The study of alternation in two-way finite automata ( 2 fas) has been quite non-systematic. Since the 1970's, various authors with a variety of motivations have studied 2 fas with various types of alternation and a variety of names, creating a fairly long list of sporadic contributions with little internal consistency. This article attempts to organize the subject into a single unifying framework. We start with a detailed account of all contributions to date, that reveals the large variety of approaches and the lack of consistency between them. We then identify and name four types of automata that these contributions have really studied over the years: general two-way Boolean finite automata ( 2 b fas); monotone 2 b fas; basic 2 b fas; and (monotone basic, or) alternating 2 b fas ( 2 a fas). Next, we identify four different ways by which authors have described how such automata compute, each offering a distinct view onto their operation: the circuit view, where computation is modeled by a circuit of Boolean gates; the formula view, where computation is modeled by a Boolean formula over configuration-variables; the run view, where decisions are determined by the existence of appropriate trees of configuration-goal pairs, called “runs”; and the (most classic) tree view, where computation is modeled by a tree of configurations. After carefully defining each 2 b fa type and each view, we prove the following. First, that within each type, every two of the four views are equivalent to each other, in the strong sense that each of them closely mimics the other at every step of the computation. Second, that not all types of 2 b fas are equivalent: although general 2 b fas are as powerful as monotone 2 b fas, and basic 2 b fas are as powerful as 2 a fas (up to polynomial differences in the number of states), general 2 b fas may need exponentially fewer states than 2 a fas.

This work presents a method for the registration of three-dimensional (3-D) shapes. The method is based on the iterative closest point (ICP) algorithm and improves it through the use of a 3-D volume containing the shapes to be registered. The Voronoi diagram of the "model" shape points is first constructed in the volume. Then this is used for the calculation of the closest point operator. This way a dramatic decrease of the computational cost is achieved.

This paper describes a method for the accurate and computationally efficient registration of 3-D shapes. The method is based on the iterative closest point (ICP) algorithm and improves it by dramatically decreasing the computational cost of the algorithm's most inefficient step, namely the implementation of the closest point operator. The decrease is achieved with the help of a 3-D volume containing the points to be registered. Prior to the implementation of the ICP algorithm, the Voronoi diagram of the "model" points is constructed in the volume, by means of the morphological Voronoi tesselation method with respect to the Euclidean distance metric. The use of the tesselated volume renders the calculation of the closest point operator extremely fast and speeds up the ICP algorithm tremendously.