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

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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!