Componenti spaziali della rappresentazione cognitiva della grandezza del numero

The humans process the numbers in a similar way to animals. There are countless studies in which similar performance between animals and humans (adults and/or children) are reported. Three models have been developed to explain the cognitive mechanisms underlying the number processing. The triple-code model (Dehaene, 1992) posits an mental number line as preferred way to represent magnitude. The mental number line has three particular effects: the distance, the magnitude and the SNARC effects. The SNARC effect shows a spatial association between number and space representations. In other words, the small numbers are related to left space while large numbers are related to right space. Recently a vertical SNARC effect has been found (Ito & Hatta, 2004; Schwarz & Keus, 2004), reflecting a space-related bottom-to-up representation of numbers. The magnitude representations horizontally and vertically could influence the subject performance in explicit and implicit digit tasks. The goal of this research project aimed to investigate the spatial components of number representation using different experimental designs and tasks. The experiment 1 focused on horizontal and vertical number representations in a within- and between-subjects designs in a parity and magnitude comparative tasks, presenting positive or negative Arabic digits (1-9 without 5). The experiment 1A replied the SNARC and distance effects in both spatial arrangements. The experiment 1B showed an horizontal reversed SNARC effect in both tasks while a vertical reversed SNARC effect was found only in comparative task. In the experiment 1C two groups of subjects performed both tasks in two different instruction-responding hand assignments with positive numbers. The results did not show any significant differences between two assignments, even if the vertical number line seemed to be more flexible respect to horizontal one. On the whole the experiment 1 seemed to demonstrate a contextual (i.e. task set) influences of the nature of the SNARC effect. The experiment 2 focused on the effect of horizontal and vertical number representations on spatial biases in a paper-and-pencil bisecting tasks. In the experiment 2A the participants were requested to bisect physical and number (2 or 9) lines horizontally and vertically. The findings demonstrated that digit 9 strings tended to generate a more rightward bias comparing with digit 2 strings horizontally. However in vertical condition the digit 2 strings generated a more upperward bias respect to digit 9 strings, suggesting a top-to-bottom number line. In the experiment 2B the participants were asked to bisect lines flanked by numbers (i.e. 1 or 7) in four spatial arrangements: horizontal, vertical, right-diagonal and left-diagonal lines. Four number conditions were created according to congruent or incongruent number line representation: 1-1, 1-7, 7-1 and 7-7. The main results showed a more reliable rightward bias in horizontal congruent condition (1-7) respect to incongruent condition (7-1). Vertically the incongruent condition (1-7) determined a significant bias towards bottom side of line respect to congruent condition (7-1). The experiment 2 suggested a more rigid horizontal number line while in vertical condition the number representation could be more flexible. In the experiment 3 we adopted the materials of experiment 2B in order to find a number line effect on temporal (motor) performance. The participants were presented horizontal, vertical, rightdiagonal and left-diagonal lines flanked by the same digits (i.e. 1-1 or 7-7) or by different digits (i.e. 1-7 or 7-1). The digits were spatially congruent or incongruent with their respective hypothesized mental representations. Participants were instructed to touch the lines either close to the large digit, or close to the small digit, or to bisected the lines. Number processing influenced movement execution more than movement planning. Number congruency influenced spatial biases mostly along the horizontal but also along the vertical dimension. These results support a two-dimensional magnitude representation. Finally, the experiment 4 addressed the visuo-spatial manipulation of number representations for accessing and retrieval arithmetic facts. The participants were requested to perform a number-matching and an addition verification tasks. The findings showed an interference effect between sum-nodes and neutral-nodes only with an horizontal presentation of digit-cues, in number-matching tasks. In the addition verification task, the performance was similar for horizontal and vertical presentations of arithmetic problems. In conclusion the data seemed to show an automatic activation of horizontal number line also used to retrieval arithmetic facts. The horizontal number line seemed to be more rigid and the preferred way to order number from left-to-right. A possible explanation could be the left-to-right direction for reading and writing. The vertical number line seemed to be more flexible and more dependent from the tasks, reflecting perhaps several example in the environment representing numbers either from bottom-to-top or from top-to-bottom. However the bottom-to-top number line seemed to be activated by explicit task demands.

Constraint based methods for allocation and scheduling of periodic applications

This work presents exact algorithms for the Resource Allocation and Cyclic Scheduling Problems (RA&CSPs). Cyclic Scheduling Problems arise in a number of application areas, such as in hoist scheduling, mass production, compiler design (implementing scheduling loops on parallel architectures), software pipelining, and in embedded system design. The RA&CS problem concerns time and resource assignment to a set of activities, to be indefinitely repeated, subject to precedence and resource capacity constraints. In this work we present two constraint programming frameworks facing two different types of cyclic problems. In first instance, we consider the disjunctive RA&CSP, where the allocation problem considers unary resources. Instances are described through the Synchronous Data-flow (SDF) Model of Computation. The key problem of finding a maximum-throughput allocation and scheduling of Synchronous Data-Flow graphs onto a multi-core architecture is NP-hard and has been traditionally solved by means of heuristic (incomplete) algorithms. We propose an exact (complete) algorithm for the computation of a maximum-throughput mapping of applications specified as SDFG onto multi-core architectures. Results show that the approach can handle realistic instances in terms of size and complexity. Next, we tackle the Cyclic Resource-Constrained Scheduling Problem (i.e. CRCSP). We propose a Constraint Programming approach based on modular arithmetic: in particular, we introduce a modular precedence constraint and a global cumulative constraint along with their filtering algorithms. Many traditional approaches to cyclic scheduling operate by fixing the period value and then solving a linear problem in a generate-and-test fashion. Conversely, our technique is based on a non-linear model and tackles the problem as a whole: the period value is inferred from the scheduling decisions. The proposed approaches have been tested on a number of non-trivial synthetic instances and on a set of realistic industrial instances achieving good results on practical size problem.

