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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, Σ 1, Σ 2, V, f, q 0, Q acc, Q rej ), V : Σ 1 x Q → Q a Σ 1 :

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Introduction Transducer definition R Σ 1 * x Σ 2 * R = {(0 m 1 m,2 m ) : m ≥ 0} Σ 1 = {0,1} Σ 2 = {2} Input: #0 m 1 m $ Output: 2 m Transducer may accept or reject input

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Introduction Transducer types Deterministic (dfst) Probabilistic (pfst) Quantum (qfst)

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Introduction Computing relations R Σ 1 * x Σ 2 * R = {(0 m 1 m,2 m ) : 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 01 α

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Introduction Computing relations R Σ 1 * x Σ 2 * R = {(0 m 1 m,2 m ) : m ≥ 0} For 0 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. 01 α ε

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Introduction Computing relations R Σ 1 * x Σ 2 * R = {(0 m 1 m,2 m ) : 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. 01 ½ 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 R 1 = {(0 m 1 m,2 m ) : 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 R 2 = {(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(0 m 1 m,2 m ) 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 R k = {(0 m 1 m,2 m ) : 0 m k} For arbitrary ε > 0 and for arbitrary k the relation R k = {(0 m 1 m,2 m ) : 0 m k} Can be computed by pfst of size 2k + const with probability 1 – ε For arbitrary dfst computing R k the number of the states is not less than k

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When deterministic fst are possible fst for R k’ = {(w2w,w) : m k, w {0, 1} m } The relation R k’ = {(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 R k’ the number of the states is not less than a k 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 relation R k’’ = {( code(w) 2 code(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 R k’’ the number of the states is not less than a k where a is a cardinality of the alphabet for w

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Outline Introduction Previous results When deterministic transducers are possible Quantum vs. probabilistic transducers

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Exponential size difference for relation R(0 m 1 n 2 k,3 m ) 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 size The relation R s’’ = {(0 m 1 n 2 k,3 m ) : 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 R s’’ with probability bounded away from ½ the number of the states is not less than a k 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 R s’’’ = {(0 m 1 n a,4 k ) : 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!

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