Exact Numerical Processing

Paper submitted to Euromicro Symposium on Digital Systems Design (DSD), Belek-Antalya, Turkey, 2003.

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Webers Law signature, and was independent of conventional number knowledge.

There is more than "more is up": Hand and foot responses reverse the vertical association of number magnitudes

Recent research in cognitive sciences shows a growing interest in spatial-numerical associations. The horizontal SNARC (spatial-numerical association of response codes) effect is defined by faster left-sided responses to small numbers and faster right-sided responses to large numbers in a parity judgment task. In this study we investigated whether there is also a SNARC effect for upper and lower responses. The grounded cognition approach suggests that the universal experience of "more is up" serves as a robust frame of reference for vertical number representation. In line with this view, lower hand responses to small numbers were faster than to large numbers (Experiment 1). Interestingly, the vertical SNARC effect reversed when the lower responses were given by foot instead of the hand (Experiments 2, 3, and 4). We found faster upper (hand) responses to small numbers and faster lower (foot) responses to large numbers. Additional experiments showed that spatial factors cannot account for the reversal of the vertical SNARC effect (Experiments 4 and 5). Our results question the view of "more is up" as a robust frame of reference for spatial-numerical associations. We discuss our results within a hierarchical framework of numerical cognition and point to a possible link between effectors and number representation.

A preliminary study of open learner model representation formats to support formative assessment

Open learner models provide a way of showing teachers and students the learning progress of a student against expectations. Creating an effective interface to present a learner model is an important part in open learner modeling. Usefulness of the learner model relies on using a clear andeffective representation format to facilitate users’understanding of the information presented. In this paper, wepropose a range of open learner model representation formatsto display students’ learning task status and achievement interms of learning outcomes. We investigate different types ofrepresentation formats and data useful for users to inspect thelearner model. An interface design prototype has been built toprovide potential users with an evaluation platform thatenables investigation of these aspects. The results obtained willallow us to develop a more effective new open learner modelvisualisation tool.

Towards flexible parsing of structured textual model representations

Existing parsers for textual model representation formats such as XMI and HUTN are unforgiving and fail upon even the smallest inconsistency between the structure and naming of metamodel elements and the contents of serialised models. In this paper, we demonstrate how a fuzzy parsing approach can transparently and automatically resolve a number of these inconsistencies, and how it can eventually turn XML into a human-readable and editable textual model representation format for particular classes of models.

An RNS based transform Architecture for H.264/AVC

This paper presents the architecture and the VHDL design of an integer 2-D DCT used in the H.264/AVC. The 2-D DCT computation is performed by exploiting it’s orthogonality and separability property. The symmetry of the forward and inverse transform is used in this implementation. To reduce the computation overhead for the addition, subtraction and multiplication operations, we analyze the suitability of carry-free position independent residue number system (RNS) for the implementation of 2-D DCT. The implementation has been carried out in VHDL for Altera FPGA. We used the negative number representation in RNS, bit width analysis of the transforms and dedicated registers present in the Logic element of the FPGA to optimize the area. The complexity and efficiency analysis show that the proposed architecture could provide higher through-put.

The Line Spectral Frequency Model of a Finite-Length Sequence

The line spectral frequency (LSF) of a causal finite length sequence is a frequency at which the spectrum of the sequence annihilates or the magnitude spectrum has a spectral null. A causal finite-length sequencewith (L + 1) samples having exactly L-LSFs, is referred as an Annihilating (AH) sequence. Using some spectral properties of finite-length sequences, and some model parameters, we develop spectral decomposition structures, which are used to translate any finite-length sequence to an equivalent set of AH-sequences defined by LSFs and some complex constants. This alternate representation format of any finite-length sequence is referred as its LSF-Model. For a finite-length sequence, one can obtain multiple LSF-Models by varying the model parameters. The LSF-Model, in time domain can be used to synthesize any arbitrary causal finite-length sequence in terms of its characteristic AH-sequences. In the frequency domain, the LSF-Model can be used to obtain the spectral samples of the sequence as a linear combination of spectra of its characteristic AH-sequences. We also summarize the utility of the LSF-Model in practical discrete signal processing systems.

A new topological index for the Changchun institute of applied chemistry C-13 NMR information system

A method to assign a single number representation for each atom (node) in a molecular graph, Atomic IDentification (AID) number, is proposed based on the counts of weighted paths terminated on that atom. Then, a new topological index, Molecular IDentification (MID) number is developed from AID. The MID is tested systematically, over half a million of structures are examined, and MID shows high discrimination for various structural isomers. Thus it can be used for documentation in the Changchun Institute of Chemistry C-13 NMR information system.

