bound]]. This number of repeats increases if ''c'' becomes closer to 1/2, but it does not depend on the input size ''n''.

More generally, if ''c'' can depend on the input size <math>n</math> polynomially, as <math>c = O(n^{-k}) </math>, then we can rerun the algorithm for <math>O(n^{2k})</math> and take the majority vote. By [[Hoeffding's inequality]], this gives us a '''BPP''' algorithm.

The important thing is that this constant ''c'' is not allowed to depend on the input. On the other hand, a '''PP''' algorithm is permitted to do something like the following:
* On a YES instance, output YES with probability 1/2&nbsp;+&nbsp;1/2<sup>''n''</sup>, where ''n'' is the length of the input.
* On a NO  instance, output YES with probability 1/2&nbsp;−&nbsp;1/2<sup>''n''</sup>.

Because these two probabilities are ''exponentially'' close together, even if we run it for a ''polynomial'' number of times it is very difficult to tell whether we are operating on a YES instance or a NO instance. Attempting to achieve a fixed desired probability level using a majority vote and the Chernoff bound requires a number of repetitions that is exponential in ''n''.

==PP compared to other complexity classes==

'''PP''' includes '''BPP''', since probabilistic algorithms described in the definition of '''BPP''' form a subset of those in the definition of '''PP'''.

'''PP''' also includes '''[[NP (complexity class)|NP]]'''. To prove this, we show that the [[NP-complete]] [[Boolean satisfiability problem|satisfiability]] problem belongs to '''PP'''. Consider a probabilistic algorithm that, given a formula ''F''(''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>) chooses an assignment ''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> 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 <math>\frac12 - \frac1{2^{n + 1}}</math> and NO with probability <math>\frac12 + \frac1{2^{n + 1}}</math>.

If the formula is unsatisfiable, the algorithm will always output YES with probability <math>\frac12 - \frac1{2^{n + 1}} < \frac12</math>. If there exists a satisfying assignment, it will output YES with probability at least
<math>\left(\frac12 - \frac1{2^{n + 1}}\right)\cdot \left(1 - \frac1{2^n}\right) + 1\cdot\frac1{2^n} = \frac12 + \frac1{2^{2n + 1}} > \frac12</math>
(exactly 1/2 if it picked an unsatisfying assignment and 1 if it picked a satisfying assignment, averaging to some number greater than 1/2). Thus, this algorithm puts satisfiability in '''PP'''. As '''SAT''' is NP-complete, and we can prefix any deterministic [[polynomial-time many-one reduction]] onto the '''PP''' algorithm, '''NP''' is included in '''PP'''. Because '''PP''' is closed under complement, it also includes '''co-NP'''.

Furthermore, '''PP''' includes '''[[Arthur–Merlin protocol|MA]]''',<ref>[http://lpcs.math.msu.su/~ver/papers/am-pp.ps N.K. Vereshchagin, "On the Power of PP"]</ref> which subsumes the previous two inclusions.

'''PP''' also includes '''[[BQP]]''', the class of decision problems solvable by efficient polynomial time [[quantum computer]]s. In fact, BQP is [[low (complexity)|low]] for '''PP''', meaning that a '''PP''' machine achieves no benefit from being able to solve '''BQP''' problems instantly. The class of polynomial time on quantum computers with [[postselection]], '''[[PostBQP]]''', is equal to '''PP'''<ref name="PostBQP = PP">{{cite journal|last=Aaronson|first=Scott|year=2005|title=Quantum computing, postselection, and probabilistic polynomial-time|journal=Proceedings of the Royal Society A|volume=461|issue=2063|pages=3473–3482|doi=10.1098/rspa.2005.1546|arxiv=quant-ph/0412187|bibcode=2005RSPSA.461.3473A|s2cid=1770389 }}</ref> (see [[#PostBQP]] below).

Furthermore, '''PP''' includes '''[[QMA]]''', which subsumes inclusions of '''MA''' and '''BQP'''.

