The Borel Complexity of Isomorphism for Some First Order Theories

In this work we consider several instances of the following problem: "how complicated can the isomorphism relation for countable models be?"' Using the Borel reducibility framework, we investigate this question with regard to the space of countable models of particular complete first-order theories. We also investigate to what extent this complexity is mirrored in the number of back-and-forth inequivalent models of the theory. We consider this question for two large and related classes of theories. First, we consider o-minimal theories, showing that if T is o-minimal, then the isomorphism relation is either Borel complete or Borel. Further, if it is Borel, we characterize exactly which values can occur, and when they occur. In all cases Borel completeness implies lambda-Borel completeness for all lambda. Second, we consider colored linear orders, which are (complete theories of) a linear order expanded by countably many unary predicates. We discover the same characterization as with o-minimal theories, taking the same values, with the exception that all finite values are possible except two. We characterize exactly when each possibility occurs, which is similar to the o-minimal case. Additionally, we extend Schirrman's theorem, showing that if the language is finite, then T is countably categorical or Borel complete. As before, in all cases Borel completeness implies lambda-Borel completeness for all lambda.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

Computational Foundations for Safe and Efficient Human-Robot Collaboration in Assembly Cells

Human and robots have complementary strengths in performing assembly operations. Humans are very good at perception tasks in unstructured environments. They are able to recognize and locate a part from a box of miscellaneous parts. They are also very good at complex manipulation in tight spaces. The sensory characteristics of the humans, motor abilities, knowledge and skills give the humans the ability to react to unexpected situations and resolve problems quickly. In contrast, robots are very good at pick and place operations and highly repeatable in placement tasks. Robots can perform tasks at high speeds and still maintain precision in their operations. Robots can also operate for long periods of times. Robots are also very good at applying high forces and torques. Typically, robots are used in mass production. Small batch and custom production operations predominantly use manual labor. The high labor cost is making it difficult for small and medium manufacturers to remain cost competitive in high wage markets. These manufactures are mainly involved in small batch and custom production. They need to find a way to reduce the labor cost in assembly operations. Purely robotic cells will not be able to provide them the necessary flexibility. Creating hybrid cells where humans and robots can collaborate in close physical proximities is a potential solution. The underlying idea behind such cells is to decompose assembly operations into tasks such that humans and robots can collaborate by performing sub-tasks that are suitable for them. Realizing hybrid cells that enable effective human and robot collaboration is challenging. This dissertation addresses the following three computational issues involved in developing and utilizing hybrid assembly cells: - We should be able to automatically generate plans to operate hybrid assembly cells to ensure efficient cell operation. This requires generating feasible assembly sequences and instructions for robots and human operators, respectively. Automated planning poses the following two challenges. First, generating operation plans for complex assemblies is challenging. The complexity can come due to the combinatorial explosion caused by the size of the assembly or the complex paths needed to perform the assembly. Second, generating feasible plans requires accounting for robot and human motion constraints. The first objective of the dissertation is to develop the underlying computational foundations for automatically generating plans for the operation of hybrid cells. It addresses both assembly complexity and motion constraints issues. - The collaboration between humans and robots in the assembly cell will only be practical if human safety can be ensured during the assembly tasks that require collaboration between humans and robots. The second objective of the dissertation is to evaluate different options for real-time monitoring of the state of human operator with respect to the robot and develop strategies for taking appropriate measures to ensure human safety when the planned move by the robot may compromise the safety of the human operator. In order to be competitive in the market, the developed solution will have to include considerations about cost without significantly compromising quality. - In the envisioned hybrid cell, we will be relying on human operators to bring the part into the cell. If the human operator makes an error in selecting the part or fails to place it correctly, the robot will be unable to correctly perform the task assigned to it. If the error goes undetected, it can lead to a defective product and inefficiencies in the cell operation. The reason for human error can be either confusion due to poor quality instructions or human operator not paying adequate attention to the instructions. In order to ensure smooth and error-free operation of the cell, we will need to monitor the state of the assembly operations in the cell. The third objective of the dissertation is to identify and track parts in the cell and automatically generate instructions for taking corrective actions if a human operator deviates from the selected plan. Potential corrective actions may involve re-planning if it is possible to continue assembly from the current state. Corrective actions may also involve issuing warning and generating instructions to undo the current task.

Time-Reversal Massive Multipath Effect and Bandwidth Heterogeneity

The proliferation of new mobile communication devices, such as smartphones and tablets, has led to an exponential growth in network traffic. The demand for supporting the fast-growing consumer data rates urges the wireless service providers and researchers to seek a new efficient radio access technology, which is the so-called 5G technology, beyond what current 4G LTE can provide. On the other hand, ubiquitous RFID tags, sensors, actuators, mobile phones and etc. cut across many areas of modern-day living, which offers the ability to measure, infer and understand the environmental indicators. The proliferation of these devices creates the term of the Internet of Things (IoT). For the researchers and engineers in the field of wireless communication, the exploration of new effective techniques to support 5G communication and the IoT becomes an urgent task, which not only leads to fruitful research but also enhance the quality of our everyday life. Massive MIMO, which has shown the great potential in improving the achievable rate with a very large number of antennas, has become a popular candidate. However, the requirement of deploying a large number of antennas at the base station may not be feasible in indoor scenarios. Does there exist a good alternative that can achieve similar system performance to massive MIMO for indoor environment? In this dissertation, we address this question by proposing the time-reversal technique as a counterpart of massive MIMO in indoor scenario with the massive multipath effect. It is well known that radio signals will experience many multipaths due to the reflection from various scatters, especially in indoor environments. The traditional TR waveform is able to create a focusing effect at the intended receiver with very low transmitter complexity in a severe multipath channel. TR's focusing effect is in essence a spatial-temporal resonance effect that brings all the multipaths to arrive at a particular location at a specific moment. We show that by using time-reversal signal processing, with a sufficiently large bandwidth, one can harvest the massive multipaths naturally existing in a rich-scattering environment to form a large number of virtual antennas and achieve the desired massive multipath effect with a single antenna. Further, we explore the optimal bandwidth for TR system to achieve maximal spectral efficiency. Through evaluating the spectral efficiency, the optimal bandwidth for TR system is found determined by the system parameters, e.g., the number of users and backoff factor, instead of the waveform types. Moreover, we investigate the tradeoff between complexity and performance through establishing a generalized relationship between the system performance and waveform quantization in a practical communication system. It is shown that a 4-bit quantized waveforms can be used to achieve the similar bit-error-rate compared to the TR system with perfect precision waveforms. Besides 5G technology, Internet of Things (IoT) is another terminology that recently attracts more and more attention from both academia and industry. In the second part of this dissertation, the heterogeneity issue within the IoT is explored. One of the significant heterogeneity considering the massive amount of devices in the IoT is the device heterogeneity, i.e., the heterogeneous bandwidths and associated radio-frequency (RF) components. The traditional middleware techniques result in the fragmentation of the whole network, hampering the objects interoperability and slowing down the development of a unified reference model for the IoT. We propose a novel TR-based heterogeneous system, which can address the bandwidth heterogeneity and maintain the benefit of TR at the same time. The increase of complexity in the proposed system lies in the digital processing at the access point (AP), instead of at the devices' ends, which can be easily handled with more powerful digital signal processor (DSP). Meanwhile, the complexity of the terminal devices stays low and therefore satisfies the low-complexity and scalability requirement of the IoT. Since there is no middleware in the proposed scheme and the additional physical layer complexity concentrates on the AP side, the proposed heterogeneous TR system better satisfies the low-complexity and energy-efficiency requirement for the terminal devices (TDs) compared with the middleware approach.