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PROBABILISTIC TURING
MACHINES
Stephany Coffman-Wolph
Wednesday, March 28, 2007
PROBABILISTIC TURING MACHINE

There are several popular definitions:
A nondeterministic Turing Machine (TM) which
randomly chooses between available transitions at
each point according to some probability distribution
 A type of nondeterministic TM where each
nondeterministic step is called a coin-flip step and
has two legal next moves
 A Turing Machine in which some transitions are
random choices among finitely many alternatives


Also known as a Randomized Turing Machine
TM SPECIFICS

There are (at least) three tapes
1st Tape holds the input
 2nd Tape (also known as the random tape) is covered
randomly (and independently) with 0’s and 1’s

½ probability of a 0
 ½ probability of a 1


3rd Tape is used as the scratch tape
WHEN A PROBABILISTIC TM RECOGNIZES
A LANGUAGE
Accept all strings in the language
 Reject all strings not in the language


However, a probabilistic TM will have a
probability of error
PROBABILISTIC TM FACTS
Each “branch” in the TMs computation has a
probability
 Can have stochastic results
 Hence, on a given input it:

May have different run times
 May not halt

Therefore, it may accept the input in a given
execution, but reject in another execution
 Time and space complexity can be measured
using the worst case computation branch

PROBABILISTIC ALGORITHM
Also known as Randomized Algorithms
 An algorithm designed to use the outcome of a
random process
 In other words, part of the logic for the algorithm
uses randomness
 Often the algorithm has access to a pseudorandom number generator
 The algorithm uses random bits to help make
choices (in hope of getting better performance)

WHY USE PROBABILISTIC ALGORITHMS?

Probabilistic algorithms are useful because
It is time consuming to calculate the “best” answer
 Estimation could introduce an unwanted bias that
invalidates the results


For example: Random Sampling
Random sampling is used to obtain information about
individuals in a large population
 Asking everyone would take too long
 Querying a not random selected subset might
influence (or bias) the results

BPP
Bounded error Probability in Polynomial time
 Definition:



The class of languages that are recognized by
probabilistic polynomial time TM with an error
probability of 1/3 (or less)
Or another way say it:

The class of languages that a probabilistic TM halts
in polynomial time with either a accept or reject
answer at least 2/3 of the time
A PROBLEM IN BPP:
Can be solved by an algorithm that is allowed to
make random decisions (called coin-flips)
 Guaranteed to run in polynomial time
 On a given run of the algorithm, it has a (at
most) 1/3 probability of giving an incorrect
answer
 These algorithms are known as probabilistic
algorithms

WHY 1/3?
Actually, this is arbitrary
 In fact can be any constant between 0 and ½ (as
long as it is independent of the input)
 Why?

If the algorithm is run many times, the probability of
the probabilistic TM being wrong the majority of the
time decreases exponentially
 Therefore, these kinds of algorithms can become
more accurate by running it several times (and then
taking the majority vote of the results)

TO ILLUSTRATE THE CONCEPT
Let the error probability be 1/3
 We have a box containing many red and blue
balls

2/3 of the balls are one color
 1/3 of the balls are the other color
 (But, we don’t know which color is 2/3 or which color
is 1/3)

To find out, we start taking samples at random
and keep track of which color ball we pulled from
the box
 The color that comes up most frequently during a
large sampling will most likely be the majority
color originally in the box

HOW THIS RELATES…
The blue and red balls correspond to branches in
a probabilistic (polynomial time) TM. Lets call it
M1
 We can assign each color:

Red = accepting
 Blue = rejecting

The sampling can be done by running M1 using
another probabilistic TM (lets call it M2) with a
better error probability
 M2’s error probability is exponentially small if it
runs M1 a polynomial number of times and
outputs the result that occurs most often

FORMALLY:
Let the error probability (Є) be a fixed constant
strictly between 0 and ½
 Let poly(n) be any polynomial
 For any poly(n), a probabilistic polynomial time
TM M1 that operates with error probability Є has
an equivalent probabilistic polynomial time TM
M2
 This TM M2 has an error probability of 2-poly(n)

RP
Randomized Polynomial time
 A class of problems that will run in polynomial
time on a probabilistic TM with the following
properties: If the correct answer is

no, always return no
 yes, return yes with probability at least ½



Otherwise, returns no
Formally

The class of languages for which membership can be
determined in polynomial time by a probabilistic TM
with no false acceptances and less than half of the
rejections are false rejections
FACTS ABOUT RP
If the algorithm returns a yes answer, then yes is
the correct answer
 If the algorithm returns a no answer, then it may
or may not be correct
 The ½ in the definition is arbitrary



Like we saw in the BPP class, running the algorithm
addition repetitions will decrease the chance of the
algorithm giving the wrong answer
Often referred to as a Monte-Carlo Algorithm (or
Monte-Carlo Turing Machine)
MONTE CARLO ALGORITHM
A numerical Monte Carlo method used to find
solutions to problems that cannot easily to solved
using standard numerical methods
 Often relies on random (or pseudo-random)
numbers
 Is stochastic or nondeterministic in some manner

CO-RP

A class of problems that will run in polynomial
time on a probabilistic TM with the following
properties: If the correct answer is
yes, always return yes
 no, return no with probability at least ½



Otherwise, returns a yes
In other words:
If the algorithm returns a no answer, then no is the
correct answer
 If the algorithm returns a yes answer, then it may or
may not be correct

