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6.4 Probabilistic Polynomial Time

Probabilistic Time Complexity

Probabilistic Complexity ClassesRelationships between Probabilistic and Nonprobabilistic Complexity Classes

As in the case of deterministic and nondeterministic Turing transducers, each move of aprobabilistic Turing transducer is assumed to take one unit of time. The time that a computation

takes is assumed to be equal to the number of moves made during the computation. Thespace

the computation takes is assumed to equal the number of locations visited in the auxiliary work

tape, which has the maximal such number.

Probabilistic Time Complexity

A probabilistic Turing transducer M is said to be T(n) time-bounded, or oftime complexity T(n),if M halts within T(n) time in each computation on each input of length n. If T(n) is apolynomial, then M is also said to bepolynomially time-bounded, or to havepolynomial timecomplexity.

M is said to be T(n) expected time-bounded, or ofexpected time complexity T(n), if for each

input x of M the function T(n) satisfies

If T(n) is a polynomial, then M is said to bepolynomially expected time-bounded, or ofpolynomially expected time complexity.

Arguments similar to those given for Church's Thesis in Section 4.1, and for the sequential

computation thesis in Section 5.1, also apply for the following thesis.

The Probabilistic Computation Thesis A function that is computable mechanically with the

aid of probabilistic choices can also be computed by a probabilistic Turing transducer ofpolynomially related time complexity and polynomially related, expected time complexity.

Probabilistic Complexity Classes

The tractability of problems with respect to probabilistic time is determined by the existence of

bounded-error probabilistic Turing transducers of polynomial time complexity for solving theproblems. In light of this observation, the following classes of language recognition problems are

of interest here.

BPP-- the class of membership problems for the languages in

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{ L | L is a language accepted by a bounded-error probabilistic Turing machine of

polynomial time complexity }.

RP-- the class of membership problems for the languages in

{ L | L is a language accepted by a polynomially time-bounded, probabilistic Turingmachine M, which satisfies the following two conditions for some constant < 1.

a. On input x from L, M has an accepting computation with probability 1 - e(x) 1- .

b. On input x not from L, M has only nonaccepting computations. }ZPP-- the class of membership problems for the languages in

{ L | L is a language accepted by a probabilistic Turing machine, which has zero error

probability and polynomially expected time complexity. }

Relationships between Probabilistic and Nonprobabilistic

Complexity Classes

The relationship between the different classes of problems, as well as their relationship to the

classes studied in Chapter 5, is illustrated in Figure6.4.1.

Figure 6.4.1A hierarchy of some classes of problems.

None of the inclusions is known to be proper. The relationship is proved below.

Theorem 6.4.1 BPPis included inPSPACE.

Proof Consider any problem K inBPP. Let L denote the language that K induces. By the

definition ofBPPthere exists a bounded-error, polynomially time-bounded, probabilistic Turingmachine M1 that accepts L. Let < 1/2 be a constant that bounds the error probability of M1, and

let p(n) be the time complexity of M1.

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With no loss of generality it is assumed that M1 has a constant k, such that in each probabilistic

move, M1 has exactly k options. (Any probabilistic Turing machine can be modified to have such

a property, with k being the least common multiple of the number of options in the differentmoves of the Turing machine.) In addition, it is assumed that M1 has some polynomial q(n), such

that in each computation on each input x it makes exactly q(|x|) probabilistic moves.

Consequently, M1 on each input x has exactly k

q(|x|)

possible computations, with eachcomputation having an equal probability of occurring.

From M1, a deterministic Turing machine M2 can be constructed to accept the language L. M2relies on the following two properties of M1.

a. If x is in L, then M1 has at least probability 1- > 1/2 of having an accepting computationon input x.

b. If x is not in L, then M1 has at least probability 1 - > 1/2 of having a nonacceptingcomputation on input x.

On a given input x, M2 determines which of the above properties holds, and accordingly decideswhether to accept or reject the input.

Given an input x, the Turing machine M2 starts its computation by computing p(|x|). Then one at

a time, M2 lists all the sequences of transition rules of M1 whose lengths are at most p(|x|). Foreach such sequence, M2 checks whether the sequence corresponds to a computation of M1. M2determines whether each computation of M1 is accepting or rejecting. In addition, M2 counts the

number ma of accepting computations, and the number mrof nonaccepting computations.

M2 accepts the input x if it determines that the probability ma/(ma + mr) of M1 accepting x is

greater than 1/2, that is, if ma > mr. M2 rejects x if it determines that the probability mr/(ma + mr)

of M1 rejecting x is greater than 1/2, that is, if mr> ma.

The nonprimality problem is an example of a problem in the classRP(see Example6.2.1). For

RPthe following result holds.

Theorem 6.4.2 RPis inBPP NP.