Mathematical skills: intergenerational features and relationships with cognitive and linguistic abilities

This thesis aimed to investigate the cognitive underpinnings of math skills, with particular reference to cognitive, and linguistic markers, core mechanisms of number processing and environmental variables. In particular, the issue of intergenerational transmission of math skills has been deepened, comparing parents’ and children’s basic and formal math abilities. This pattern of relationships amongst these has been considered in two different age ranges, preschool and primary school children. In the first chapter, a general introduction on mathematical skills is offered, with a description of some seminal works up to recent studies and latest findings. The first chapter concludes with a review of studies about the influence of environmental variables. In particular, a review of studies about home numeracy and intergenerational transmission is examined. The first study analyzed the relationship between mathematical skills of children attending primary school and those of their mothers. The objective of this study was to understand the influence of mothers' math abilities on those of their children. In the second study, the relationship between parents’ and children numerical processing has been examined in a sample of preschool children. The goal was to understand how mathematical skills of parents were relevant for the development of the numerical skills of children, taking into account children’s cognitive and linguistic skills as well as the role of home numeracy. The third study had the objective of investigating whether the verbal and nonverbal cognitive skills presumed to underlie arithmetic are also related to reading. Primary school children were administered measures of reading and arithmetic to understand the relationships between these two abilities and testing for possible shared cognitive markers. Finally, in the general discussion a summary of main findings across the study is presented, together with clinical and theoretical implications.

Flexible Computing Systems For AI Acceleration At The Extreme Edge Of The IoT

Embedding intelligence in extreme edge devices allows distilling raw data acquired from sensors into actionable information, directly on IoT end-nodes. This computing paradigm, in which end-nodes no longer depend entirely on the Cloud, offers undeniable benefits, driving a large research area (TinyML) to deploy leading Machine Learning (ML) algorithms on micro-controller class of devices. To fit the limited memory storage capability of these tiny platforms, full-precision Deep Neural Networks (DNNs) are compressed by representing their data down to byte and sub-byte formats, in the integer domain. However, the current generation of micro-controller systems can barely cope with the computing requirements of QNNs. This thesis tackles the challenge from many perspectives, presenting solutions both at software and hardware levels, exploiting parallelism, heterogeneity and software programmability to guarantee high flexibility and high energy-performance proportionality. The first contribution, PULP-NN, is an optimized software computing library for QNN inference on parallel ultra-low-power (PULP) clusters of RISC-V processors, showing one order of magnitude improvements in performance and energy efficiency, compared to current State-of-the-Art (SoA) STM32 micro-controller systems (MCUs) based on ARM Cortex-M cores. The second contribution is XpulpNN, a set of RISC-V domain specific instruction set architecture (ISA) extensions to deal with sub-byte integer arithmetic computation. The solution, including the ISA extensions and the micro-architecture to support them, achieves energy efficiency comparable with dedicated DNN accelerators and surpasses the efficiency of SoA ARM Cortex-M based MCUs, such as the low-end STM32M4 and the high-end STM32H7 devices, by up to three orders of magnitude. To overcome the Von Neumann bottleneck while guaranteeing the highest flexibility, the final contribution integrates an Analog In-Memory Computing accelerator into the PULP cluster, creating a fully programmable heterogeneous fabric that demonstrates end-to-end inference capabilities of SoA MobileNetV2 models, showing two orders of magnitude performance improvements over current SoA analog/digital solutions.

Towards a logical foundation of randomized computation

This dissertation investigates the relations between logic and TCS in the probabilistic setting. It is motivated by two main considerations. On the one hand, since their appearance in the 1960s-1970s, probabilistic models have become increasingly pervasive in several fast-growing areas of CS. On the other, the study and development of (deterministic) computational models has considerably benefitted from the mutual interchanges between logic and CS. Nevertheless, probabilistic computation was only marginally touched by such fruitful interactions. The goal of this thesis is precisely to (start) bring(ing) this gap, by developing logical systems corresponding to specific aspects of randomized computation and, therefore, by generalizing standard achievements to the probabilistic realm. To do so, our key ingredient is the introduction of new, measure-sensitive quantifiers associated with quantitative interpretations. The dissertation is tripartite. In the first part, we focus on the relation between logic and counting complexity classes. We show that, due to our classical counting propositional logic, it is possible to generalize to counting classes, the standard results by Cook and Meyer and Stockmeyer linking propositional logic and the polynomial hierarchy. Indeed, we show that the validity problem for counting-quantified formulae captures the corresponding level in Wagner's hierarchy. In the second part, we consider programming language theory. Type systems for randomized \lambda-calculi, also guaranteeing various forms of termination properties, were introduced in the last decades, but these are not "logically oriented" and no Curry-Howard correspondence is known for them. Following intuitions coming from counting logics, we define the first probabilistic version of the correspondence. Finally, we consider the relationship between arithmetic and computation. We present a quantitative extension of the language of arithmetic able to formalize basic results from probability theory. This language is also our starting point to define randomized bounded theories and, so, to generalize canonical results by Buss.