10 research outputs found
Potential of quantum finite automata with exact acceptance
The potential of the exact quantum information processing is an interesting,
important and intriguing issue. For examples, it has been believed that quantum
tools can provide significant, that is larger than polynomial, advantages in
the case of exact quantum computation only, or mainly, for problems with very
special structures. We will show that this is not the case.
In this paper the potential of quantum finite automata producing outcomes not
only with a (high) probability, but with certainty (so called exactly) is
explored in the context of their uses for solving promise problems and with
respect to the size of automata. It is shown that for solving particular
classes of promise problems, even those without some
very special structure, that succinctness of the exact quantum finite automata
under consideration, with respect to the number of (basis) states, can be very
small (and constant) though it grows proportional to in the case
deterministic finite automata (DFAs) of the same power are used. This is here
demonstrated also for the case that the component languages of the promise
problems solvable by DFAs are non-regular. The method used can be applied in
finding more exact quantum finite automata or quantum algorithms for other
promise problems.Comment: We have improved the presentation of the paper. Accepted to
International Journal of Foundation of Computer Scienc
From Quantum Query Complexity to State Complexity
State complexity of quantum finite automata is one of the interesting topics
in studying the power of quantum finite automata. It is therefore of importance
to develop general methods how to show state succinctness results for quantum
finite automata. One such method is presented and demonstrated in this paper.
In particular, we show that state succinctness results can be derived out of
query complexity results.Comment: Some typos in references were fixed. To appear in Gruska Festschrift
(2014). Comments are welcome. arXiv admin note: substantial text overlap with
arXiv:1402.7254, arXiv:1309.773
Unary probabilistic and quantum automata on promise problems
We continue the systematic investigation of probabilistic and quantum finite
automata (PFAs and QFAs) on promise problems by focusing on unary languages. We
show that bounded-error QFAs are more powerful than PFAs. But, in contrary to
the binary problems, the computational powers of Las-Vegas QFAs and
bounded-error PFAs are equivalent to deterministic finite automata (DFAs).
Lastly, we present a new family of unary promise problems with two parameters
such that when fixing one parameter QFAs can be exponentially more succinct
than PFAs and when fixing the other parameter PFAs can be exponentially more
succinct than DFAs.Comment: Minor correction
Generalizations of the distributed Deutsch-Jozsa promise problem
In the {\em distributed Deutsch-Jozsa promise problem}, two parties are to
determine whether their respective strings are at the {\em
Hamming distance} or . Buhrman et al. (STOC' 98)
proved that the exact {\em quantum communication complexity} of this problem is
while the {\em deterministic communication complexity} is
. This was the first impressive (exponential) gap between
quantum and classical communication complexity.
In this paper, we generalize the above distributed Deutsch-Jozsa promise
problem to determine, for any fixed , whether
or , and show that an exponential gap between exact
quantum and deterministic communication complexity still holds if is an
even such that , where is given. We also deal with a promise version of the
well-known {\em disjointness} problem and show also that for this promise
problem there exists an exponential gap between quantum (and also
probabilistic) communication complexity and deterministic communication
complexity of the promise version of such a disjointness problem. Finally, some
applications to quantum, probabilistic and deterministic finite automata of the
results obtained are demonstrated.Comment: we correct some errors of and improve the presentation the previous
version. arXiv admin note: substantial text overlap with arXiv:1309.773
New results on classical and quantum counter automata
We show that one-way quantum one-counter automaton with zero-error is more
powerful than its probabilistic counterpart on promise problems. Then, we
obtain a similar separation result between Las Vegas one-way probabilistic
one-counter automaton and one-way deterministic one-counter automaton.
We also obtain new results on classical counter automata regarding language
recognition. It was conjectured that one-way probabilistic one blind-counter
automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz:
Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and
Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also
show several separation results for blind/non-blind counter automata.Comment: 21 page