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Size of Quantum Finite State Transducers
Ruben Agadzanyan, Rusins Freivalds
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Outline
Introduction Previous results When deterministic transducers
are possible Quantum vs. probabilistic
transducers
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Introduction
Probabilistic transducer definition Computing relations Quantum transducer definition
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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
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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
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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
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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
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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
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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
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Outline
Introduction Previous results When deterministic transducers
are possible Quantum vs. probabilistic
transducers
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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
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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.
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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.
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Outline
Introduction Previous results When deterministic
transducers are possible Quantum vs. probabilistic
transducers
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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
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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
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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.
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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
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Outline
Introduction Previous results When deterministic transducers
are possible Quantum vs. probabilistic
transducers
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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
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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
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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
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Conclusion
Comparing transducers by size: probabilistic smaller than
deterministic quantum smaller than
probabilistic and deterministic
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Thank you!