III: Small: Sampling Techniques in Computational Logic

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Constrained sampling and counting are two fundamental problems in data analysis. In constrained sampling the task is to sample randomly from among possible solutions to a Boolean formula. A related problem is that of constrained counting, determining the number of possible solutions to a Boolean formula. These problems have applications in machine learning, probabilistic reasoning, and planning, among other areas In particular, te project looks at the electronic-design-automation industry to determine what practical solutions to the problems require. Both problems can be viewed as aspects of one of the most fundamental problems in artificial intelligence, which is to understand the structure of the solution space of a given set of constraints.This project focuses on the development of new algorithmic techniques for constrained sampling and counting, based on a universal hashing - a classical algorithmic technique in theoretical computer science. Many of the ideas underlying the proposed approach go back to the 1980s, but they have never been reduced to practice. This project builds on recent progress in Boolean reasoning to develop methods to reduce these algorithmic ideas to practice. Methods for approximations with formal guarantees provide opportunities to scale what is fundamentally a computationally intractable problem. Pruning techniques can also reduce "waste" in hashed solutions, but introduce challenges in ensuring samples are independently distributed. This work has potential for breakthrough results in constrained sampling and counting, providing a new algorithmic toolbox in machine learning, probabilistic reasoning, and the like.