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Size of Quantum Finite State Transducers. Ruben Agadzanyan, Rusins Freivalds. Outline. Introduction Previous results When deterministic transducers are possible Quantum vs. probabilistic transducers. Introduction. Probabilistic transducer definition Computing relations - PowerPoint PPT Presentation
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Size of Quantum Finite State Transducers
Ruben Agadzanyan, Rusins Freivalds
Outline
Introduction Previous results When deterministic transducers
are possible Quantum vs. probabilistic
transducers
Introduction
Probabilistic transducer definition Computing relations Quantum transducer definition
Introduction Transducer definition
Finite state transducer (fst) is a tuple
T = (Q, Σ1, Σ2, V, f, q0, Qacc, Qrej),
V : Σ1 x Q → Q
a Σ1 :
nnnnn
n
n
n
pppp
pppp
pppp
pppp
...
...............
...
...
...
210
2222120
1121110
0020100
n210
n
2
1
0
q.....qqq
q
...
q
q
q
Introduction Transducer definition
R Σ1* x Σ2
*
R = {(0m1m,2m) : m ≥ 0} Σ1 = {0,1} Σ2 = {2} Input: #0m1m$ Output: 2m
Transducer may accept or reject input
Introduction Transducer types
Deterministic (dfst)
Probabilistic (pfst)
Quantum (qfst)
nnnnn
n
n
n
pppp
pppp
pppp
pppp
...
...............
...
...
...
210
2222120
1121110
0020100
n210
n
2
1
0
q.....qqq
q
...
q
q
q
0...100
...............
1...000
0...001
0...010
8/3...8/38/20
...............
1...000
0...4/304/1
2/1...02/10
2/12/1
2/12/1
Introduction Computing relations
R Σ1* x Σ2
*
R = {(0m1m,2m) : m ≥ 0}
For α > 1/2 we say that T computes the relation R with probability α if for all v, whenever (v, w) R, then T (w|v) ≥ α, and whenever (v, w) R, then T (w|v) 1 - α
0 1α
Introduction Computing relations
R Σ1* x Σ2
*
R = {(0m1m,2m) : m ≥ 0}
For 0 < α < 1 we say that T computes the relation R with isolated cutpoint α if there exists ε > 0 such that for all v, whenever (v, w) R, then T (w|v) ≥ α + ε, but whenever (v, w) R, then T (w|v) α - ε.
0 1α
ε
Introduction Computing relations
R Σ1* x Σ2
*
R = {(0m1m,2m) : m ≥ 0}
We say that T computes the relation R with probability bounded away from ½ if there exists ε > 0 such that for all v, whenever (v, w) R, then T (w|v) ≥ ½ + ε, but whenever (v, w) R, then T (w|v) ½ - ε.
0 1½
ε
Outline
Introduction Previous results When deterministic transducers
are possible Quantum vs. probabilistic
transducers
Previous results
Probabilistic transducers are more powerful than the deterministic ones (can compute more relations)
Computing relations with quantum and deterministic transducers
Computing a relation with probability 2/3
Previous results pfst and qfst more powerful than dfst?
For arbitrary ε > 0 the relation R1 = {(0m1m,2m) : m ≥ 0}
can be computed by a pfst with probability 1 – ε.
can be computed by a qfst with probability 1 – ε.
cannot be computed by a dfst.
Previous results other useful relation
The relation R2 = {(w2w, w) : w {0, 1}*}
can be computed by a pfst and qfst with probability 2/3.
Outline
Introduction Previous results When deterministic
transducers are possible Quantum vs. probabilistic
transducers
When deterministic transducers are possible
Comparing sizes of probabilistic and deterministic transducers
Not a big difference for relation R(0m1m,2m)
Exponential size difference for relation R(w2w,w), probability of correct answer: 2/3
Relation with exponential size difference and probability: 1-ε
When deterministic fst are possible fst for Rk = {(0m1m,2m) : 0 m k}
For arbitrary ε > 0 and for arbitrary k the relation
Rk = {(0m1m,2m) : 0 m k} Can be computed by pfst of size
2k + const with probability 1 – ε
For arbitrary dfst computing Rk the number of the states is not less than k
When deterministic fst are possible fst for Rk’ = {(w2w,w) : m k, w {0, 1}m}
The relationRk’ = {(w2w,w) : m k, w {0, 1}m} Can be computed by pfst of size
2k + const with probability 2/3 (can’t be improved)
For arbitrary dfst computing Rk’ the number of the states is not less than ak
where a is a cardinality of the alphabet for w.
When deterministic fst are possible improving probability
For arbitrary ε > 0 and k the relationRk’’ = {(code(w)2code(w),w) :m k, w {0, 1}m} Can be computed by pfst of size
2k + const with probability 1 - ε
For arbitrary dfst computing Rk’’ the number of the states is not less than ak
where a is a cardinality of the alphabet for w
m
mm wwwwwwwcode 223
22
2121 3...333),...,,(
321
Outline
Introduction Previous results When deterministic transducers
are possible Quantum vs. probabilistic
transducers
Quantum vs. probabilistic transducers
Exponential size difference for relation R(0m1n2k,3m)
Relation which can be computed with an isolated cutpoint, but not with a probability bouded away from 1/2
Quantum vs. probabilistic fst exponential difference in sizeThe relation Rs’’ = {(0m1n2k,3m) : n k & (m = k V m =
n) & m s & n s & k s} Can be computed by qfst of size
const with probability 4/7 – ε, ε > 0
For arbitrary pfst computing Rs’’ with probability bounded away from ½ the number of the states is not less than ak
where a is a cardinality of the alphabet for w
Quantum vs. probabilistic fst qfst with probability bounded away from 1/2?The relation Rs’’’ = {(0m1na,4k) : m s & n s &
(a = 2 → k = m) & (a = 3 → k = n)} Can be computed by pfst and by
qfst of size s + const with an isolated cutpoint, but not with a probability bounded away from ½
Conclusion
Comparing transducers by size: probabilistic smaller than
deterministic quantum smaller than
probabilistic and deterministic
Thank you!