Finite Ambiguity Research Articles

Distance automata having large finite distance or finite ambiguity

A distance automaton is a (nondeterministic finite) automaton which is equipped with a nonnegative cost function on its transitions. The distance of a word recognized by such a machine quantifies the expenses associated with the recognition of this word. The distance of a distance automaton is the maximal distance of a word recognized by this machine or is infinite, depending on whether or not a maximum exists. We present distance automata havingn states and distance 2 n − 2. As a by-product we obtain regular languages having exponential finite order. Given a finitely ambiguous distance automaton withn states, we show that either its distance is at most 3 n − 1, or the growth of the distance in this machine is linear in the input length. The infinite distance problem for these distance automata is NP-hard and solvable in polynomial space. The infinite-order problem for regular languages is PSPACE-complete.

Shape from shading as a partially well-constrained problem

For general objects, and for illumination from a general direction, we study the constraints on shape imposed by shading. Assuming generalized Lambertian reflectance, we argue that, for a typical image, shading determines shape essentially up to a finite ambiguity. Thus regularization is often unnecessary, and should be avoided. More conjectural arguments imply that shape is typically determined with little ambiguity. However, it is pointed out that the degree to which shape is constrained depends on the image. Some images uniquely determine the imaged surface, while, for others, shape can be uniquely determined over most of the image, but infinitely ambiguous in small regions bordering the image boundary, even though the image contains singular points. For these images, shape from shading is a partially well-constrained problem. The ambiguous regions may cause shape reconstruction to be unstable at the image boundary. Our main result is that, contrary to previous belief, the image of the occluding boundary does not strongly constrain the surface solution. Also, it is shown that characteristic strips are curves of steepest ascent on the imaged surface. Finally, a theorem characterizing the properties of generic images is presented.

On the classification of G‐manifolds up to finite ambiguity

On the classification of <i>G</i>‐manifolds up to finite ambiguity

Minimal Seifert manifolds and the knot finiteness theorem

It is proved that the Alexander modules determine the stable type of a knot up to finite ambiguity. The proof uses a new existence theorem of minimal Seifert surfaces for multidimensional knots of codimension two.

Rational homotopy equivalences of Lie type

Consider K⊂G, a proper pair of equal rank compact connected Lie groups. It is known (see [6], proof of theorem 1·1) that the group of self-homotopy equivalences of the rationalization of G/K is (anti)isomorphic, in a natural way, to the group of graded algebra automorphisms of H*(G/K; ℚ). On one side of the matter there is the fact that AutH*(G/K; ℚ) almost gives the integral picture of the group of homotopy classes of the self-homotopy equivalences of G/K, that is, up to a finite ambiguity and up to grading automorphisms of H*(G/K; ℚ) (i.e. those acting on H2i as λi.id, for some non-zero λ∈ℚ): see also [6]. On the other side there is the classical description by Borel[2] of H*(G/K; ℚ) in terms of invariants of Weyl groups, which gives hope for a satisfactory understanding of AutH*(G/K; ℚ).

A note on finite-valued and finitely ambiguous transducers

We look at some decision problems concerning nondeterministic finite transducers. The problems concern finite-valuedness, finite ambiguity, equivalence, etc. For a fixedk, we give a polynomial time algorithm for deciding whether or not a transducer isk-valued. The result holds when “valued” is replaced by “ambiguous”. In fact, the following problems are decidable: 1) Given a transducer, is itk-ambiguous for somek? 2) Given two finitely ambiguous transducers, are they equivalent? For unambiguous transducers, equivalence is decidable in polynomial time.