Alvaro Velasquez, PhD, is a Program Manager at Defense Advanced Research Projects Agency. He is also a Visiting Professor in the Department of Computer Science at the University of Colorado Boulder.

Houbing Herbert Song, PhD, is a Tenured Associate Professor, Director of the NSF Center for Aviation Big Data Analytics (Planning), an Associate Director for Leadership of the DOT Transportation Cybersecurity Center for Advanced Research and Education, and Director of Security and Optimization for the Networked Globe Laboratory at the University of Maryland.

Pradeep Ravikumar, PhD, is an Assistant Professor in the Department of Computer Science at the University of Texas at Austin.

S. Shankar Sastry, PhD, is a Professor of Electrical Engineering and Computer Sciences, Bio-Engineering, and Mechanical Engineering at the University of California, Berkeley.

Sandeep Neema, PhD, is a Professor in the Department of Computer Science and the Director of the Institute for Software Integrated Systems at Vanderbilt University.

List of Contributors xv About the Authors xxi Part I Fundamentals 1 1 What Is Neurosymbolic AI? An Overview and Frontier Problems 3Alvaro Velasquez, Lucas White, Patrick Cooper, Antony Zhao, and Lekai Chen 1.1 Introduction 3 1.2 Neurosymbolic Artificial Intelligence 4 1.2.1 Explicit to Implicit: From Symbolic Representations to Neural Networks 5 1.2.2 Implicit to Explicit: From Neural Networks to Symbolic Representations 6 1.3 Frontiers Problems 7 1.3.1 Neurosymbolic AI for Synthetic Biology 7 1.3.2 Neurosymbolic AI for Robust Autonomy 9 1.3.3 Neurosymbolic AI for Creative Scientific Discovery 11 1.4 Conclusion 11 References 12 2 Reasoning in Neurosymbolic AI 15Son Tran, Edjard Mota, and Artur d'Avila Garcez 2.1 What Is Reasoning in Neural Networks? 15 2.1.1 Reasoning in LLMs 16 2.1.2 AI from a Neurosymbolic Perspective 19 2.2 Background: Logic and RBMs 21 2.2.1 Illustrating Logical Reasoning with the Sudoku Puzzle 23 2.2.2 Sudoku with Strategies of Sampling 26 2.2.3 Restricted Boltzmann Machines 27 2.3 Symbolic Reasoning with Energy-based Neural Networks 28 2.3.1 Related Work 28 2.3.2 Knowledge Representation in RBMs 30 2.3.3 Reasoning in RBMs 33 2.3.4 Logical Boltzmann Machines 36 2.3.5 Experimental Results 39 2.3.6 Extensions of LBMs 43 2.4 LBMs for MaxSAT 49 2.4.1 LBM with Dual Annealing 52 2.4.2 Experimental Results of LBM for MaxSAT 52 2.5 Integrating Learning and Reasoning in LBMs 54 2.6 Challenges for Neurosymbolic AI 57 2.6.1 Nonmonotonic Logic 58 2.6.2 Planning 58 2.6.3 Learning from Its Mistakes 59 2.7 Conclusion 60 References 62 3 Neurosymbolic Assurance Using Concept Probes in Foundation Models 69Ramneet Kaur, Anirban Roy, and Susmit Jha 3.1 Introduction 69 3.2 Neural Features and Concept Probes 71 3.3 Foundation Models as Specification Lens 72 3.4 Symbolic Specification of ML Models Using Concept Probes 75 3.5 Implementation and Evaluation 78 3.6 Conclusion and Open Challenges 86 References 87 4 Toward Assured Autonomy Using Neurosymbolic Components and Systems 89Abhishek Dubey, Taylor T. Johnson, Xenofon Koutsoukos, Baiting Luo, Diego Manzanas Lopez, Miklos Maroti, Ayan Mukhopadhyay, Nicholas Potteiger, Serena Serbinowska, Daniel Stojcsics, Yunuo Zhang, and Gabor Karsai 4.1 Introduction 89 4.2 Problem Formulation and Challenges: Maneuver Control for Autonomous Vehicles 90 4.3 Software Architecture: Components and Interactions 91 4.4 Probabilistic World Model 93 4.4.1 Obstacle Map Calculation 94 4.4.2 Reward Map Calculation 96 4.5 Planner 97 4.5.1 Formalization 98 4.5.2 Online Planning Through Monte Carlo Search 98 4.5.3 Scalability Through Hierarchical Planning 100 4.5.4 Evaluation