28,815 research outputs found
Some Problems in Automata Theory Which Depend on the Models of Set Theory
We prove that some fairly basic questions on automata reading infinite words
depend on the models of the axiomatic system ZFC. It is known that there are
only three possibilities for the cardinality of the complement of an
omega-language accepted by a B\"uchi 1-counter automaton . We prove
the following surprising result: there exists a 1-counter B\"uchi automaton
such that the cardinality of the complement of the omega-language
is not determined by ZFC: (1). There is a model of ZFC in which
is countable. (2). There is a model of ZFC in which has
cardinal . (3). There is a model of ZFC in which
has cardinal with . We prove a very
similar result for the complement of an infinitary rational relation accepted
by a 2-tape B\"uchi automaton . As a corollary, this proves that the
Continuum Hypothesis may be not satisfied for complements of 1-counter
omega-languages and for complements of infinitary rational relations accepted
by 2-tape B\"uchi automata. We infer from the proof of the above results that
basic decision problems about 1-counter omega-languages or infinitary rational
relations are actually located at the third level of the analytical hierarchy.
In particular, the problem to determine whether the complement of a 1-counter
omega-language (respectively, infinitary rational relation) is countable is in
. This is rather surprising if
compared to the fact that it is decidable whether an infinitary rational
relation is countable (respectively, uncountable).Comment: To appear in the journal RAIRO-Theoretical Informatics and
Application
Spin networks, quantum automata and link invariants
The spin network simulator model represents a bridge between (generalized)
circuit schemes for standard quantum computation and approaches based on
notions from Topological Quantum Field Theories (TQFT). More precisely, when
working with purely discrete unitary gates, the simulator is naturally modelled
as families of quantum automata which in turn represent discrete versions of
topological quantum computation models. Such a quantum combinatorial scheme,
which essentially encodes SU(2) Racah--Wigner algebra and its braided
counterpart, is particularly suitable to address problems in topology and group
theory and we discuss here a finite states--quantum automaton able to accept
the language of braid group in view of applications to the problem of
estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and
Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200
The Complexity of Infinite Computations In Models of Set Theory
We prove the following surprising result: there exist a 1-counter B\"uchi
automaton and a 2-tape B\"uchi automaton such that the \omega-language of the
first and the infinitary rational relation of the second in one model of ZFC
are \pi_2^0-sets, while in a different model of ZFC both are analytic but non
Borel sets.
This shows that the topological complexity of an \omega-language accepted by
a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by
a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC.
We show that a similar result holds for the class of languages of infinite
pictures which are recognized by B\"uchi tiling systems.
We infer from the proof of the above results an improvement of the lower
bound of some decision problems recently studied by the author
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
State succinctness of two-way finite automata with quantum and classical states
{\it Two-way quantum automata with quantum and classical states} (2QCFA) were
introduced by Ambainis and Watrous in 2002. In this paper we study state
succinctness of 2QCFA.
For any and any , we show that:
{enumerate} there is a promise problem which can be solved by a
2QCFA with one-sided error in a polynomial expected running time
with a constant number (that depends neither on nor on ) of
quantum states and classical states,
whereas the sizes of the corresponding {\it deterministic finite automata}
(DFA), {\it two-way nondeterministic finite automata} (2NFA) and polynomial
expected running time {\it two-way probabilistic finite automata} (2PFA) are at
least , , and , respectively; there
exists a language over the alphabet
which can be recognized by a 2QCFA with one-sided error
in an exponential expected running time with a constant number of
quantum states and classical states,
whereas the sizes of the corresponding DFA, 2NFA and polynomial expected
running time 2PFA are at least , , and ,
respectively; {enumerate} where is a constant.Comment: 26pages, comments and suggestions are welcom
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