Causal language modeling using the Transformer architecture has yielded remarkable capabilities in Large Language Models (LLMs) over the last few years. However, the extent to which fundamental search and reasoning capabilities emerged within LLMs remains a topic of ongoing debate. In this work, we study if causal language modeling can learn a complex task such as solving Sudoku puzzles. To solve a Sudoku, the model is first required to search over all empty cells of the puzzle to decide on a cell to fill and then apply an appropriate strategy to fill the decided cell. Sometimes, the application of a strategy only results in thinning down the possible values in a cell rather than concluding the exact value of the cell. In such cases, multiple strategies are applied one after the other to fill a single cell. We observe that Transformer models trained on this synthetic task can indeed learn to solve Sudokus (our model solves $94.21\%$ of the puzzles fully correctly) when trained on a logical sequence of steps taken by a solver. We find that training Transformers with the logical sequence of steps is necessary and without such training, they fail to learn Sudoku. We also extend our analysis to Zebra puzzles (known as Einstein puzzles) and show that the model solves $92.04 \%$ of the puzzles fully correctly. In addition, we study the internal representations of the trained Transformer and find that through linear probing, we can decode information about the set of possible values in any given cell from them, pointing to the presence of a strong reasoning engine implicit in the Transformer weights.
In this paper, we explore the capability of an agent to construct a logical sequence of action steps, thereby assembling a strategic procedural plan. This plan is crucial for navigating from an initial visual observation to a target visual outcome, as depicted in real-life instructional videos. Existing works have attained partial success by extensively leveraging various sources of information available in the datasets, such as heavy intermediate visual observations, procedural names, or natural language step-by-step instructions, for features or supervision signals. However, the task remains formidable due to the implicit causal constraints in the sequencing of steps and the variability inherent in multiple feasible plans. To tackle these intricacies that previous efforts have overlooked, we propose to enhance the capabilities of the agent by infusing it with procedural knowledge. This knowledge, sourced from training procedure plans and structured as a directed weighted graph, equips the agent to better navigate the complexities of step sequencing and its potential variations. We coin our approach KEPP, a novel Knowledge-Enhanced Procedure Planning system, which harnesses a probabilistic procedural knowledge graph extracted from training data, effectively acting as a comprehensive textbook for the training domain. Experimental evaluations across three widely-used datasets under settings of varying complexity reveal that KEPP attains superior, state-of-the-art results while requiring only minimal supervision.
Knowledge is a network of interconnected concepts. Yet, precisely how the topological structure of knowledge constrains its acquisition remains unknown, hampering the development of learning enhancement strategies. Here we study the topological structure of semantic networks reflecting mathematical concepts and their relations in college-level linear algebra texts. We hypothesize that these networks will exhibit structural order, reflecting the logical sequence of topics that ensures accessibility. We find that the networks exhibit strong core-periphery architecture, where a dense core of concepts presented early is complemented with a sparse periphery presented evenly throughout the exposition; the latter is composed of many small modules each reflecting more narrow domains. Using tools from applied topology, we find that the expositional evolution of the semantic networks produces and subsequently fills knowledge gaps, and that the density of these gaps tracks negatively with community ratings of each textbook. Broadly, our study lays the groundwork for future efforts developing optimal design principles for textbook exposition and teaching in a classroom setting.