This book constitutes the proceedings of the 15th International Symposium on NASA Formal Methods, NFM 2023, held in Houston, Texas, USA, during May 16-18, 2023. The 26 full and 3 short papers presented in this volume were carefully reviewed and selected from 75 submissions. The papers deal with advances in formal methods, formal methods techniques, and formal methods in practice.  

Neurosymbolic programming is an emerging area that bridges the areas of deep learning and program synthesis. As in classical machine learning, the goal is to learn functions from data. However, these functions are represented as programs that can use neural modules in addition to symbolic primitives and are induced using a combination of symbolic search and gradient-based optimization. Neurosymbolic programming can offer multiple advantages over end-to-end deep learning. Programs can sometimes naturally represent long-horizon, procedural tasks that are difficult to perform using deep networks. Neurosymbolic representations are also, commonly, easier to interpret and formally verify than neural networks. The restrictions of a programming language can serve as a form of regularization and lead to more generalizable and data-efficient learning. Compositional programming abstractions can also be a natural way of reusing learned modules across learning tasks.In this monograph, the authors illustrate these potential benefits with concrete examples from recent work on neurosymbolic programming. They also categorize the main ways in which symbolic and neural learning techniques come together in this area and conclude with a discussion of the open technical challenges in the field. The comprehensive review of neurosymbolic programming introduces the reader to the topic and provides an insightful treatise on an increasingly important topic at the intersection of programming languages and machine learning.

The two-volume set LNCS 9779 and LNCS 9780 constitutes the refereed proceedings of the 28th International Conference on Computer Aided Verification, CAV 2016, held in Toronto, ON, USA, in July 2016. The total of 46 full and 12 short papers presented in the proceedings was carefully reviewed and selected from 195 submissions. The papers were organized in topical sections named: probabilistic systems; synthesis; constraint solving; model checking; program analysis; timed and hybrid systems; verification in practice; concurrency; and automata and games.

The two-volume set LNCS 9779 and LNCS 9780 constitutes the refereed proceedings of the 28th International Conference on Computer Aided Verification, CAV 2016, held in Toronto, ON, USA, in July 2016. The total of 46 full and 12 short papers presented in the proceedings was carefully reviewed and selected from 195 submissions. The papers were organized in topical sections named: probabilistic systems; synthesis; constraint solving; model checking; program analysis; timed and hybrid systems; verification in practice; concurrency; and automata and games.