Size%20of%20Quantum%20Finite%20State%20Transducers

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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, S1, S2, V, f, q0, Qacc, Qrej),
• V S1 x Q ? Q
• a ? S1

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Introduction Transducer definition

• R? S1 x S2
• R (0m1m,2m) m 0
• S1 0,1
• S2 2
• Input 0m1m
• Output 2m
• 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? S1 x S2
• R (0m1m,2m) m 0
• For a gt 1/2 we say that T computes the relation R
  with probability a if for all v, whenever (v, w)
  ? R, then T (wv) a, and whenever (v, w) ? R,
  then T (wv) ?? 1 - a

0
1
a
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Introduction Computing relations

• R? S1 x S2
• R (0m1m,2m) m 0
• For 0 lt a lt 1 we say that T computes the relation
  R with isolated cutpoint a if there exists e gt 0
  such that for all v, whenever (v, w) ? R, then T
  (wv) a e, but whenever (v, w) ? R, then T
  (wv) ?? a - e.

e
0
1
a
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Introduction Computing relations

• R? S1 x S2
• R (0m1m,2m) m 0
• We say that T computes the relation R with
  probability bounded away from ½ if there exists e
  gt 0 such that for all v, whenever (v, w) ? R,
  then T (wv) ½ e, but whenever (v, w) ? R,
  then T (wv) ?? ½ - e.

e
0
1
½
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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 e gt 0 the relation
  R1 (0m1m,2m) m 0
• can be computed by a pfst with probability 1 e.
  
• can be computed by a qfst with probability 1 e.
  
• 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-e

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When deterministic fst are possible fst for Rk
(0m1m,2m) 0 ? m ? k

• For arbitrary e gt 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 e
• 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, 1m

• The relation
• Rk (w2w,w) ??m ? k, w ? 0, 1m
• Can be computed by pfst of size
  2k const with probability 2/3 (cant be
  improved)
• For arbitrary dfst computing Rk the number of
  the states is not less than
  akwhere a is a cardinality of the alphabet for w.

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When deterministic fst are possible improving
probability

• For arbitrary e gt 0 and k the relation
• Rk (code(w)2code(w),w) ??m ? k, w ? 0,
  1m
• Can be computed by pfst of size
  2k const with probability 1 - e
• For arbitrary dfst computing Rk the number of
  the states is not less than
  akwhere 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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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 size

• The 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 e, e gt 0
• For arbitrary pfst computing Rs with
  probability bounded away from ½ the number of the
  states is not less than akwhere
  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!