Size of Quantum Finite State Transducers

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

IntroductionTransducer definition • Finite state transducer (fst) is a tuple T = (Q, Σ1, Σ2, V, f, q0, Qacc, Qrej), V : Σ1 x Q → Q a  Σ1 :

IntroductionTransducer 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

IntroductionTransducer types • Deterministic (dfst) • Probabilistic (pfst) • Quantum (qfst)

IntroductionComputing 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 1 α

IntroductionComputing relations R Σ1* x Σ2* R = {(0m1m,2m) : m ≥ 0} • For 0 < α < 1 we say that T computes the relation R with isolated cutpointαif 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. ε 0 1 α

IntroductionComputing 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. ε 0 1 ½

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 resultspfst 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 resultsother useful relation The relation R2 = {(w2w, w) : w  {0, 1}*} • can be computed by a pfst and qfst with probability 2/3.

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 possiblefst 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 + constwith probability 1 – ε • For arbitrary dfst computing Rk the number of the states is not less than k

When deterministic fst are possiblefst for Rk’ = {(w2w,w) : m  k, w  {0, 1}m} The relation Rk’ = {(w2w,w) : m  k, w  {0, 1}m} • Can be computed by pfst of size 2k + constwith probability 2/3 (can’t 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.

When deterministic fst are possibleimproving probability For arbitrary ε > 0 and k the relation Rk’’ = {(code(w)2code(w),w) :m  k, w  {0, 1}m} • Can be computed by pfst of size 2k + constwith probability 1 - ε • For arbitrary dfst computing Rk’’ the number of the states is not less than akwhere a is a cardinality of the alphabet for w

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 fstexponential 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 constwith 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 akwhere a is a cardinality of the alphabet for w

Quantum vs. probabilistic fstqfst 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!

Size of Quantum Finite State Transducers