PROBABILISTIC COMPUTATION By Remanth Dabbati

INDEX  Probabilistic Turing Machine  Probabilistic Complexity Classes  Probabilistic Algorithms

PROBABILISTIC TURING MACHINE It is a turing machine with ability to make decisions based on the unbiased coin tosses.  A probabilistic machine is defined by assigning the each input, the output which occurs with probability greater than the ½  If the probabilities are equal for transitions then it is deterministic turing machine with additional write instruction that gives 1 or 0 on the tape, simply it is a deterministic turing machine with added tape that contains full of random bits called Random Tape  A probabilistic turing machine is Non deterministic turing machine which randomly chooses between the available transitions according to the some probability distribution

PROBABILISTIC COMPLEXITY CLASSES Overall there are 16 complexity classes they are 1) AM - Arthur Merlin Protocol 2) AWPP - Almost Wide Probabilistic Protocol 3) BPL - Bounded error probabilistic logarithmic space 4) BPP - Bounded error probabilistic polynomial time 5) Co-RP 6) IP - Interactive Polynomial time 7) MA - Merlin Arthur protocol 8) PostBQP

9) PP - Probabilistic turing machine 10) QCMA – Quantum classical merlin Arthur 11) QIP – Quantum interactive polynomial time 12) QMA – Quantum merlin Arthur 13) RL - Randomized Logarithmic-space 14) RP - Randomized polynomial time 15) ZPP- zero-error probabilistic polynomial time 16) BQP - Bounded error quantum polynomial time

Arthur-Merlin protocol This class is set of decision problems that decides in polynomial with AM protocol with 2 messages Arthur tosses some random coins and sends the output of all coins to merlin, merlin replies with some purported proofs and then the Arthur verifies the proof Arthur is only allowed to send outcomes of coin tosses to Merlin, and in the final stage Arthur must decide whether to accept or reject using only his previously generated random coin flips and Merlin's message

Merlin-Arthur Protocol It is 1 message protocol Merlin sends Arthur a message, and then Arthur decides whether to accept or not by running a probabilistic polynomial time computation Merlin does not have access to Arthur's coin tosses in this protocol, since it is a single-message protocol and Arthur tosses his coins only after receiving Merlin's message.

 AM contains the problem of deciding if two graphs are not isomorphic. The protocol using private coins is the following and can be transformed to a public coin protocol. Given two graphs G and H, Arthur randomly chooses one of them, and chooses a random permutation of its vertices, presenting the permuted graph I to Merlin. Merlin has to answer if I was created from G or H. If the graphs are non isomorphic, Merlin will be able to answer with full certainty (by checking if I is isomorphic to G). However, if the graphs are isomorphic, it is both possible that G or H was used to create I, and equally likely. In this case, Merlin has no way to tell them apart and can convince Arthur with probability at most 1/2, and this can be amplified to 1/4 by repetition.

BQP -bounded error quantum polynomial time It 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  here the quantum computer uses quantum algorithm, that solves the decision problem with high probability and is guaranteed to run in polynomial time. On any given run of the algorithm, it has a probability of at most 1/3 that it will give the wrong answer.

QMA I t is the set of decision problems for which when the answer is YES- a polynomial-size quantum proof (a quantum state) which convinces a polynomial-time quantum verifier of the fact with high probability. The answer is NO, every polynomial-size quantum state is rejected by the verifier with high probability.

Probabilistic Algorithm Karger’s Algorithm : It is invented by david Karger  It is a randomized algorithm to compute a minimum cut of connected graph.  It is based on contraction of edge(u,v) in an undirected graph G. it simply merges the nodes u, v to uv as one.  Karger's basic algorithm iteratively contracts randomly chosen edges until only two nodes remain; those nodes represent a cut in the original graph

Example for the krager algorithm.

ATLANTIC CITY ALGORITHM Atlantic City algorithm is a probabilistic polynomial- time algorithm that answers correctly at least 75% of the time, in some versions its greater than 50%,some value. The two common classes of probabilistic classes are Monte Carlo algorithm : fast, probably correct Las Vegas algorithm : correct, probably fast But this algorithm is probably correct and probably fast.

FREIVALD’S Algorithm It is a probabilistic randomized algorithm used to verify matrix multiplication. When there are three matrices A, B, C of m×n. a general problem checking A×B = C Naïve’s algorithm would do the same process in O(n ) time where freivald’s algorithm reduces to O(n 2 ) time

REFERENCES   classes 

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