Separation for dot-depth two

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results

Separation for dot-depth two

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results

Separating regular languages with two quantifier alternations

We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second. Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C. This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes. Here, we are interested in classes within concatenation hierarchies. Such hierarchies are built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations. The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages definable in first-order logic. Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks. Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory. Our main theorem is generic: we show that separation is decidable for the level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in the special case of the dot-depth, we push this result to the level 5/2. In logical terms, this solves separation for $\Sigma_3$: first-order sentences having at most three quantifier blocks starting with an existential one

Stereoscopic depth aftereffects without retinal position correspondence between adaptation and test stimuli

AbstractTo clarify whether stereo-slant aftereffects are independent of stimulated retinal position, two experiments compared the magnitude of aftereffects between the following two conditions: when the adaptation and test stimulus fell on (1) the same retinal position, and (2) on different retinal positions separated by 0.5°–20°. In Experiment 1, disc- or ring-shaped surface consisting of random-dots was presented at the central or peripheral visual fields. In Experiment 2, rectangular surface was presented at the upper or lower visual fields. After two minutes inspection of a random-dot stereogram depicting a ±30° slanted surface, the observer adjusted the slant of the test stimulus to appear fronto-parallel. The results of the experiments showed that significant aftereffects were observed similarly in both conditions. Moreover, the separation nor the stimulus shape scarcely affected the magnitude of the aftereffects. Based on these results we concluded that the depth processing mechanism which operates independently from the stimulated retinal position is responsible for the depth aftereffects we found

Displaying 3D images: algorithms for single-image random-dot

A new, simple, and symmetric algorithm can be implemented that results in higher levels of detail in solid objects than previously possible with autostereograms. In a stereoscope, an optical instrument similar to binoculars, each eye views a different picture and thereby receives the specific image that would have arisen naturally. An early suggestion for a color stereo computer display involved a rotating filter wheel held in front of the eyes. In contrast, this article describes a method for viewing on paper or on an ordinary computer screen without special equipment, although it is limited to the display of 3D monochromatic objects. (The image can be colored, say, for artistic reasons, but the method we describe does not allow colors to be allocated in a way that corresponds to an arbitrary coloring of the solid object depicted.) The image can easily be constructed by computer from any 3D scene or solid object description

Neural Models of Motion Integration, Segmentation, and Probablistic Decision-Making

When brain mechanism carry out motion integration and segmentation processes that compute unambiguous global motion percepts from ambiguous local motion signals? Consider, for example, a deer running at variable speeds behind forest cover. The forest cover is an occluder that creates apertures through which fragments of the deer's motion signals are intermittently experienced. The brain coherently groups these fragments into a trackable percept of the deer in its trajectory. Form and motion processes are needed to accomplish this using feedforward and feedback interactions both within and across cortical processing streams. All the cortical areas V1, V2, MT, and MST are involved in these interactions. Figure-ground processes in the form stream through V2, such as the seperation of occluding boundaries of the forest cover from the boundaries of the deer, select the motion signals which determine global object motion percepts in the motion stream through MT. Sparse, but unambiguous, feauture tracking signals are amplified before they propogate across position and are intergrated with far more numerous ambiguous motion signals. Figure-ground and integration processes together determine the global percept. A neural model predicts the processing stages that embody these form and motion interactions. Model concepts and data are summarized about motion grouping across apertures in response to a wide variety of displays, and probabilistic decision making in parietal cortex in response to random dot displays.National Science Foundation (SBE-0354378); Office of Naval Research (N00014-01-1-0624