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 $\lambda$-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.Comment: Updated journal pre-prin

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

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

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 $\mathbb{Z}^{4 \times 4}$ and decidable for $\mathbb{Z}^{2 \times 2}$. 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

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

Quasilocal Conservation Laws: Why We Need Them

We argue that conservation laws based on the local matter-only stress-energy-momentum tensor (characterized by energy and momentum per unit volume) cannot adequately explain a wide variety of even very simple physical phenomena because they fail to properly account for gravitational effects. We construct a general quasi}local conservation law based on the Brown and York total (matter plus gravity) stress-energy-momentum tensor (characterized by energy and momentum per unit area), and argue that it does properly account for gravitational effects. As a simple example of the explanatory power of this quasilocal approach, consider that, when we accelerate toward a freely-floating massive object, the kinetic energy of that object increases (relative to our frame). But how, exactly, does the object acquire this increasing kinetic energy? Using the energy form of our quasilocal conservation law, we can see precisely the actual mechanism by which the kinetic energy increases: It is due to a bona fide gravitational energy flux that is exactly analogous to the electromagnetic Poynting flux, and involves the general relativistic effect of frame dragging caused by the object's motion relative to us.Comment: 20 pages, 1 figur

Mapping the Galactic Halo I. The `Spaghetti' Survey

We describe a major survey of the Milky Way halo designed to test for kinematic substructure caused by destruction of accreted satellites. We use the Washington photometric system to identify halo stars efficiently for spectroscopic followup. Tracers include halo giants (detectable out to more than 100 kpc), blue horizontal branch stars, halo stars near the main sequence turnoff, and the ``blue metal-poor stars'' of Preston et al (1994). We demonstrate the success of our survey by showing spectra of stars we have identified in all these categories, including giants as distant as 75 kpc. We discuss the problem of identifying the most distant halo giants. In particular, extremely metal-poor halo K dwarfs are present in approximately equal numbers to the distant giants for V fainter than 18, and we show that our method will distinguish reliably between these two groups of metal-poor stars. We plan to survey 100 square degrees at high galactic latitude, and expect to increase the numbers of known halo giants, BHB stars and turnoff stars by more than an order of magnitude. In addition to the strong test that this large sample will provide for the question `was the Milky Way halo accreted from satellite galaxies?', we will improve the accuracy of mass measurements of the Milky Way beyond 50 kpc via the kinematics of the many distant giants and BHB stars we will find. We show that one of our first datasets constrains the halo density law over galactocentric radii of 5-20 kpc and z heights of 2-15 kpc. The data support a flattened power-law halo with b/a of 0.6 and exponent -3.0. More complex models with a varying axial ratio may be needed with a larger dataset.Comment: 55 pages, 22 figures, to appear in the Astronomical Journa