2048 is a stochastic single-player game involving 16 cells on a 4 by 4 grid, where a player chooses a direction among up, down, left, and right to obtain a score by merging two tiles with the same number located in neighboring cells along the chosen direction. This paper presents that a variant 2048-4x3 12 cells on a 4 by 3 board, one row smaller than the original, has been strongly solved. In this variant, the expected score achieved by an optimal strategy is about $50724.26$ for the most common initial states: ones with two tiles of number 2. The numbers of reachable states and afterstates are identified to be $1,152,817,492,752$ and $739,648,886,170$, respectively. The key technique is to partition state space by the sum of tile numbers on a board, which we call the age of a state. An age is invariant between a state and its successive afterstate after any valid action and is increased two or four by stochastic response from the environment. Therefore, we can partition state space by ages and enumerate all (after)states of an age depending only on states with the recent ages. Similarly, we can identify (after)state values by going along with ages in decreasing order.

Grid-based storage systems with uniformly shaped loads (e.g., containers, pallets, totes) are commonplace in logistics, industrial, and transportation domains. A key performance metric for such systems is the maximization of space utilization, which requires some loads to be placed behind or below others, preventing direct access to them. Consequently, dense storage settings bring up the challenge of determining how to place loads while minimizing costly rearrangement efforts necessary during retrieval. This paper considers the setting involving an inbound phase, during which loads arrive, followed by an outbound phase, during which loads depart. The setting is prevalent in distribution centers, automated parking garages, and container ports. In both phases, minimizing the number of rearrangement actions results in more optimal (e.g., fast, energy-efficient, etc.) operations. In contrast to previous work focusing on stack-based systems, this effort examines the case where loads can be freely moved along the grid, e.g., by a mobile robot, expanding the range of possible motions. We establish that for a range of scenarios, such as having limited prior knowledge of the loads' arrival sequences or grids with a narrow opening, a (best possible) rearrangement-free solution always exists, including when the loads fill the grid to its capacity. In particular, when the sequences are fully known, we establish an intriguing characterization showing that rearrangement can always be avoided if and only if the open side of the grid (used to access the storage) is at least 3 cells wide. We further discuss useful practical implications of our solutions.

Quantum Tiq-Taq-Toe is a well-known benchmark and playground for both quantum computing and machine learning. Despite its popularity, no reinforcement learning (RL) methods have been applied to Quantum Tiq-Taq-Toe. Although there has been some research on Quantum Chess this game is significantly more complex in terms of computation and analysis. Therefore, we study the combination of quantum computing and reinforcement learning in Quantum Tiq-Taq-Toe, which may serve as an accessible testbed for the integration of both fields. Quantum games are challenging to represent classically due to their inherent partial observability and the potential for exponential state complexity. In Quantum Tiq-Taq-Toe, states are observed through Measurement (a 3x3 matrix of state probabilities) and Move History (a 9x9 matrix of entanglement relations), making strategy complex as each move can collapse the quantum state.

We introduce and analyze a model for self-reconfigurable robots made up of unit-cube modules. Compared to past models, our model aims to newly capture two important practical aspects of real-world robots. First, modules often do not occupy an exact unit cube, but rather have features like bumps extending outside the allotted space so that modules can interlock. Thus, for example, our model forbids modules from squeezing in between two other modules that are one unit distance apart. Second, our model captures the practical scenario of many passive modules assembled by a single robot, instead of requiring all modules to be able to move on their own. We prove two universality results. First, with a supply of auxiliary modules, we show that any connected polycube structure can be constructed by a carefully aligned plane sweep. Second, without additional modules, we show how to construct any structure for which a natural notion of external feature size is at least a constant; this property largely consolidates forbidden-pattern properties used in previous works on reconfigurable modular robots.

