Finite Ambiguity Research Articles

Equivalence, Unambiguity, and Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the equivalence, unambiguity, and sequentiality problems are decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant and it is called unambiguous if there exists at most one accepting run on every tree. For the equivalence problem, we show that for two finitely ambiguous max-plus tree automata, it is decidable whether both assign the same weight to every tree. For the unambiguity and sequentiality problems, we show that for every finitely ambiguous max-plus tree automaton, both the existence of an equivalent unambiguous automaton and the existence of an equivalent deterministic automaton are decidable.

On the power of finite ambiguity in Büchi complementation

In this work, we exploit the power of finite ambiguity for the complementation problem of Büchi automata by using reduced run directed acyclic graphs (DAGs) over infinite words, in which each vertex has at most one predecessor; these reduced run DAGs have only a finite number of infinite runs, thus obtaining the finite ambiguity in Büchi complementation. We show how to use this type of reduced run DAGs as a unified tool to optimize both rank-based and slice-based complementation constructions for Büchi automata with a finite degree of ambiguity. As a result, given a Büchi automaton with n states and a finite degree of ambiguity, the number of states in the complementary Büchi automaton constructed by the classical rank-based and slice-based complementation constructions can be improved from 2O(nlog⁡n) and O((3n)n) to O(6n)⊆2O(n) and O(4n), respectively. We further show how to construct such reduced run DAGs for limit deterministic Büchi automata and obtain a specialized complementation algorithm, thus demonstrating the generality of the power of finite ambiguity.

Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups

For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.

Combinatorial iterated integrals and the harmonic volume of graphs

Let Γ be a connected bridgeless metric graph, and fix a point v of Γ. We define combinatorial iterated integrals on Γ along closed paths at v, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of Γ at v and the tensor algebra on the first homology of Γ, ∫:Zπ1(Γ,v)×TH1(Γ,R)→R. We show that this pairing on the two-step unipotent quotient of the group algebra allows one to recover the base-point v up to well-understood finite ambiguity. We encode the data of this structure as the combinatorial harmonic volume which is valued in the tropical intermediate Jacobian. We also give a potential-theoretic characterization for hyperellipticity for graphs.

A Pattern Logic for Automata with Outputs

We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric. We then consider three particular automata models (finite automata, transducers and automata weighted by integers here called sum-automata) and instantiate the generic logic for each of them. We give tight complexity results for the three logics with respect to the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. As a consequence of our complexity results, we directly obtain that these classical properties can be decided in polynomial time.

Profinite rigidity for twisted Alexander polynomials

Abstract We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a ℤ {{\mathbb{Z}}} -cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

The rational homotopy type of (n−1)‐connected manifolds of dimension up to 5n−3

We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity.

Minimal determination of a pure qutrit state and four-measurement protocol for pure qudit state

Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. Peres conjectured that two unbiased bases may be sufficient to determine a pure state up to some finite ambiguities. Here we find that the Peres conjecture is correct in the case d = 3 where a pure state is determined up to at most six candidates states and is incorrect in the case of d = 4 for which a counterexample is constructed. We observe that the target state can be picked out from candidates states by an adaptive two-outcome measurement. We thus provided a minimal qutrit tomography protocol constituted of three measurements in contrast to the four measurements required in previous non-adaptive method. We also reduce a five-measurement protocol for a pure qudit state into three full-dimensional measurements and one two-outcome measurement protocol, which exhibits robustness to the usual white noise thus also applies to nearly pure state.

Joint Blind Parameter Estimation of Non-cooperative High-Order Modulated PCMA Signals

A joint blind parameter estimation algorithm based on minimum channel stability function aimed at the non-cooperative high-order modulated paired carrier multiple access (PCMA) signals is proposed. The method, which uses hierarchical search to estimate time delay, amplitude and frequency offset and the estimation of phase offset, including finite ambiguity, is presented simultaneously based on the derivation of the channel stability function. In this work, the structure of hierarchical iterative processing is used to enhance the performance of the algorithm, and the improved algorithm is used to reduce complexity. Compared with existing data-aided algorithms, this algorithm does not require a priori information. Therefore, it has significant advantage in solving the problem of blind parameter estimation of non-cooperative high-order modulated PCMA signals. Simulation results show the performance of the proposed algorithm is similar to the modified Cramer-Rao bound (MCRB) when the signal-to-noise ratio is larger than 16 dB. The simulation results also verify the practicality of the proposed algorithm.

