Programming Examples Needing Polymorphic Recursion

Inferring types for polymorphic recursive function definitions (abbreviated to polymorphic recursion) is a recurring topic on the mailing lists of popular typed programming languages. This is despite the fact that type inference for polymorphic recursion using for all-types has been proved undecidable. This report presents several programming examples involving polymorphic recursion and determines their typability under various type systems, including the Hindley-Milner system, an intersection-type system, and extensions of these two. The goal of this report is to show that many of these examples are typable using a system of intersection types as an alternative form of polymorphism. By accomplishing this, we hope to lay the foundation for future research into a decidable intersection-type inference algorithm. We do not provide a comprehensive survey of type systems appropriate for polymorphic recursion, with or without type annotations inserted in the source language. Rather, we focus on examples for which types may be inferred without type annotations.

An Online Distributed Algorithm for Inferring Policy Routing Configurations

We present an online distributed algorithm, the Causation Logging Algorithm (CLA), in which Autonomous Systems (ASes) in the Internet individually report route oscillations/flaps they experience to a central Internet Routing Registry (IRR). The IRR aggregates these reports and may observe what we call causation chains where each node on the chain caused a route flap at the next node along the chain. A chain may also have a causation cycle. The type of an observed causation chain/cycle allows the IRR to infer the underlying policy routing configuration (i.e., the system of economic relationships and constraints on route/path preferences). Our algorithm is based on a formal policy routing model that captures the propagation dynamics of route flaps under arbitrary changes in topology or path preferences. We derive invariant properties of causation chains/cycles for ASes which conform to economic relationships based on the popular Gao-Rexford model. The Gao-Rexford model is known to be safe in the sense that the system always converges to a stable set of paths under static conditions. Our CLA algorithm recovers the type/property of an observed causation chain of an underlying system and determines whether it conforms to the safe economic Gao-Rexford model. Causes for nonconformity can be diagnosed by comparing the properties of the causation chains with those predicted from different variants of the Gao-Rexford model.

On the Complexity of Quantum ACC

For any q > 1, let MOD_q be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q,t > 1, MOD_q is equivalent to MOD_t (up to constant depth). Based on the case q=2, Moore has shown that quantum analogs of AC^(0), ACC[q], and ACC, denoted QAC^(0)_wf, QACC[2], QACC respectively, define the same class of operators, leaving q > 2 as an open question. Our result resolves this question, implying that QAC^(0)_wf = QACC[q] = QACC for all q. We also prove the first upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC (both for arbitrary complex amplitudes) and BQACC (for rational number amplitudes) and show that they are all contained in TC^(0). To do this, we show that a TC^(0) circuit can keep track of the amplitudes of the state resulting from the application of a QACC operator using a constant width polynomial size tensor sum. In order to accomplish this, we also show that TC^(0) can perform iterated addition and multiplication in certain field extensions.

A Neural Model of Surface Perception: Lightness, Anchoring, and Filling-in

This article develops a neural model of how the visual system processes natural images under variable illumination conditions to generate surface lightness percepts. Previous models have clarified how the brain can compute the relative contrast of images from variably illuminate scenes. How the brain determines an absolute lightness scale that "anchors" percepts of surface lightness to us the full dynamic range of neurons remains an unsolved problem. Lightness anchoring properties include articulation, insulation, configuration, and are effects. The model quantatively simulates these and other lightness data such as discounting the illuminant, the double brilliant illusion, lightness constancy and contrast, Mondrian contrast constancy, and the Craik-O'Brien-Cornsweet illusion. The model also clarifies the functional significance for lightness perception of anatomical and neurophysiological data, including gain control at retinal photoreceptors, and spatioal contrast adaptation at the negative feedback circuit between the inner segment of photoreceptors and interacting horizontal cells. The model retina can hereby adjust its sensitivity to input intensities ranging from dim moonlight to dazzling sunlight. A later model cortical processing stages, boundary representations gate the filling-in of surface lightness via long-range horizontal connections. Variants of this filling-in mechanism run 100-1000 times faster than diffusion mechanisms of previous biological filling-in models, and shows how filling-in can occur at realistic speeds. A new anchoring mechanism called the Blurred-Highest-Luminance-As-White (BHLAW) rule helps simulate how surface lightness becomes sensitive to the spatial scale of objects in a scene. The model is also able to process natural images under variable lighting conditions.