A polynomial time Turing machine with a PP [[oracle machine|oracle]] ('''P<sup>PP</sup>''') can solve all problems in  '''[[PH (complexity)|PH]]''', the entire [[polynomial hierarchy]]. This result was shown by [[Seinosuke Toda]] in 1989 and is known as [[Toda's theorem]]. This is evidence of how hard it is to solve problems in '''PP'''. The class [[Sharp-P|#P]] is in some sense about as hard, since '''P'''<sup>'''#P'''</sup> = '''P<sup>PP</sup>''' and therefore '''P'''<sup>'''#P'''</sup> includes '''PH''' as well.<ref>{{cite journal
 | last = Toda | first = Seinosuke | author-link = Seinosuke Toda
 | doi = 10.1137/0220053
 | issue = 5
 | journal = [[SIAM Journal on Computing]]
 | mr = 1115655
 | pages = 865–877
 | title = PP is as hard as the polynomial-time hierarchy
 | volume = 20
 | year = 1991}}</ref>

'''PP''' strictly includes uniform '''[[TC0|TC<sup>0</sup>]]''', the class of constant-depth, unbounded-fan-in [[boolean circuit]]s with [[majority gate]]s that are uniform (generated by a polynomial-time algorithm).<ref>Allender 1996, as cited in Burtschick 1999</ref>

'''PP''' is included in '''[[PSPACE]]'''. This can be easily shown by exhibiting a polynomial-space algorithm for '''MAJSAT''', defined below; simply try all assignments and count the number of satisfying ones.

'''PP''' is not included in '''SIZE'''(n<sup>k</sup>) for any k, by [[Karp-Lipton theorem#Application to circuit lower bounds – Kannan's theorem|Kannan's theorem]].

==Complete problems and other properties==
Unlike '''BPP''', '''PP''' is a syntactic rather than semantic class. Any polynomial-time probabilistic machine recognizes some language in '''PP'''. In contrast, given a description of a polynomial-time probabilistic machine, it is undecidable in general to determine if it recognizes a language in '''BPP'''.

'''PP''' has natural complete problems, for example, '''MAJSAT'''.<ref name=gill/> '''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 ''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> make F true and NO otherwise.

{{Anchor|complement}}

===Proof that PP is closed under complement===
Let ''L'' be a language in '''PP'''. Let <math>L^c</math> denote the complement of ''L''. By the definition of '''PP''' there is a polynomial-time probabilistic algorithm ''A'' with the property that 

:<math>x \in L \Rightarrow \Pr[A \text{ accepts } x] > \frac{1}{2} \quad \text{and} \quad x \not\in L \Rightarrow \Pr[A \text{ accepts } x] \le \frac{1}{2}.</math>

We claim that [[without loss of generality]], the latter inequality is always strict; the theorem can be deduced from this claim: let <math>A^c</math> denote the machine which is the same as ''A'' except that <math>A^c</math> accepts when ''A'' would reject, and vice versa. Then 

:<math>x \in L^c \Rightarrow \Pr[A^c \text{ accepts } x] > \frac{1}{2} \quad \text{and} \quad x \not\in L^c \Rightarrow \Pr[A^c \text{ accepts } x] <  \frac{1}{2},</math>

which implies that <math>L^c</math> is in '''PP'''.

Now we justify our without loss of generality assumption. Let <math>f(|x|)</math> be the polynomial upper bound on the running time of ''A'' on input ''x''. Thus ''A'' makes at most <math>f(|x|)</math> random coin flips during its execution. In particular the probability of acceptance is an integer multiple of <math>2^{-f(|x|)}</math> and we have:

:<math>x \in L \Rightarrow \Pr[A \text{ accepts } x] \ge \frac{1}{2} + \frac{1}{2^{f(|x|)}}.</math> 

Define a machine ''A''′ as follows: on input ''x'', ''A''′ runs ''A'' as a subroutine, and rejects if ''A'' would reject; otherwise, if ''A'' would accept, ''A''′ flips <math>f(|x|)+1</math> coins and rejects if they are all heads, and accepts otherwise. Then 

:<math>x \not\in L \Rightarrow \Pr[A' \text{ accepts } x] \le \frac{1}{2} \cdot \left (1- \frac{1}{2^{f(|x|)+1}} \right ) < \frac{1}{2}</math>
and
:<math> x \in L \Rightarrow \Pr[A' \text{ accepts } x] \ge \left (\frac{1}{2}+\frac{1}{2^{f(|x|)}} \right )\cdot \left ( 1-\frac{1}{2^{f(|x|)+1}} \right ) > \frac{1}{2}.</math>

This justifies the assumption (since ''A''′ is still a polynomial-time probabilistic algorithm) and completes the proof.