ZPP
Zero-error Probabilistic Polynomial
 The class of languages for which a probabilistic
TM halts in polynomial time with no false
acceptances or rejections, but sometimes gives an
“I don’t know” answer
 In other words:

It always returns a guaranteed correct yes or no
answer
 It might return an “I don’t know” answer

FACTS ABOUT ZPP

The running time is unbounded
But it is polynomial on average (for any input)
 It is expected to halt in polynomial time


Similar to definition of P except:
ZPP allows the TM to have “randomness”
 The expected running time is measured (instead of
the worst-case)


Often referred to as a Las-Vegas algorithm (or
Las-Vegas Turning Machine)
LAS VEGAS ALGORITHM
A randomized algorithm that never gives an
incorrect result. It either produces a result or
fails
 Therefore, it is said that the algorithm “does not
gamble” with it’s result. It only “gambles” with
the resources used for computation

L, ¬ L, AND ZPP

If L is in ZPP, then ¬ L is in ZPP


Where ¬ L represents the complement of L
Why?
If L is accepted by TM M that is in ZPP.
 We can alter M to accept ¬ L by

Turning the acceptance by M into halting without
acceptance
 If M halted without accepting before, instead we accept and
halt

RELATIONSHIP BETWEEN RP AND ZPP


ZPP = RP  co-RP
Proof Part 1: RP  co-RP is in ZPP






Let L be a language recognized by RP algorithm A and coRP algorithm B
Let w be in L
Run w on A. If A returns yes, the answer must be yes. If A
returns no, run w on B. If B returns no, then the answer
must be no. Otherwise, repeat.
Only one of the algorithms can ever give a wrong answer.
The chance of an algorithm giving the wrong answer is
50%.
The chance of having the kth repetition shrinks
exponentially. Therefore, the expected running time is
polynomial
Hence, RP intersect co-RP is contained in ZPP
RELATIONSHIP BETWEEN RP AND ZPP


ZPP = RP  co-RP
Proof Part 2: ZPP is contained in RP  co-RP
Let C be an algorithm in ZPP
 Construct the RP algorithm using C:




Run C for (at least) double its expected running time.
If it gives an answer, that must be the answer
If it doesn’t given an answer before the algorithm stops, then
the answer is no
The chance that algorithm C produces an answer before it
is stopped is ½ (and hence fitting the definition of an RP
algorithm)
 The co-RP algorithm is almost identical, but it gives a yes
answer if C does produce an answer.
 Therefore, we can conclude that ZPP is contained in RP 
co-RP

WHAT WE CAN ALSO CONCLUDE

As seen in the proof of ZPP = RP  co-RP we can
conclude that
ZPP  RP
 ZPP  co-RP

RELATIONSHIP BETWEEN P AND ZPP
P  ZPP
 Proof


Any deterministic, polynomial time bounded TM is
also a probabilistic TM that ignores its special
feature that allows it to make random choices
RELATIONSHIP BETWEEN NP AND RP
RP  NP
 Proof

Let M1 be a probabilistic TM in RP for language L
 Construct a nondeterministic TM M2 for L


Both of these TMs are bounded by the same polynomial
When M1 examines a random bit for the first time,
M2 chooses both possible values for the bit and writes
it on a tape
 M2 will accept whenever M1 accepts. M2 will not
accept otherwise

RELATIONSHIP BETWEEN NP AND RP

Proof continued
Let w be in L
 M1 has a 50% probability of accepting w.




There must be some sequence of bits on the random tape
that leads to the acceptance of w
M2 will choose that sequence of bits and accepts
when the choice is made. Thus, w is in the language
of M2
If w is not in L, then there is no sequence of random
bits that will make M1 accept. Therefore, M2 cannot
choose a sequence of bits that leads to acceptance.
Thus, w is not in the language of M2
DIAGRAM SHOWING RELATIONSHIP OF
PROBLEM CLASSES
NP
RP
P
ZPP
Co-RP
Co-NP
WHERE DOES BPP FIT IN?
It is still an open question whether NP is a
subset of BPP or BPP is a subset of NP
 However, it is believed that RP is a subset of BPP

DIAGRAM SHOWING RELATIONSHIP OF
PROBLEM CLASSES
BPP
RP
ZPP
P
WHY STUDY PROBABILISTIC TM?

To attempt to answer the question:
Does randomness add power?
 Or putting it another way:


Are there problems that can be solved by a probabilistic TM
(in polynomial time) but these same problems cannot be
solved by a deterministic TM in polynomial time?
RESOURCES





Introduction of the Theory of Computation by Michael
Sipser, PWS Publishing Company, 1997
Introduction to Automata Theory, Languages, and
Computation by Hopcroft, Motwani, and Ullman,
Person Education, Inc, 2006
Introduction to the Theory of Computation by Eitan
Gurari, Computer Science Press, 1989
(http://www.cse.ohio-state.edu/~gurari/theorybk/theory-bk.html)
“Dictionary of Algorithms and Data Structures”, NIST
website (http://www.nist.gove/dads/)
“Probabilistic Turing Machines”, “Randomized
algorithm”, “BPP”, “ZPP”, “CP”, “Monte Carlo
algorithm”, and “Las Vegas algorithm”, Wikipedia
website (http://en.wikipedia.org/)