6,375 research outputs found
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
Decision Questions for Probabilistic Automata on Small Alphabets
We study the emptiness and -reachability problems for unary and
binary Probabilistic Finite Automata (PFA) and characterise the complexity of
these problems in terms of the degree of ambiguity of the automaton and the
size of its alphabet. Our main result is that emptiness and
-reachability are solvable in EXPTIME for polynomially ambiguous unary
PFA and if, in addition, the transition matrix is binary, we show they are in
NP. In contrast to the Skolem-hardness of the -reachability and
emptiness problems for exponentially ambiguous unary PFA, we show that these
problems are NP-hard even for finitely ambiguous unary PFA. For binary
polynomially ambiguous PFA with fixed and commuting transition matrices, we
prove NP-hardness of the -reachability (dimension 9), nonstrict
emptiness (dimension 37) and strict emptiness (dimension 40) problems.Comment: Updated journal pre-prin
Decision Questions for Probabilistic Automata on Small Alphabets
We study the emptiness and ?-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and ?-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is over {0, 1}, we show they are in NP. In contrast to the Skolem-hardness of the ?-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with commuting transition matrices, we prove NP-hardness of the ?-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems
Decidability of Cutpoint Isolation for Probabilistic Finite Automata on Letter-Bounded Inputs
We show the surprising result that the cutpoint isolation problem is decidable for probabilistic finite automata where input words are taken from a letter-bounded context-free language. A context-free language ? is letter-bounded when ? ? a?^* a?^* ? a_?^* for some finite ? > 0 where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems
Acceptance Ambiguity for Quantum Automata
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post\u27s correspondence problem
On the Identity and Group Problems for Complex Heisenberg Matrices
We study the Identity Problem, the problem of determining if a finitely
generated semigroup of matrices contains the identity matrix; see Problem 3
(Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control
Theory'' by Blondel and Megretski (2004). This fundamental problem is known to
be undecidable for and decidable for . The Identity Problem has been recently shown to be in polynomial
time by Dong for the Heisenberg group over complex numbers in any fixed
dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula.
We develop alternative proof techniques for the problem making a step forward
towards more general problems such as the Membership Problem. We extend our
techniques to show that the fundamental problem of determining if a given set
of Heisenberg matrices generates a group, can also be decided in polynomial
time
Decision Questions for Probabilistic Automata on Small Alphabets
We study the emptiness and -reachability problems for unary and
binary Probabilistic Finite Automata (PFA) and characterise the complexity of
these problems in terms of the degree of ambiguity of the automaton and the
size of its alphabet. Our main result is that emptiness and
-reachability are solvable in EXPTIME for polynomially ambiguous unary
PFA and if, in addition, the transition matrix is binary, we show they are in
NP. In contrast to the Skolem-hardness of the -reachability and
emptiness problems for exponentially ambiguous unary PFA, we show that these
problems are NP-hard even for finitely ambiguous unary PFA. For binary
polynomially ambiguous PFA with fixed and commuting transition matrices, we
prove NP-hardness of the -reachability (dimension 9), nonstrict
emptiness (dimension 37) and strict emptiness (dimension 40) problems
Decision Questions for Probabilistic Automata on Small Alphabets
We study the emptiness and lambda-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and lambda-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is binary, we show they are in NP. In contrast to the Skolem-hardness of the lambda-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with fixed and commuting transition matrices, we prove NP-hardness of the lambda-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems
On injectivity of quantum finite automata
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the injectivity problem of determining if the acceptance probability function of a MO-QFA is injective over all input words, i.e., giving a distinct probability for each input word. We show that the injectivity problem is undecidable for 8 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial state vector is real algebraic. We also show undecidability of this problem when the initial vector is rational, although with a huge increase in the number of states. We utilize properties of quaternions, free rotation groups, representations of tuples of rationals as linear sums of radicals and a reduction of the mixed modification of Post's correspondence problem, as well as a new result on rational polynomial packing functions which may be of independent interest.</div
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