A Neuromorphic Model for Achromatic and Chromatic Surface Representation of Natural Images

This study develops a neuromorphic model of human lightness perception that is inspired by how the mammalian visual system is designed for this function. It is known that biological visual representations can adapt to a billion-fold change in luminance. How such a system determines absolute lightness under varying illumination conditions to generate a consistent interpretation of surface lightness remains an unsolved problem. Such a process, called "anchoring" of lightness, has properties including articulation, insulation, configuration, and area effects. The model quantitatively simulates such psychophysical lightness data, as well as other data such as discounting the illuminant, the double brilliant illusion, and lightness constancy and contrast effects. The model retina embodies gain control at retinal photoreceptors, and spatial contrast adaptation at the negative feedback circuit between mechanisms that model the inner segment of photoreceptors and interacting horizontal cells. The model can thereby adjust its sensitivity to input intensities ranging from dim moonlight to dazzling sunlight. A new anchoring mechanism, called the Blurred-Highest-Luminance-As-White (BHLAW) rule, helps simulate how surface lightness becomes sensitive to the spatial scale of objects in a scene. The model is also able to process natural color images under variable lighting conditions, and is compared with the popular RETINEX model.

Orientational Harmonic Model for Rotation, Translation, and Scale Invariant Pattern Representation in Biological Vision

The proposed model, called the combinatorial and competitive spatio-temporal memory or CCSTM, provides an elegant solution to the general problem of having to store and recall spatio-temporal patterns in which states or sequences of states can recur in various contexts. For example, fig. 1 shows two state sequences that have a common subsequence, C and D. The CCSTM assumes that any state has a distributed representation as a collection of features. Each feature has an associated competitive module (CM) containing K cells. On any given occurrence of a particular feature, A, exactly one of the cells in CMA will be chosen to represent it. It is the particular set of cells active on the previous time step that determines which cells are chosen to represent instances of their associated features on the current time step. If we assume that typically S features are active in any state then any state has K^S different neural representations. This huge space of possible neural representations of any state is what underlies the model's ability to store and recall numerous context-sensitive state sequences. The purpose of this paper is simply to describe this mechanism.

Fuzzy ART

Adaptive Resonance Theory (ART) models are real-time neural networks for category learning, pattern recognition, and prediction. Unsupervised fuzzy ART and supervised fuzzy ARTMAP synthesize fuzzy logic and ART networks by exploiting the formal similarity between the computations of fuzzy subsethood and the dynamics of ART category choice, search, and learning. Fuzzy ART self-organizes stable recognition categories in response to arbitrary sequences of analog or binary input patterns. It generalizes the binary ART 1 model, replacing the set-theoretic: intersection (∩) with the fuzzy intersection (∧), or component-wise minimum. A normalization procedure called complement coding leads to a symmetric: theory in which the fuzzy inter:>ec:tion and the fuzzy union (∨), or component-wise maximum, play complementary roles. Complement coding preserves individual feature amplitudes while normalizing the input vector, and prevents a potential category proliferation problem. Adaptive weights :otart equal to one and can only decrease in time. A geometric interpretation of fuzzy AHT represents each category as a box that increases in size as weights decrease. A matching criterion controls search, determining how close an input and a learned representation must be for a category to accept the input as a new exemplar. A vigilance parameter (p) sets the matching criterion and determines how finely or coarsely an ART system will partition inputs. High vigilance creates fine categories, represented by small boxes. Learning stops when boxes cover the input space. With fast learning, fixed vigilance, and an arbitrary input set, learning stabilizes after just one presentation of each input. A fast-commit slow-recode option allows rapid learning of rare events yet buffers memories against recoding by noisy inputs. Fuzzy ARTMAP unites two fuzzy ART networks to solve supervised learning and prediction problems. A Minimax Learning Rule controls ARTMAP category structure, conjointly minimizing predictive error and maximizing code compression. Low vigilance maximizes compression but may therefore cause very different inputs to make the same prediction. When this coarse grouping strategy causes a predictive error, an internal match tracking control process increases vigilance just enough to correct the error. ARTMAP automatically constructs a minimal number of recognition categories, or "hidden units," to meet accuracy criteria. An ARTMAP voting strategy improves prediction by training the system several times using different orderings of the input set. Voting assigns confidence estimates to competing predictions given small, noisy, or incomplete training sets. ARPA benchmark simulations illustrate fuzzy ARTMAP dynamics. The chapter also compares fuzzy ARTMAP to Salzberg's Nested Generalized Exemplar (NGE) and to Simpson's Fuzzy Min-Max Classifier (FMMC); and concludes with a summary of ART and ARTMAP applications.