Risk-informed decision-making in the presence of epistemic uncertainty

An important issue in risk analysis is the distinction between epistemic and aleatory uncertainties. In this paper, the use of distinct representation formats for aleatory and epistemic uncertainties is advocated, the latter being modelled by sets of possible values. Modern uncertainty theories based on convex sets of probabilities are known to be instrumental for hybrid representations where aleatory and epistemic components of uncertainty remain distinct. Simple uncertainty representation techniques based on fuzzy intervals and p-boxes are used in practice. This paper outlines a risk analysis methodology from elicitation of knowledge about parameters to decision. It proposes an elicitation methodology where the chosen representation format depends on the nature and the amount of available information. Uncertainty propagation methods then blend Monte Carlo simulation and interval analysis techniques. Nevertheless, results provided by these techniques, often in terms of probability intervals, may be too complex to interpret for a decision-maker and we, therefore, propose to compute a unique indicator of the likelihood of risk, called confidence index. It explicitly accounts for the decisionmaker’s attitude in the face of ambiguity. This step takes place at the end of the risk analysis process, when no further collection of evidence is possible that might reduce the ambiguity due to epistemic uncertainty. This last feature stands in contrast with the Bayesian methodology, where epistemic uncertainties on input parameters are modelled by single subjective probabilities at the beginning of the risk analysis process.

SYSTOLIC IIR FILTERS WITH BIT LEVEL PIPELINING.

A novel bit-level systolic array architecture for implementing first-order IIR filter sections is presented. A latency of only two clock cycles is achieved by using a radix-4 redundant number representation, performing the recursive computation most-significant-digit first, and feeding back each digit of the result as soon as it is available.

Bit-level systolic arrays for IIR filtering.

A novel bit-level systolic array architecture for implementing IIR (infinite-impulse response) filter sections is presented. A first-order section achieves a latency of only two clock cycles by using a radix-2 redundant number representation, performing the recursive computation most significant digit first, and feeding back each digit of the result as soon as it is available. The design is extended to produce a building block from which second- and higher-order sections can be connected.

A longitudinal analysis of estimation, counting skills, and mathematical ability across the first school year

In response to claims that the quality (and in particular linearity) of children's mental representation of number acts as a constraint on number development, we carried out a longitudinal assessment of the relationships between number line estimation, counting, and mathematical abilities. Ninety-nine 5-year-olds were tested on 4 occasions at 3 monthly intervals. Correlations between the 3 types of ability were evident, but while the quality of children's estimations changed over time and performance on the mathematical tasks improved over the same period, changes in one were not associated with changes in the other. In contrast to the earlier claims that the linearity of number representation is potentially a unique contributor to children's mathematical development, the data suggest that this variable is not significantly privileged in its impact over and above simple procedural number skills. We propose that both early arithmetic success and estimating skill are bound closely to developments in counting ability.

A high performance IIR digital filter chip

The design of a high-performance IIR (infinite impulse response) digital filter is described. The chip architecture operates on 11-b parallel, two's complement input data with a 12-b parallel two's complement coefficient to produce a 14-b two's complement output. The chip is implemented in 1.5-µm, double-layer-metal CMOS technology, consumes 0.5 W, and can operate up to 15 Msample/s. The main component of the system is a fine-grained systolic array that internally is based on a signed binary number representation (SBNR). Issues addressed include testing, clock distribution, and circuitry for conversion between two's complement and SBNR.

Agraphs: definição, implementação e suas ferramentas

Programs manipulate information. However, information is abstract in nature and needs to be represented, usually by data structures, making it possible to be manipulated. This work presents the AGraphs, a representation and exchange format of the data that uses typed directed graphs with a simulation of hyperedges and hierarchical graphs. Associated to the AGraphs format there is a manipulation library with a simple programming interface, tailored to the language being represented. The AGraphs format in ad-hoc manner was used as representation format in tools developed at UFRN, and, to make it more usable in other tools, an accurate description and the development of support tools was necessary. These accurate description and tools have been developed and are described in this work. This work compares the AGraphs format with other representation and exchange formats (e.g ATerms, GDL, GraphML, GraX, GXL and XML). The main objective this comparison is to capture important characteristics and where the AGraphs concepts can still evolve

Discalculia do desenvolvimento: avaliação da representação numérica pela ZAREKI-R

O presente estudo investigou aspectos da representação numérica (processamento numérico e cálculo) e memória operacional de crianças com transtornos de aprendizagem. Participaram 30 crianças de idade entre 9 e 10 anos, ambos os gêneros, divididas em dois grupos: sem dificuldade em aritmética (SDA; N=11) e com dificuldade em aritmética (CDA; N=19), avaliadas pela ZAREKI-R, Matrizes Coloridas de Raven, o Blocos de Corsi e o BCPR. Crianças CDA exibiram escores levemente mais baixos que as SDA quanto ao nível intelectual e nos Blocos de Corsi. Na ZAREKI-R apresentaram prejuízo nos subtestes ditado de números, cálculo mental, problemas aritméticos e total. Crianças CDA apresentaram déficits específicos em memória operacional visuoespacial e comprometimento em processamento numérico e cálculo, compatível com discalculia do desenvolvimento.