and Analysis 101 4.5.5 Neurosymbolic Extensions for Planning Under Partial Observability 101 4.6 Trajectory Control with Evolving Behavior Trees (EBTs) 103 4.6.1 Safe Autonomous UAV Navigation 103 4.6.2 Safe EBTs for Navigation 104 4.6.3 Evaluation 106 4.7 Assurance for Neurosymbolic Systems 108 4.7.1 Neurosymbolic Verification with BehaVerify 109 4.7.2 Assurance on Grid Abstractions 111 4.7.3 Timing Results and Conclusions 112 4.7.4 Future Work 113 4.8 Conclusions 114 References 115 5 Safe Neurosymbolic Learning and Control 119Somil Bansal and Jaime F. Fisac 5.1 Problem Setup 119 5.1.1 Dynamical Safety Problem 120 5.1.2 Running Example: Air Collision Avoidance 122 5.2 Hamilton-Jacobi (HJ) Reachability 123 5.2.1 Methods to Solve HJI-VI and Compute Unsafe Set 126 5.2.2 Running Example: Air Collision Avoidance 127 5.3 A Neurosymbolic Perspective on Learning Safe Controllers 129 5.3.1 Self-supervised Neurosymbolic Learning for Synthesizing Safe Controllers 129 5.3.2 Neurosymbolic Reinforcement Learning for Synthesizing Safe Controllers 135 5.3.3 Connections Between Reinforcement and Self-supervised Neurosymbolic Learning 143 5.4 Safety Assurances for Learned Controllers 144 5.4.1 Probabilistic Safety Assurances Through Conformal Prediction 145 5.4.2 Robust Safety Assurances Through Forward Reachability 148 5.5 Frontiers, Open Questions, and Promising Directions 150 References 151 6 Controllable Generation via Locally Constrained Resampling 159Kareem Ahmed, Kai-Wei Chang, and Guy Van den Broeck 6.1 Introduction 159 6.2 Background 160 6.2.1 Notation and Preliminaries 160 6.2.2 A Probability Distribution over Sentences 161 6.2.3 The State of Conditional Sampling 162 6.3 Locally Constrained Resampling: A Tale of Two Distributions 163 6.3.1 Inducing a Local Tractable Distribution 164 6.3.2 Tractable Operations via Compilation 165 6.3.3 Intermezzo: Constraint Circuits and DFAs 168 6.3.4 Correcting Sample Bias: Importance Sampling... and Resampling 168 6.4 Related Work 170 6.5 Experimental Evaluation 171 6.6 Conclusion and Future Work 175 Appendix A Controllable Generation via Locally Constrained Resampling 175 A. 1 Language Detoxification 175 A. 2 Sudoku 176 A. 3 Warcraft Shortest Path 176 A. 4 Broader Impact 177 References 177 7 Tractable and Expressive Generative Modeling with Probabilistic Flow Circuits 183Sahil Sidheekh and Sriraam Natarajan 7.1 Introduction 183 7.2 Tractable Probabilistic Modeling 188 7.2.1 Inference Queries 189 7.2.2 The Expressivity-tractability Trade-off 190 7.3 Probabilistic Circuits 191 7.3.1 Defining a Probabilistic Circuit 192 7.3.2 Structural Properties 193 7.3.3 Tractable Inference with PCs 194 7.3.4 Parameter Learning for PCs 195 7.3.5 Structure Learning for PCs 195 7.4 Normalizing Flows: A Primer 197 7.4.1 Sampling and Inference in Flows 199 7.5 Integrating Normalizing Flows and PC 200 7.5.1 The Challenge 200 7.5.2 -Decomposability 201 7.6 Probabilistic Flow Circuits 205 7.7 Experiments and Results 210 7.7.1 Modeling Complex 3D Manifolds 211 7.7.2 Scaling to High-dimensional Data 212 7.7.3 Sample Generation and Inference 215 7.7.4 Ablation: Influence of PC Complexity 215 7.8 Conclusion and Discussion 216 7.8.1 Key Takeaways 217 7.8.2 Limitations and Future Directions 217 Acknowledgements 218 References 219 8 Toward Verifiable and Scalable In-context Fine-tuning in Neurosymbolic AI 223Neel P. Bhatt, Alvaro Velasquez, Zhangyang Wang, and Ufuk Topcu 8.1 Introduction 223 8.2 Neurosymbolic Fine-tuning Using Automated Feedback from Formal Verification 225 8.2.1 Introduction 225 8.2.2 Preliminaries 226 8.2.3 Methodology 227 8.2.4 Experimental Results 232 8.2.5 Conclusion 240 8.3 Uncertainty-aware