On dot-depth two

Etant donnés des entiers positifs m1, …, mk, on définit des congruences ~(m1, …, mk) en relation avec une version du jeu de Ehrenfeucht-Fraissé, et qui correspondent au niveau k de la hiérarchie de concaténation de Straubing. Etant donné un alphabet fini A, une condition nécessaire et suffisante est donnée pour que les monoïdes définis par ces congruences soient de dot-delpth exactement

On a conjecture concerning dot-depth two languages

AbstractIn this paper, we study the second level of the dot-depth hierarchy for star-free regular languages. We investigate a necessary condition stated by Straubing for a language to have dot-depth two, and prove that it is sufficient for languages whose syntactic monoid is inverse with three inverse generators. Also we disprove a conjecture according to which Straubing's condition would be equivalent to both dot-depth two and another condition expressed in terms of two-sided semidirect product

Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

On a complete set of generators for dot-depth two

AbstractA complete set of generators for Straubing's dot-depth-two monoids has been characterized as a set of quotients of the form A∗/∼(n,m), where n and m denote positive integers, A∗ denotes the free monoid generated by a finite alphabet A, and ∼(n,m) denote congruences related to a version of the Ehrenfeucht—Fraïssé game. This paper studies combinatorial properties of the ∼(n,m)'s and in particular the inclusion relations between them. Several decidability and inclusion consequences are discussed

The Role of Attention in Ambiguous Reversals of Structure-From-Motion

Multiple dots moving independently back and forth on a flat screen induce a compelling illusion of a sphere rotating in depth (structure-from-motion). If all dots simultaneously reverse their direction of motion, two perceptual outcomes are possible: either the illusory rotation reverses as well (and the illusory depth of each dot is maintained), or the illusory rotation is maintained (but the illusory depth of each dot reverses). We investigated the role of attention in these ambiguous reversals. Greater availability of attention – as manipulated with a concurrent task or inferred from eye movement statistics – shifted the balance in favor of reversing illusory rotation (rather than depth). On the other hand, volitional control over illusory reversals was limited and did not depend on tracking individual dots during the direction reversal. Finally, display properties strongly influenced ambiguous reversals. Any asymmetries between ‘front’ and ‘back’ surfaces – created either on purpose by coloring or accidentally by random dot placement – also shifted the balance in favor of reversing illusory rotation (rather than depth). We conclude that the outcome of ambiguous reversals depends on attention, specifically on attention to the illusory sphere and its surface irregularities, but not on attentive tracking of individual surface dots

Motion transparency : depth ordering and smooth pursuit eye movements

When two overlapping, transparent surfaces move in different directions, there is ambiguity with respect to the depth ordering of the surfaces. Little is known about the surface features that are used to resolve this ambiguity. Here, we investigated the influence of different surface features on the perceived depth order and the direction of smooth pursuit eye movements. Surfaces containing more dots, moving opposite to an adapted direction, moving at a slower speed, or moving in the same direction as the eyes were more likely to be seen in the back. Smooth pursuit eye movements showed an initial preference for surfaces containing more dots, moving in a non-adapted direction, moving at a faster speed, and being composed of larger dots. After 300 to 500 ms, smooth pursuit eye movements adjusted to perception and followed the surface whose direction had to be indicated. The differences between perceived depth order and initial pursuit preferences and the slow adjustment of pursuit indicate that perceived depth order is not determined solely by the eye movements. The common effect of dot number and motion adaptation suggests that global motion strength can induce a bias to perceive the stronger motion in the back

Reionization after Planck: the derived growth of the cosmic ionizing emissivity now matches the growth of the galaxy UV luminosity density

Thomson optical depth tau measurements from Planck provide new insights into the reionization of the universe. In pursuit of model-independent constraints on the properties of the ionising sources, we determine the empirical evolution of the cosmic ionizing emissivity. We use a simple two-parameter model to map out the evolution in the emissivity at z>~6 from the new Planck optical depth tau measurements, from the constraints provided by quasar absorption spectra and from the prevalence of Ly-alpha emission in z~7-8 galaxies. We find the redshift evolution in the emissivity dot{N}_{ion}(z) required by the observations to be d(log Nion)/dz=-0.15(-0.11)(+0.08), largely independent of the assumed clumping factor C_{HII} and entirely independent of the nature of the ionising sources. The trend in dot{N}_{ion}(z) is well-matched by the evolution of the galaxy UV-luminosity density (dlog_{10} rho_UV/dz=-0.11+/-0.04) to a magnitude limit >~-13 mag, suggesting that galaxies are the sources that drive the reionization of the universe. The role of galaxies is further strengthened by the conversion from the UV luminosity density rho_UV to dot(N)_{ion}(z) being possible for physically-plausible values of the escape fraction f_{esc}, the Lyman-continuum photon production efficiency xi_{ion}, and faint-end cut-off $M_{lim}$ to the luminosity function. Quasars/AGN appear to match neither the redshift evolution nor normalization of the ionizing emissivity. Based on the inferred evolution in the ionizing emissivity, we estimate that the z~10 UV-luminosity density is 8(-4)(+15)x lower than at $z~6, consistent with the observations. The present approach of contrasting the inferred evolution of the ionizing emissivity with that of the galaxy UV luminosity density adds to the growing observational evidence that faint, star-forming galaxies drive the reionization of the universe.Comment: 20 pages, 12 figures, 5 tables, Astrophysical Journal, updated to match version in press, Figure 6 shows the main result of the pape

Coneheads: Hierarchy Aware Attention

Attention networks such as transformers have achieved state-of-the-art performance in many domains. These networks rely heavily on the dot product attention operator, which computes the similarity between two points by taking their inner product. However, the inner product does not explicitly model the complex structural properties of real world datasets, such as hierarchies between data points. To remedy this, we introduce cone attention, a drop-in replacement for dot product attention based on hyperbolic entailment cones. Cone attention associates two points by the depth of their lowest common ancestor in a hierarchy defined by hyperbolic cones, which intuitively measures the divergence of two points and gives a hierarchy aware similarity score. We test cone attention on a wide variety of models and tasks and show that it improves task-level performance over dot product attention and other baselines, and is able to match dot-product attention with significantly fewer parameters. Our results suggest that cone attention is an effective way to capture hierarchical relationships when calculating attention