Proof Consider any problem K inRP. Let L be the language that K induces. By the definition

ofRP, it follows that there exist a constant < 1, and a polynomially time-bounded Turingmachine M1, that satisfy the following conditions.

a.

If x is in L, then M1 has a probability 1 - > 0 of having an accepting computation on x.b. If x is not in L, then M1 has only nonaccepting computations on x.L is accepted by a nondeterministic Turing machine M2 similar to M1 and of identical timecomplexity. The only difference is that M2 considers each probabilistic move of M1 as

nondeterministic. Consequently,RPis inNP.

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M1 can also be simulated by a bounded-error probabilistic Turing machine M3 of similar time

complexity. Specifically, let k be any constant such thatk

< 1/2. Then M3 simulates k

computations of M1 on a given input x. M3 accepts x if M1 accepts x in any of the simulatedcomputations. Otherwise, M3 rejects x. It follows thatRPis also inBPP.

Finally, forZPPthe following result is shown.

Theorem 6.4.3 ZPPis contained inRP.

Proof Consider any probabilistic Turing machine M1 that has 0 error probability. Let (n)

denote the expected time complexity of M1. Assume that (n) is some polynomial in n.

From M1, a probabilistic Turing machine M2 of the following form can be constructed. Given an

input x, the probabilistic Turing machine M2 starts its computation by evaluating (|x|). Then

M2 simulates c (|x|) moves of M1 on input x for some constant c > 1. M2 halts in an acceptingstate if during the simulation it reaches an accepting state of M1. Otherwise, M2 halts in a

nonaccepting state.

By construction M2 has no accepting computation on input x if x is not in L(M1). On the other

hand, if x is in L(M1), then M2 halts in a nonaccepting state, with probability equal to that of M1

having an accepting computation that requires more than c (|x|) moves. That is, the error

probability e(x) is equal to i=c (|x|)+1 pi, where pi denotes the probability that, on x, M1 will

have a computation that takes exactly i steps.

Now

(|x|) (x)

= p0 0 + p1 1 + + pc (|x|) c (|x|) +

p0 0 + p1 1 + + pc (|x|) c (|x|) + i=c (|x|)+1 p i

= p0 0 + p1 1 + + pc (|x|) c (|x|) + (c (|x|) + 1)e(x)

= e(x) +

Consequently, M2 accepts x with probability

1 - e(x)

1 -

1 - 1/c

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6.4 Probabilistic Polynomial Time

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Probabilistic Time Complexity

Probabilistic Complexity Classes

Relationships between Probabilistic and Nonprobabilistic Complexity Classes

As in the case of deterministic and nondeterministic Turing transducers, each move of a

probabilistic Turing transducer is assumed to take one unit of time. The time that a computationtakes is assumed to be equal to the number of moves made during the computation. Thespace

the computation takes is assumed to equal the number of locations visited in the auxiliary work

tape, which has the maximal such number.

Probabilistic Time Complexity

A probabilistic Turing transducer M is said to be T(n) time-bounded, or oftime complexity T(n),

if M halts within T(n) time in each computation on each input of length n. If T(n) is a

polynomial, then M is also said to bepolynomially time-bounded, or to havepolynomial time

complexity.

M is said to be T(n) expected time-bounded, or ofexpected time complexity T(n), if for eachinput x of M the function T(n) satisfies

If T(n) is a polynomial, then M is said to bepolynomially expected time-bounded, or of

polynomially expected time complexity.

Arguments similar to those given for Church's Thesis in Section 4.1, and for the sequential

computation thesis in Section 5.1, also apply for the following thesis.

The Probabilistic Computation Thesis A function that is computable mechanically with the

aid of probabilistic choices can also be computed by a probabilistic Turing transducer of

polynomially related time complexity and polynomially related, expected time complexity.

Probabilistic Complexity Classes

The tractability of problems with respect to probabilistic time is determined by the existence of

bounded-error probabilistic Turing transducers of polynomial time complexity for solving the

problems. In light of this observation, the following classes of language recognition problems are

of interest here.

BPP-- the class of membership problems for the languages in

{ L | L is a language accepted by a bounded-error probabilistic Turing machine ofpolynomial time complexity }.

RP-- the class of membership problems for the languages in

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{ L | L is a language accepted by a polynomially time-bounded, probabilistic Turing

machine M, which satisfies the following two conditions for some constant < 1.

a. On input x from L, M has an accepting computation with probability 1 - e(x) 1- .

b.

On input x not from L, M has only nonaccepting computations. }

ZPP-- the class of membership problems for the languages in

{ L | L is a language accepted by a probabilistic Turing machine, which has zero errorprobability and polynomially expected time complexity. }

Relationships between Probabilistic and Nonprobabilistic

Complexity Classes

The relationship between the different classes of problems, as well as their relationship to the

classes studied in Chapter 5, is illustrated in Figure6.4.1.

