PP (complexity)
Encyclopedia
In complexity theory, PP 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.
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 less than or equal to 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.
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.
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
In both definitions, "less than or equal" can be changed to "less than" (see below), and the treshold 1/2 can be relaced by any fixed rational number in (0,1), without changing the class.
. This number of repeats increases if c becomes closer to 1/2, but it does not depend on the input size n.
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:
Because these two probabilities are so close together, especially for large n, even if we run it a large 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 also contains NP. To prove this, we show that the NP-complete
satisfiability
problem 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 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 contained in PP. Because PP is closed under complement, it also contains co-NP.
Furthermore, PP contains MA
, which subsumes previous two inclusions.
PP also contains BQP
, the class of decision problems solvable by efficient polynomial time quantum computer
s. In fact, BQP is 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 (see #Quantum computation reformulation below).
Futhermore, PP contains QMA
, which subsumes inclusions of MA and BQP.
A polynomial time Turing machine with a PP oracle
(PPP) can solve all problems in 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 #P
is in some sense about as hard, since P#P = PPPand therefore P#P contains PH as well.
PP strictly contains TC0
, the class of constant-depth, unbounded-fan-in uniform boolean circuit
s with majority gates. (Allender 1996, as cited in Burtschick 1999).
PP is contained 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 contained in SIZE(nk) for any k (proof).
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.
Proof that PP is closed under complement
Let L be a language in PP. Let 
denote the complement of L. By the definition of PP there is a polynomial-time probabilistic algorithm A with the property that 
and 
. We claim that without loss of generality
, the latter inequality is always strict; once we do this the theorem can be proved as follows. Let
denote the machine which is the same as A except that 
accepts when A would reject, and vice-versa. Then 
and 
, which implies that 
is in PP.
Now we justify our without loss of generality assumption. Let
be the polynomial upper bound on the running time of A on input x. Thus A makes at most 
random coin flips during its execution. In particular, 
since the probability of acceptance is an integer multiple of 
. 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 
coins and rejects if they are all heads, and accepts otherwise. Then
and
. 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 that PP is closed under symmetric difference
. It was an open problem
for 14 years whether PP was closed under union
and intersection
; this was settled in the affirmative by Beigel, Reingold, and Spielman. Alternate proofs were later given by Li and Aaronson (see #Quantum computation reformulation below).
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. Scott Aaronson
showed in 2004 that PostBQP is equal to PP. 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
for PP, and that QMA
is contained in PP.
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...
s solvable by a probabilistic Turing machine
Probabilistic Turing machine
In computability theory, a probabilistic Turing machine is a non-deterministic Turing machine which randomly chooses between the available transitions at each point according to some probability distribution....
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 less than or equal to 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.
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.
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 strictly greater than 1/2
- For all x not in L, M outputs 1 with probability less than or equal to 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 strictly greater than 1/2
- For all x in not in L, the fraction of strings y of length p(|x|) which satisfy M(x,y) = 1 is less than or equal to 1/2.
In both definitions, "less than or equal" can be changed to "less than" (see below), and the treshold 1/2 can be relaced by any fixed rational number in (0,1), without changing the class.
PP vs BPP
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 greater than 1/2, such as 2/3 or 501/1000. If this is the case, then we can run the algorithm a polynomial number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff boundChernoff bound
In probability theory, the Chernoff bound, named after Herman Chernoff, gives exponentially decreasing bounds on tail distributions of sums of independent random variables...
. This number of repeats increases if c becomes closer to 1/2, but it does not depend on the input size n.
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 + 1/2n, where n is the length of the input.
- On a NO instance, output YES with probability 1/2 - 1/2n.
Because these two probabilities are so close together, especially for large n, even if we run it a large 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 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 this, we show that the NP-complete
NP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
satisfiability
Boolean satisfiability problem
In computer science, satisfiability is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE...
problem 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 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 contained in PP. Because PP is closed under complement, it also contains co-NP.
Furthermore, PP contains MA
Arthur-Merlin protocol
In computational complexity theory, an Arthur–Merlin protocol is an interactive proof system in which the verifier's coin tosses are constrained to be public . This notion was introduced by...
, which subsumes previous two inclusions.
PP also contains BQP
BQP
In computational complexity theory BQP is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances...
, the class of decision problems solvable by efficient polynomial time quantum computer
Quantum computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...
s. In fact, BQP is low
Low (complexity)
In computational complexity theory, a complexity class B is said to be low for a complexity class A if AB = A; that is, A with an oracle for B is equal to A. Such a statement implies that an abstract machine which solves problems in A achieves no additional power if it is given the ability to solve...
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
Postselection
In probability theory, to postselect is to condition a probability space upon the occurrence of a given event. In symbols, once we postselect for an event E, the probability of some other event F changes from Pr[F] to the conditional probability Pr[F|E].For a discrete probability space, Pr[F|E] =...
, PostBQP
PostBQP
PostBQP is a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with postselection and bounded error ....
