3,533 research outputs found
The complexity properties of probabilistic automata with isolated cut point
A probabilistic automaton (PA) which accepts a language with e-isolated cut point 1 2 corresponds to a PA which computes with ( 1 2-e) bounded error probability. Let P(L, e) be the minimal number of states of a PA necessary for accepting a language L with e-isolated cut point 1 2. It is shown that there are languages Lk, 1 < k < ∞ and an infinite sequence of numbers 0 < e1 < e2 < ... < 1 2 such that for all i ≥1, P(Lk,ei) P(Lk, ei+1) → 0 when k→∞. It is also shown that the probabilistic recognition of the language Wk is more effective than that of the Lk. © 1988
Mean-payoff Automaton Expressions
Quantitative languages are an extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. When the mode of branching of the automaton
is deterministic, nondeterministic, or alternating, the corresponding class of
quantitative languages is not robust as it is not closed under the pointwise
operations of max, min, sum, and numerical complement. Nondeterministic and
alternating mean-payoff automata are not decidable either, as the quantitative
generalization of the problems of universality and language inclusion is
undecidable.
We introduce a new class of quantitative languages, defined by mean-payoff
automaton expressions, which is robust and decidable: it is closed under the
four pointwise operations, and we show that all decision problems are decidable
for this class. Mean-payoff automaton expressions subsume deterministic
mean-payoff automata, and we show that they have expressive power incomparable
to nondeterministic and alternating mean-payoff automata. We also present for
the first time an algorithm to compute distance between two quantitative
languages, and in our case the quantitative languages are given as mean-payoff
automaton expressions
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
Computation in Finitary Stochastic and Quantum Processes
We introduce stochastic and quantum finite-state transducers as
computation-theoretic models of classical stochastic and quantum finitary
processes. Formal process languages, representing the distribution over a
process's behaviors, are recognized and generated by suitable specializations.
We characterize and compare deterministic and nondeterministic versions,
summarizing their relative computational power in a hierarchy of finitary
process languages. Quantum finite-state transducers and generators are a first
step toward a computation-theoretic analysis of individual, repeatedly measured
quantum dynamical systems. They are explored via several physical systems,
including an iterated beam splitter, an atom in a magnetic field, and atoms in
an ion trap--a special case of which implements the Deutsch quantum algorithm.
We show that these systems' behaviors, and so their information processing
capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous
corrections and update
Deciding the value 1 problem for probabilistic leaktight automata
The value 1 problem is a decision problem for probabilistic automata over
finite words: given a probabilistic automaton, are there words accepted with
probability arbitrarily close to 1? This problem was proved undecidable
recently; to overcome this, several classes of probabilistic automata of
different nature were proposed, for which the value 1 problem has been shown
decidable. In this paper, we introduce yet another class of probabilistic
automata, called leaktight automata, which strictly subsumes all classes of
probabilistic automata whose value 1 problem is known to be decidable. We prove
that for leaktight automata, the value 1 problem is decidable (in fact,
PSPACE-complete) by constructing a saturation algorithm based on the
computation of a monoid abstracting the behaviours of the automaton. We rely on
algebraic techniques developed by Simon to prove that this abstraction is
complete. Furthermore, we adapt this saturation algorithm to decide whether an
automaton is leaktight. Finally, we show a reduction allowing to extend our
decidability results from finite words to infinite ones, implying that the
value 1 problem for probabilistic leaktight parity automata is decidable
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
Polynomially Ambiguous Probabilistic Automata on Restricted Languages
We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with sub-exponential ambiguity. We show that the emptiness problem for non-strict cut-points of polynomially ambiguous PFA remains undecidable even when the input word is over a bounded language and all PFA transition matrices are commutative. In doing so, we introduce a new technique based upon the Turakainen construction of a PFA from a Weighted Finite Automata which can be used to generate PFA of lower dimensions and of subexponential ambiguity. We also study freeness/injectivity problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases
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