Probabilistic Turing Machines

1. Probabilistic Turing Machines Stephany Coffman-Wolph Wednesday, March 28, 2007

2. 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

3. 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

4. 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

5. 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

6. 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 pseudo-random number generator • The algorithm uses random bits to help make choices (in hope of getting better performance)

7. 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

8. 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

9. 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

10. 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)

11. 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

12. 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

13. 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)

14. 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

15. 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)

16. 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

17. 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

18. 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

19. 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)

20. 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

21. 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

22. 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 co-RP 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

23. 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

24. What We Can Also Conclude • As seen in the proof of ZPP = RP  co-RP we can conclude that • ZPP  RP • ZPP  co-RP

25. 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

26. 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

27. 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

28. Diagram Showing Relationship of Problem Classes

29. 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

30. Diagram Showing Relationship of Problem Classes

31. 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?

32. 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 EitanGurari, Computer Science Press, 1989 (http://www.cse.ohio-state.edu/~gurari/theory-bk/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/)