David Russo proved in his 1985 Ph.D. thesis<ref>{{cite thesis| title = Structural Properties of Complexity Classes| type = Ph.D Thesis| publisher=University of California, Santa Barbara | year = 1985 | author = David Russo}}</ref> that '''PP''' is closed under [[symmetric difference]]. It was an [[open problem]] for 14 years whether '''PP''' was closed under [[union (set theory)|union]] and [[Intersection (set theory)|intersection]]; this was settled in the affirmative by Beigel, Reingold, and Spielman.<ref>R. Beigel, N. Reingold, and D. A. Spielman, "[http://citeseer.ist.psu.edu/152946.html PP is closed under intersection]", ''Proceedings of ACM Symposium on Theory of Computing 1991'', pp. 1–9, 1991.</ref> Alternate proofs were later given by Li<ref>{{cite thesis| title = On the Counting Functions| type= Ph.D Thesis  | publisher=University of Chicago| year = 1993| author = Lide Li}}</ref> and Aaronson (see [[#PostBQP]] below).

==Other equivalent complexity classes==

===PostBQP===
The quantum complexity class '''[[BQP]]''' is the class of problems solvable in [[polynomial time]] on a [[quantum Turing machine]]. By adding [[postselection]], a larger class called '''[[PostBQP]]''' is obtained. Informally, postselection gives the computer the following power: whenever some event (such as measuring a qubit in a certain state) has nonzero probability, you are allowed to assume that it takes place.<ref>{{cite web|url=http://www.scottaaronson.com/talks/postselect.ppt|title=The Amazing Power of Postselection|last=Aaronson|first=Scott|access-date=2009-07-27}}</ref> [[Scott Aaronson]] showed in 2004 that '''PostBQP''' is equal to '''PP'''.<ref name="PostBQP = PP"/><ref>{{cite web|url=http://weblog.fortnow.com/2004/01/complexity-class-of-week-pp-by-guest.html|title=Complexity Class of the Week: PP|last=Aaronson|first=Scott|date=2004-01-11|work=Computational Complexity Weblog|access-date=2008-05-02}}</ref> This reformulation of '''PP''' makes it easier to show certain results, such as that '''PP''' is closed under intersection (and hence, under union), that '''BQP''' is [[low (complexity)|low]] for '''PP''', and that '''[[QMA]]''' is included in '''PP'''.

===PQP===
'''PP''' is also equal to another quantum complexity class known as '''PQP''', which is the unbounded error analog of '''BQP'''. It denotes the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of less than 1/2 for all instances. Even if all amplitudes used for '''PQP'''-computation are drawn from algebraic numbers, still '''PQP''' coincides with '''PP'''.<ref>{{cite journal|last1=Yamakami|first1=Tomoyuki|title=Analysis of Quantum Functions|journal=Int. J. Found. Comput. Sci.|volume=14|issue=5|pages=815–852|bibcode=1999quant.ph..9012Y|year=1999|doi=10.1142/S0129054103002047 |arxiv=quant-ph/9909012|s2cid=3265603 }}</ref>

==Notes==
{{reflist}}

==References==
*{{Cite book|first=C.|last=Papadimitriou|author-link=Christos Papadimitriou|title=Computational Complexity|chapter=chapter 11|publisher=Addison-Wesley|year=1994}}.
*{{Cite book|first=E.|last=Allender|contribution=A note on uniform circuit lower bounds for the counting hierarchy|title=Proceedings 2nd International Computing and Combinatorics Conference (COCOON)|volume=1090|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|pages=127–135|year=1996}}.
*{{cite journal
 | last1 = Burtschick | first1 = Hans-Jörg
 | last2 = Vollmer | first2 = Heribert
 | doi = 10.1142/S0129054198000180
 | id={{ECCC|1996|96|005}}
 | issue = 3
 | journal = Int. J. Found. Comput. Sci.
 | pages = 277–294
 | title = Lindström quantifiers and leaf language definability
 | volume = 9
 | year = 1998}}

==External links==
* {{CZoo|PP|P#pp}}

{{ComplexityClasses}}

[[Category:Probabilistic complexity classes]]
[[Category:Quantum complexity theory]]