Fine-tuning and Inference for Multimodal Foundation Models 242 8.3.1 Introduction 242 8.3.2 Conformal Prediction 242 8.3.3 Perception Uncertainty 244 8.3.4 Decision Uncertainty 245 8.3.5 Estimating Decision Uncertainty Score 248 8.3.6 Targeted Interventions 248 8.3.7 Experiments 251 8.3.8 Automated Refinement 253 8.3.9 Conclusion 257 8.4 Toward a Hybrid Architecture: Dynamic Interleaving of Neural and Symbolic Reasoning 257 8.5 Conclusion and Future Directions 260 8.5.1 Extending the Scope: Symbolic Tool Use for Mathematical Reasoning 261 References 262 Part II Advanced Topics 267 9 Physics-informed Deep Learning 269Nithin Chalapathi, Yiheng Du, Sanjeev Raja, and Aditi S. Krishnapriyan 9.1 Introduction 269 9.1.1 Data Generation in Physics-informed Machine Learning 271 9.1.2 Architectures 274 9.1.3 Training Objectives 282 9.1.4 Open Challenges 288 9.1.5 Connections to Atomistic Modeling 289 References 291 10 Causal Representation Learning 307Burak Varc, Chandler Squires, and Pradeep Ravikumar 10.1 Introduction 307 10.2 Background 310 10.2.1 Model Classes and Identifiability 311 10.2.2 Causal Graphical Models and Interventions 312 10.2.3 Causal Representation Models 314 10.2.4 CRL Identifiability and Equivalence Classes 315 10.3 Interventional CRL 317 10.4 CRL with Linear SCMs 320 10.4.1 Linear Mixing on Linear Latent SCMs 321 10.4.2 General Mixing on Linear Latent SCMs 323 10.5 CRL with General SCMs 324 10.5.1 Linear Mixing on General Latent SCMs 326 10.5.2 Multi-node Interventions 330 10.5.3 General Mixing on General Latent SCMs 332 10.6 Experiments 335 10.6.1 Linear Mixing with Synthetic Data 336 10.6.2 Experiments on Image Data 337 10.7 Other Approaches 339 10.8 Summary 340 References 341 11 Neurosymbolic Computing: Hardware-Software Co-design 347Xiaoxuan Yang, Zhangyang Wang, Miroslav Pajic, Hai "Helen" Li, Yiran Chen, X. Sharon Hu, Chris H. Kim, Shimeng Yu, and Rajit Manohar 11.1 Introduction 347 11.2 Background 348 11.2.1 Neurosymbolic Artificial Intelligence 348 11.2.2 Emerging Hardware Computing Platforms 350 11.3 Trends and Challenges 351 11.3.1 Enhance Reasoning and Generalization 351 11.3.2 Enable Compositionality 352 11.3.3 Handle Uncertainty 353 11.3.4 Improve System Efficiency 354 11.3.5 Demonstrate Full-stack NeSy Systems 354 11.4 Applications and Future Topics 355 11.5 Conclusions 356 References 356 12 Programmatic Reinforcement Learning 365Swarat Chaudhuri 12.1 Introduction 365 12.2 Programmatic RL 367 12.3 Imitation-projected Policy Gradients 369 12.4 Related Work 373 12.5 Conclusion 374 References 376 Part III Applications 381 13 From Symbolic to Neurosymbolic Information Extraction 383Mihai Surdeanu, Marco A. Valenzuela-Escárcega, Gus Hahn-Powell, Robert Vacareanu, Gwendolen Herongrove, Enrique Noriega-Atala, Özgün Babur, Emek Demir, and Clayton T. Morrison 13.1 Motivation and Overview 383 13.2 An Example of Symbolic IE 386 13.2.1 Introduction 386 13.2.2 Approach 387 13.2.3 Intrinsic Evaluation: Machine Reading Performance 394 13.2.4 Extrinsic Evaluation: Discovery of Biological Hypotheses 396 13.2.5 Conclusion 401 13.3 Problems of Symbolic IE Systems 401 13.4 Generating Rules 402 13.4.1 Introduction 402 13.4.2 Approach 403 13.4.3 Evaluation 405 13.4.4 Conclusion 409 13.5 Matching Rules 409 13.5.1 Introduction 409 13.5.2 Approach 411 13.5.3 Evaluation 415 13.5.4 Conclusion 421 13.6 Take Away 421 References 422 14 Neurosymbolic AI for Legal AI-TRISM: Trustworthy, Reliable, Interpretable, Safe Models 429Deepa Tilwani, Yash Saxena, Ankur Padia, Srinivasan Parthasarathy, and Manas Gaur 14.1 Introduction 429 14.1.1 Neurosymbolic RAG 431 14.1.2 Advantages of Using Neurosymbolic RAG 432 14.2...