Quantum and probabilistic finite multitape automata

Ginta Garkaje and Rusins Freivalds

Riga, Latvia

First, we discuss the following 2-tape language

L1 = {(0n1m,2k)| n=m=k }

Theorem. The language L1 can be recognized with

arbitrary probability 1-ε by a probabilistic 2-tape finite automaton.

2n + 3m = 5k3n + 6m = 9k

2n + 9m = 11k

1 1 1 1 1 1 0 0 0 0 0

2 2 2 2 2 2 2

SOFSEM 2009

Theorem. There exists no quantum finite 2-tape automaton which recognizes the language L41 with bounded error.

For arbitrary positive ε, there exists a probabilistic finite 2-tape automaton recognizing the language L41 with a probability 1-ε.

There exists no probabilistic finite 2-tape automaton which recognizes language L42 with a bounded error.There exists a quantum finite 2-tape automaton recognizing the language L42 with a probability 1-ε.

zy,x,||z)y,{(xL42

SOFSEM 2009

kkksssmnmnmn

41 smn||)}2...2020,0102...20102010{(0L k21kk2211 }

are binary words and eitherx=y or y=z but not both of them.}

Theorem. For arbitrary r, there exists a quantum finite 2-tape automaton recognizing the language L43 with the probability 1.

For arbitrary r, there exists no quantum finite 2-tape automaton with 2 Ω(r/log r) states which recognizes the language L43 with abounded error.

For arbitrary r, and for arbitrary positive ε there exists a probabilistic finite 2-tape automaton with const. r states recognizing the language L43 with probability 1- ε.

them}ofboth not but zyor y either x and

r,zyx ds,binary wor are zy,x,||z)y,{(xL44

Theorem. For arbitrary r, there exists quantum finite 2-tape automaton with 2 Ω(r/log r) states which recognizes the language L44 with the probability 1.

For arbitrary r, there exists no probabilistic finite 2-tape automaton with 2 Ω(r/log r) states which recognizes the language L44 with the probability 1.

SOFSEM 2009

kkkssmnmn

43 smn&rk||)}2...20,0102...2010{(0L k1kk11 }