Algebraic structures with unbounded Chern numbers

We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in dimension four.

VACUUM ENERGY OF SCHRÖDINGER OPERATORS ON METRIC GRAPHS

We present an integral formulation of the vacuum energy of Schrödinger operators on finite metric graphs. Local vertex matching conditions on the graph are classified according to the general scheme of Kostrykin and Schrader. While the vacuum energy of the graph can contain finite ambiguities the Casimir force on a bond with compactly supported potential is well defined. The vacuum energy is determined from the zeta function of the graph Schrödinger operator which is derived from an appropriate secular equation via the argument principle. A quantum graph has an associated probabilistic classical dynamics which is generically both ergodic and mixing. The results therefore present an analytic formulation of the vacuum energy of this quasi-one-dimensional quantum system which is classically chaotic.

Classes of two-dimensional languages and recognizability conditions

The paper deals with some classes of two-dimensional recognizable languages of “high complexity”, in a sense specified in the paper and motivated by some necessary conditions holding for recognizable and unambiguous languages. For such classes we can solve some open questions related to unambiguity, finite ambiguity and complementation. Then we reformulate a necessary condition for recognizability stated by Matz, introducing a new complexity function. We solve an open question proposed by Matz, showing that all the known necessary conditions for recognizability of a language and its complement are not sufficient. The proof relies on a family of languages defined by functions.

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction Φ on a lattice L⊂ℤd determines a lattice spin system with potential AΦ. The pressure P(AΦ) and free energy FAΦ(β)=−(1/β)P(βAΦ) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a Holder continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question ``Can you hear the shape of a drum?''.

The local defect index up to finite ambiguity

The rational Hopf invariant of the primary obstruction to a nowhere vanishing section in a vectorbundle is computed in terms of characteristic classes. This leads to a local rigidity result on the local defect index of a map.

Random Generation for Finitely Ambiguous Context-free Languages

We prove that a word of length n from a finitely ambiguous context-free language can be generated at random under uniform distribution in O(n2 log n) time by a probabilistic random access machine assuming a logarithmic cost criterion. We also show that the same problem can be solved in polynomial time for every language accepted by a polynomial time 1-NAuxPDA with polynomially bounded ambiguity.

Algebraic Invariants of Simple 4-Knots

We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.

On the homotopy types of closed 4-manifolds covered by S2 × R2

We show that if M is a closed 4-manifold such that π 2( M) ≅ Z and χ( M) ⩽ 0 then its homotopy type is determined up to a finite ambiguity by π = π 1( M). We also show that (for given π) the action of π on π 2( M) and the orientation character ω 1( M) determine each other, and find strong constraints on the possible k-invariants when χ( M) = 0. Finally we show that Wh( π) = 0.

Non-decoupling, triviality and the? parameter

The dependence of the $\rho$ parameter on the mass of the Higgs scalar and the top quark is computed non--perturbatively using the $1/N_F$ expansion in the standard model. We find an explicit expression for the $\rho$ parameter that requires the presence of a physical cutoff. This should come as no surprise since the theory is presumably trivial. By taking this cutoff into account, we find that the $\rho$ parameter can take values only within a limited range and has finite ambiguities that are suppressed by inverse powers of the cutoff scale, the so called ``scaling--violations". We find that large deviations from the perturbative results are possible, but only when the cutoff effects are also large.

An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients

For generic Sturm-Liouville problems with piecewise constant leading coefficients, the leading coefficient can be determined up to a finite ambiguity from the eigenvalues of the problem.