Monte Carlo Search gives excellent results in multiple difficult combinatorial problems. Using a prior to perform non uniform playouts during the search improves a lot the results compared to uniform playouts. Handmade heuristics tailored to the combinatorial problem are often used as priors. We propose a method to automatically compute a prior. It uses statistics on solved problems. It is a simple and general method that incurs no computational cost at playout time and that brings large performance gains. The method is applied to three difficult combinatorial problems: Latin Square Completion, Kakuro, and Inverse RNA Folding.

Deep Neural Networks have achieved great success in some of the complex tasks that humans can do with ease. These include image recognition/classification, natural language processing, game playing etc. However, modern Neural Networks fail or perform poorly when trained on tasks that can be solved easily using backtracking and traditional algorithms. Therefore, we use the architecture of the Neuro Logic Machine (NLM) and extend its functionality to solve a 9X9 game of Sudoku. To expand the application of NLMs, we generate a random grid of cells from a dataset of solved games and assign up to 10 new empty cells. The goal of the game is then to find a target value ranging from 1 to 9 and fill in the remaining empty cells while maintaining a valid configuration. In our study, we showcase an NLM which is capable of obtaining 100% accuracy for solving a Sudoku with empty cells ranging from 3 to 10. The purpose of this study is to demonstrate that NLMs can also be used for solving complex problems and games like Sudoku. We also analyze the behaviour of NLMs with a backtracking algorithm by comparing the convergence time using a graph plot on the same problem. With this study we show that Neural Logic Machines can be trained on the tasks that traditional Deep Learning architectures fail using Reinforcement Learning. We also aim to propose the importance of symbolic learning in explaining the systematicity in the hybrid model of NLMs.

We consider several algorithms for exploring and filling an unknown, connected region, by simple, airborne agents. The agents are assumed to be identical, autonomous, anonymous and to have a finite amount of memory. The region is modeled as a connected sub-set of a regular grid composed of square cells. The algorithms described herein are suited for Micro Air Vehicles (MAV) since these air vehicles enable unobstructed views of the ground below and can move freely in space at various heights. The agents explore the region by applying various action-rules based on locally acquired information Some of them may settle in unoccupied cells as the exploration progresses. Settled agents become virtual pheromones for the exploration and coverage process, beacons that subsequently aid the remaining, and still exploring, mobile agents. We introduce a backward propagating information diffusion process as a way to implement a deterministic indicator of process termination and guide the mobile agents. For the proposed algorithms, complete covering of the graph in finite time is guaranteed when the size of the region is fixed. Bounds on the coverage times are also derived. Extensive simulation results exhibit good agreement with the theoretical predictions.

In recent years the research in the field of autonomous vehicles has gained considerable momentum, and the idea of relieving the burden of driving from humans starts to lose its futuristic science fiction aura. Some people believe that autonomous traffic is "our last hope" of relief from the frequent road-jams, we now witness in even mid-size urban areas. We envision roads of the future with fully autonomous vehicles, that not only track the lane, keep safe distance and assist the driver, but essentially liberate humans from driving related activities altogether.

Existing metrics used to evaluate table structure recognition algorithms have shortcomings with regard to capturing text and empty cells alignment. In this paper, we build on prior work and propose a new metric - TEDS based IOU similarity (TEDS (IOU)) for table structure recognition which uses bounding boxes instead of text while simultaneously being robust against the above disadvantages. We demonstrate the effectiveness of our metric against previous metrics through various examples.

In this article, we are going to explain about Apache Spark and python in more detail. Further you need a glance at this Pyspark Training course that will teach you the skills you'll need for becoming a professional Python Spark developer. Let's begin by understanding Apache Spark. Apache Spark is a framework based on open source which has been making headlines since its beginnings in 2009 at UC Berkeley's AMPLab; at its base it is an engine for distributed processing of big data that could expand at will. Simply put, as the volume of data increases, it becomes increasingly important to be able to handle enormous streams of data while still processing and doing other operations like machine learning, and Apache Spark can do just that. According to several experts, it will soon become the standard platform for streaming computation.