Figure 6.4.1A hierarchy of some classes of problems.

None of the inclusions is known to be proper. The relationship is proved below.

Theorem 6.4.1 BPPis included inPSPACE.

Proof Consider any problem K inBPP. Let L denote the language that K induces. By the

definition ofBPPthere exists a bounded-error, polynomially time-bounded, probabilistic Turingmachine M1 that accepts L. Let < 1/2 be a constant that bounds the error probability of M1, and

let p(n) be the time complexity of M1.

With no loss of generality it is assumed that M1 has a constant k, such that in each probabilistic

move, M1 has exactly k options. (Any probabilistic Turing machine can be modified to have such

a property, with k being the least common multiple of the number of options in the differentmoves of the Turing machine.) In addition, it is assumed that M1 has some polynomial q(n), such

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that in each computation on each input x it makes exactly q(|x|) probabilistic moves.

Consequently, M1 on each input x has exactly kq(|x|)

possible computations, with each

computation having an equal probability of occurring.

From M1, a deterministic Turing machine M2 can be constructed to accept the language L. M2

relies on the following two properties of M1.

a. If x is in L, then M1 has at least probability 1- > 1/2 of having an accepting computationon input x.

b. If x is not in L, then M1 has at least probability 1 - > 1/2 of having a nonacceptingcomputation on input x.

On a given input x, M2 determines which of the above properties holds, and accordingly decides

whether to accept or reject the input.

Given an input x, the Turing machine M2 starts its computation by computing p(|x|). Then one at

a time, M2 lists all the sequences of transition rules of M1 whose lengths are at most p(|x|). Foreach such sequence, M2 checks whether the sequence corresponds to a computation of M1. M2determines whether each computation of M1 is accepting or rejecting. In addition, M2 counts the

number ma of accepting computations, and the number mrof nonaccepting computations.

M2 accepts the input x if it determines that the probability ma/(ma + mr) of M1 accepting x is

greater than 1/2, that is, if ma > mr. M2 rejects x if it determines that the probability mr/(ma + mr)

of M1 rejecting x is greater than 1/2, that is, if mr> ma.

The nonprimality problem is an example of a problem in the classRP(see Example6.2.1). For

RPthe following result holds.

Theorem 6.4.2 RPis inBPP NP.

Proof Consider any problem K inRP. Let L be the language that K induces. By the definition

ofRP, it follows that there exist a constant < 1, and a polynomially time-bounded Turing

machine M1, that satisfy the following conditions.

a. If x is in L, then M1 has a probability 1 - > 0 of having an accepting computation on x.b. If x is not in L, then M1 has only nonaccepting computations on x.

L is accepted by a nondeterministic Turing machine M2 similar to M1 and of identical time

complexity. The only difference is that M2 considers each probabilistic move of M1 asnondeterministic. Consequently,RPis inNP.

M1 can also be simulated by a bounded-error probabilistic Turing machine M3 of similar time

complexity. Specifically, let k be any constant such thatk

< 1/2. Then M3 simulates k

computations of M1 on a given input x. M3 accepts x if M1 accepts x in any of the simulatedcomputations. Otherwise, M3 rejects x. It follows thatRPis also inBPP.

http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sixse2.html#70002-3001t6.2.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sixse2.html#70002-3001t6.2.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sixse2.html#70002-3001t6.2.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sixse2.html#70002-3001t6.2.1

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Finally, forZPPthe following result is shown.

Theorem 6.4.3 ZPPis contained inRP.

Proof Consider any probabilistic Turing machine M1 that has 0 error probability. Let (n)

denote the expected time complexity of M1. Assume that (n) is some polynomial in n.

From M1, a probabilistic Turing machine M2 of the following form can be constructed. Given aninput x, the probabilistic Turing machine M2 starts its computation by evaluating (|x|). Then

M2 simulates c (|x|) moves of M1 on input x for some constant c > 1. M2 halts in an accepting

state if during the simulation it reaches an accepting state of M1. Otherwise, M2 halts in anonaccepting state.

By construction M2 has no accepting computation on input x if x is not in L(M1). On the other

hand, if x is in L(M1), then M2 halts in a nonaccepting state, with probability equal to that of M1having an accepting computation that requires more than c (|x|) moves. That is, the error

probability e(x) is equal to i=c (|x|)+1 pi, where pi denotes the probability that, on x, M1 willhave a computation that takes exactly i steps.

Now

(|x|) (x)

= p0 0 + p1 1 + + pc (|x|) c (|x|) +

p0 0 + p1 1 + + pc (|x|) c (|x|) + i=c (|x|)+1 p i

= p0 0 + p1 1 + + pc (|x|) c (|x|) + (c (|x|) + 1)e(x)

= e(x) +Consequently, M2 accepts x with probability

1 - e(x)

1 -

1 - 1/c

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