A Direct Algorithm for the Type Interference in the Rank 2 Fragment of the Second--Order λ-Calculus

We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the stratification of the second-order lambda-calculus by "rank" of types. We show that typability is an undecidable problem at every rank k ≥ 3 of this stratification. While it was already known that typability is decidable at rank ≤ 2, no direct and easy-to-implement algorithm was available. To design such an algorithm, we develop a new notion of reduction and show how to use it to reduce the problem of typability at rank 2 to the problem of acyclic semi-unification. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-unification.

A Note On the Statistical Difference of Small Direct Products

We demonstrate that if two probability distributions D and E of sufficiently small min-entropy have statistical difference ε, then the direct-product distributions D^l and E^l have statistical difference at least roughly ε\s√l, provided that l is sufficiently small, smaller than roughly ε^{4/3}. Previously known bounds did not work for few repetitions l, requiring l>ε^2.

Spatial Pattern Learning, Catastophic Forgetting and Optimal Rules of Synaptic Transmission

It is a neural network truth universally acknowledged, that the signal transmitted to a target node must be equal to the product of the path signal times a weight. Analysis of catastrophic forgetting by distributed codes leads to the unexpected conclusion that this universal synaptic transmission rule may not be optimal in certain neural networks. The distributed outstar, a network designed to support stable codes with fast or slow learning, generalizes the outstar network for spatial pattern learning. In the outstar, signals from a source node cause weights to learn and recall arbitrary patterns across a target field of nodes. The distributed outstar replaces the outstar source node with a source field, of arbitrarily many nodes, where the activity pattern may be arbitrarily distributed or compressed. Learning proceeds according to a principle of atrophy due to disuse whereby a path weight decreases in joint proportion to the transmittcd path signal and the degree of disuse of the target node. During learning, the total signal to a target node converges toward that node's activity level. Weight changes at a node are apportioned according to the distributed pattern of converging signals three types of synaptic transmission, a product rule, a capacity rule, and a threshold rule, are examined for this system. The three rules are computationally equivalent when source field activity is maximally compressed, or winner-take-all when source field activity is distributed, catastrophic forgetting may occur. Only the threshold rule solves this problem. Analysis of spatial pattern learning by distributed codes thereby leads to the conjecture that the optimal unit of long-term memory in such a system is a subtractive threshold, rather than a multiplicative weight.

The Role of Edges and Line-Ends in Illusory Contour Formation

Illusory contours can be induced along directions approximately collinear to edges or approximately perpendicular to the ends of lines. Using a rating scale procedure we explored the relation between the two types of inducers by systematically varying the thickness of inducing elements to result; in varying amounts of "edge-like" or "line-like" induction. Inducers for om illusory figures consisted of concentric rings with arcs missing. Observers judged the clarity and brightness of illusory figures as the number of arcs, their thicknesses, and spacings were parametrically varied. Degree of clarity and amount of induced brightness were both found to be inverted-U functions of the number of arcs. These results mandate that any valid model of illusory contour formation must account for interference effects between parallel lines or between those neural units responsible for completion of boundary signals in directions perpendicular to the ends of thin lines. Line width was found to have an effect on both clarity and brightness, a finding inconsistent with those models which employ only completion perpendicular to inducer orientation.

The Role of Edges and Line-Ends in Illusory Contour Formation

Illusory contours can be induced along direction approximately collinear to edges or approximately perpendicular to the ends of lines. Using a rating scale procedure we explored the relation between the two types of inducers by systematically varying the thickness of inducing elements to result in varying amounts of "edge-like" or "line-like" induction. Inducers for our illusory figures consisted of concentric rings with arcs missing. Observers judged the clarity and brightness of illusory figures as the number of arcs, their thicknesses, and spacings were parametrically varied. Degree of clarity and amount of induced brightness were both found to be inverted-U functions of the number of arcs. These results mandate that any valid model of illusory contour formation must account for interference effects between parallel lines or between those neural units responsible for completion of boundary signals in directions perpendicular to the ends of thin lines. Line width was found to have an efFect on both clarity and brightness, a finding inconsistent with those models which employ only completion perpendicular to inducer orientation.