, is equal to PP (see #Quantum computation reformulation below).
Futhermore, PP contains QMA
QMA
In computational complexity theory, QMA, which stands for Quantum Merlin Arthur, is the quantum analog of the deterministic complexity class NP or the probabilistic complexity class MA...
, which subsumes inclusions of MA and BQP.
A polynomial time Turing machine with a PP oracle
Oracle machine
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any...
(PPP) can solve all problems in PH, the entire polynomial hierarchy
Polynomial hierarchy
In computational complexity theory, the polynomial hierarchy is a hierarchy of complexity classes that generalize the classes P, NP and co-NP to oracle machines...
. This result was shown by Seinosuke Toda
Seinosuke Toda
is a computer scientist working at the Nihon University in Tokyo. He was a recipient of the 1998 Gödel Prize for proving Toda's theorem.-External links:* at the Nihon University....
in 1989 and is known as Toda's theorem
Toda's theorem
Toda's theorem was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy" and was given the 1998 Gödel Prize. The theorem states that the entire polynomial hierarchy PH is contained in PPP; this implies a closely related statement, that PH is contained in P#P...
. This is evidence of how hard it is to solve problems in PP. The class #P
Sharp-P
In computational complexity theory, the complexity class #P is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ƒ," where ƒ is the number of accepting paths of a nondeterministic...
is in some sense about as hard, since P#P = PPPand therefore P#P contains PH as well.
PP strictly contains TC0
TC0
TC0 is a complexity class used in circuit complexity. It is the first class in the hierarchy of TC classes.TC0 contains all languages which are decided by Boolean circuits with constant depth and polynomial size, containing only unbounded-fanin AND gates, OR gates, and majority gates...
, the class of constant-depth, unbounded-fan-in uniform boolean circuit
Boolean circuit
A Boolean circuit is a mathematical model of computation used in studying computational complexity theory. Boolean circuits are the main object of study in circuit complexity and are a special kind of circuits; a formal language can be decided by a family of Boolean circuits, one circuit for each...
s with majority gates. (Allender 1996, as cited in Burtschick 1999).
PP is contained in PSPACE
PSPACE
In computational complexity theory, PSPACE is the set of all decision problems which can be solved by a Turing machine using a polynomial amount of space.- Formal definition :...
. 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 contained in SIZE(nk) for any k (proof).
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. 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.
Proof that PP is closed under complementComplement (set theory)In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
Let L be a language in PP. Let denote the complement of L. By the definition of PP there is a polynomial-time probabilistic algorithm A with the property that and . We claim that without loss of generalityWithout loss of generality
Without loss of generality is a frequently used expression in mathematics...
, the latter inequality is always strict; once we do this the theorem can be proved as follows. Let
denote the machine which is the same as A except that accepts when A would reject, and vice-versa. Then and , which implies that is in PP.Now we justify our without loss of generality assumption. Let
be the polynomial upper bound on the running time of A on input x. Thus A makes at most random coin flips during its execution. In particular, since the probability of acceptance is an integer multiple of . 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 coins and rejects if they are all heads, and accepts otherwise. Thenand
. 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 that PP is closed under symmetric difference
Symmetric difference
In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
. It was an open problem
Open problem
In science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved . Some questions remain unanswered for centuries before solutions are found...
for 14 years whether PP was closed under union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
and intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
; this was settled in the affirmative by Beigel, Reingold, and Spielman. Alternate proofs were later given by Li and Aaronson (see #Quantum computation reformulation below).
PostBQP
The quantum complexity class BQPBQP
In computational complexity theory BQP is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances...
is the class of problems solvable in polynomial time on a quantum Turing machine
Quantum Turing machine
A quantum Turing machine , also a universal quantum computer, is an abstract machine used to model the effect of a quantum computer. It provides a very simple model which captures all of the power of quantum computation. Any quantum algorithm can be expressed formally as a particular quantum...
. By adding postselection
Postselection
In probability theory, to postselect is to condition a probability space upon the occurrence of a given event. In symbols, once we postselect for an event E, the probability of some other event F changes from Pr[F] to the conditional probability Pr[F|E].For a discrete probability space, Pr[F|E] =...
, a larger class called PostBQP
PostBQP
PostBQP is a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with postselection and bounded error ....
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. Scott Aaronson
Scott Aaronson
Scott Joel Aaronson is a theoretical computer scientist and faculty member in the Electrical Engineering and Computer Science department at the Massachusetts Institute of Technology.-Education:...
showed in 2004 that PostBQP is equal to PP. 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
Low (complexity)
In computational complexity theory, a complexity class B is said to be low for a complexity class A if AB = A; that is, A with an oracle for B is equal to A. Such a statement implies that an abstract machine which solves problems in A achieves no additional power if it is given the ability to solve...
for PP, and that QMA
QMA
In computational complexity theory, QMA, which stands for Quantum Merlin Arthur, is the quantum analog of the deterministic complexity class NP or the probabilistic complexity class MA...
is contained in PP.