413 research outputs found
Implications of quantum automata for contextuality
We construct zero-error quantum finite automata (QFAs) for promise problems
which cannot be solved by bounded-error probabilistic finite automata (PFAs).
Here is a summary of our results:
- There is a promise problem solvable by an exact two-way QFA in exponential
expected time, but not by any bounded-error sublogarithmic space probabilistic
Turing machine (PTM).
- There is a promise problem solvable by an exact two-way QFA in quadratic
expected time, but not by any bounded-error -space PTMs in
polynomial expected time. The same problem can be solvable by a one-way Las
Vegas (or exact two-way) QFA with quantum head in linear (expected) time.
- There is a promise problem solvable by a Las Vegas realtime QFA, but not by
any bounded-error realtime PFA. The same problem can be solvable by an exact
two-way QFA in linear expected time but not by any exact two-way PFA.
- There is a family of promise problems such that each promise problem can be
solvable by a two-state exact realtime QFAs, but, there is no such bound on the
number of states of realtime bounded-error PFAs solving the members this
family.
Our results imply that there exist zero-error quantum computational devices
with a \emph{single qubit} of memory that cannot be simulated by any finite
memory classical computational model. This provides a computational perspective
on results regarding ontological theories of quantum mechanics \cite{Hardy04},
\cite{Montina08}. As a consequence we find that classical automata based
simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently
powerful to simulate quantum contextuality. We conclude by highlighting the
interplay between results from automata models and their application to
developing a general framework for quantum contextuality.Comment: 22 page
Two Variable vs. Linear Temporal Logic in Model Checking and Games
Model checking linear-time properties expressed in first-order logic has
non-elementary complexity, and thus various restricted logical languages are
employed. In this paper we consider two such restricted specification logics,
linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is
more expressive but FO2 can be more succinct, and hence it is not clear which
should be easier to verify. We take a comprehensive look at the issue, giving a
comparison of verification problems for FO2, LTL, and various sublogics thereof
across a wide range of models. In particular, we look at unary temporal logic
(UTL), a subset of LTL that is expressively equivalent to FO2; we also consider
the stutter-free fragment of FO2, obtained by omitting the successor relation,
and the expressively equivalent fragment of UTL, obtained by omitting the next
and previous connectives. We give three logic-to-automata translations which
can be used to give upper bounds for FO2 and UTL and various sublogics. We
apply these to get new bounds for both non-deterministic systems (hierarchical
and recursive state machines, games) and for probabilistic systems (Markov
chains, recursive Markov chains, and Markov decision processes). We couple
these with matching lower-bound arguments. Next, we look at combining FO2
verification techniques with those for LTL. We present here a language that
subsumes both FO2 and LTL, and inherits the model checking properties of both
languages. Our results give both a unified approach to understanding the
behaviour of FO2 and LTL, along with a nearly comprehensive picture of the
complexity of verification for these logics and their sublogics.Comment: 37 pages, to be published in Logical Methods in Computer Science
journal, includes material presented in Concur 2011 and QEST 2012 extended
abstract
Classical and quantum Merlin-Arthur automata
We introduce Merlin-Arthur (MA) automata as Merlin provides a single
certificate and it is scanned by Arthur before reading the input. We define
Merlin-Arthur deterministic, probabilistic, and quantum finite state automata
(resp., MA-DFAs, MA-PFAs, MA-QFAs) and postselecting MA-PFAs and MA-QFAs
(resp., MA-PostPFA and MA-PostQFA). We obtain several results using different
certificate lengths.
We show that MA-DFAs use constant length certificates, and they are
equivalent to multi-entry DFAs. Thus, they recognize all and only regular
languages but can be exponential and polynomial state efficient over binary and
unary languages, respectively. With sublinear length certificates, MA-PFAs can
recognize several nonstochastic unary languages with cutpoint 1/2. With linear
length certificates, MA-PostPFAs recognize the same nonstochastic unary
languages with bounded error. With arbitrarily long certificates, bounded-error
MA-PostPFAs verify every unary decidable language. With sublinear length
certificates, bounded-error MA-PostQFAs verify several nonstochastic unary
languages. With linear length certificates, they can verify every unary
language and some NP-complete binary languages. With exponential length
certificates, they can verify every binary language.Comment: 14 page
On the state complexity of semi-quantum finite automata
Some of the most interesting and important results concerning quantum finite
automata are those showing that they can recognize certain languages with
(much) less resources than corresponding classical finite automata
\cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}.
This paper shows three results of such a type that are stronger in some sense
than other ones because (a) they deal with models of quantum automata with very
little quantumness (so-called semi-quantum one- and two-way automata with one
qubit memory only); (b) differences, even comparing with probabilistic
classical automata, are bigger than expected; (c) a trade-off between the
number of classical and quantum basis states needed is demonstrated in one case
and (d) languages (or the promise problem) used to show main results are very
simple and often explored ones in automata theory or in communication
complexity, with seemingly little structure that could be utilized.Comment: 19 pages. We improve (make stronger) the results in section
Uncountable realtime probabilistic classes
We investigate the minimum cases for realtime probabilistic machines that can
define uncountably many languages with bounded error. We show that logarithmic
space is enough for realtime PTMs on unary languages. On binary case, we follow
the same result for double logarithmic space, which is tight. When replacing
the worktape with some limited memories, we can follow uncountable results on
unary languages for two counters.Comment: 12 pages. Accepted to DCFS201
Game Characterization of Probabilistic Bisimilarity, and Applications to Pushdown Automata
We study the bisimilarity problem for probabilistic pushdown automata (pPDA)
and subclasses thereof. Our definition of pPDA allows both probabilistic and
non-deterministic branching, generalising the classical notion of pushdown
automata (without epsilon-transitions). We first show a general
characterization of probabilistic bisimilarity in terms of two-player games,
which naturally reduces checking bisimilarity of probabilistic labelled
transition systems to checking bisimilarity of standard (non-deterministic)
labelled transition systems. This reduction can be easily implemented in the
framework of pPDA, allowing to use known results for standard
(non-probabilistic) PDA and their subclasses. A direct use of the reduction
incurs an exponential increase of complexity, which does not matter in deriving
decidability of bisimilarity for pPDA due to the non-elementary complexity of
the problem. In the cases of probabilistic one-counter automata (pOCA), of
probabilistic visibly pushdown automata (pvPDA), and of probabilistic basic
process algebras (i.e., single-state pPDA) we show that an implicit use of the
reduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and
2-EXPTIME upper bounds, respectively, like for the respective non-probabilistic
versions. The bisimilarity problems for OCA and vPDA are known to have matching
lower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively);
we show that these lower bounds also hold for fully probabilistic versions that
do not use non-determinism
Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata
We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm} of cyclic languages, where Lm = {akm|k 08 N}. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language Lm with isolated cut point on one-way probabilistic finite automata is p1\u3b11 + p2\u3b12 + ef + ps\u3b1s, with p1\u3b11p2\u3b12 ef ps\u3b1s being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept Lm with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms
Photonic realization of a quantum finite automaton
We describe a physical implementation of a quantum finite automaton that recognizes a well-known family of periodic languages. The realization exploits the polarization degree of freedom of single photons and their manipulation through linear optical elements. We use techniques of confidence amplification to reduce the acceptance error probability of the automaton. It is worth remarking that the quantum finite automaton we physically realize is not only interesting per se but it turns out to be a crucial building block in many quantum finite automaton design frameworks theoretically settled in the literature
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