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 xvAbout the Authors xxiPart I Fundamentals 11 What Is Neurosymbolic AI? An Overview and Frontier Problems 3Alvaro Velasquez, Lucas White, Patrick Cooper, Antony Zhao, and Lekai Chen1.1 Introduction 31.2 Neurosymbolic Artificial Intelligence 41.2.1 Explicit to Implicit: From Symbolic Representations to Neural Networks 51.2.2 Implicit to Explicit: From Neural Networks to Symbolic Representations 61.3 Frontiers Problems 71.3.1 Neurosymbolic AI for Synthetic Biology 71.3.2 Neurosymbolic AI for Robust Autonomy 91.3.3 Neurosymbolic AI for Creative Scientific Discovery 111.4 Conclusion 11References 122 Reasoning in Neurosymbolic AI 15Son Tran, Edjard Mota, and Artur d’Avila Garcez2.1 What Is Reasoning in Neural Networks? 152.1.1 Reasoning in LLMs 162.1.2 AI from a Neurosymbolic Perspective 192.2 Background: Logic and RBMs 212.2.1 Illustrating Logical Reasoning with the Sudoku Puzzle 232.2.2 Sudoku with Strategies of Sampling 262.2.3 Restricted Boltzmann Machines 272.3 Symbolic Reasoning with Energy-based Neural Networks 282.3.1 Related Work 282.3.2 Knowledge Representation in RBMs 302.3.3 Reasoning in RBMs 332.3.4 Logical Boltzmann Machines 362.3.5 Experimental Results 392.3.6 Extensions of LBMs 432.4 LBMs for MaxSAT 492.4.1 LBM with Dual Annealing 522.4.2 Experimental Results of LBM for MaxSAT 522.5 Integrating Learning and Reasoning in LBMs 542.6 Challenges for Neurosymbolic AI 572.6.1 Nonmonotonic Logic 582.6.2 Planning 582.6.3 Learning from Its Mistakes 592.7 Conclusion 60References 623 Neurosymbolic Assurance Using Concept Probes in Foundation Models 69Ramneet Kaur, Anirban Roy, and Susmit Jha3.1 Introduction 693.2 Neural Features and Concept Probes 713.3 Foundation Models as Specification Lens 723.4 Symbolic Specification of ML Models Using Concept Probes 753.5 Implementation and Evaluation 783.6 Conclusion and Open Challenges 86References 874 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 Karsai4.1 Introduction 894.2 Problem Formulation and Challenges: Maneuver Control for Autonomous Vehicles 904.3 Software Architecture: Components and Interactions 914.4 Probabilistic World Model 934.4.1 Obstacle Map Calculation 944.4.2 Reward Map Calculation 964.5 Planner 974.5.1 Formalization 984.5.2 Online Planning Through Monte Carlo Search 984.5.3 Scalability Through Hierarchical Planning 1004.5.4 Evaluation and Analysis 1014.5.5 Neurosymbolic Extensions for Planning Under Partial Observability 1014.6 Trajectory Control with Evolving Behavior Trees (EBTs) 1034.6.1 Safe Autonomous UAV Navigation 1034.6.2 Safe EBTs for Navigation 1044.6.3 Evaluation 1064.7 Assurance for Neurosymbolic Systems 1084.7.1 Neurosymbolic Verification with BehaVerify 1094.7.2 Assurance on Grid Abstractions 1114.7.3 Timing Results and Conclusions 1124.7.4 Future Work 1134.8 Conclusions 114References 1155 Safe Neurosymbolic Learning and Control 119Somil Bansal and Jaime F. Fisac5.1 Problem Setup 1195.1.1 Dynamical Safety Problem 1205.1.2 Running Example: Air Collision Avoidance 1225.2 Hamilton–Jacobi (HJ) Reachability 1235.2.1 Methods to Solve HJI-VI and Compute Unsafe Set 1265.2.2 Running Example: Air Collision Avoidance 1275.3 A Neurosymbolic Perspective on Learning Safe Controllers 1295.3.1 Self-supervised Neurosymbolic Learning for Synthesizing Safe Controllers 1295.3.2 Neurosymbolic Reinforcement Learning for Synthesizing Safe Controllers 1355.3.3 Connections Between Reinforcement and Self-supervised Neurosymbolic Learning 1435.4 Safety Assurances for Learned Controllers 1445.4.1 Probabilistic Safety Assurances Through Conformal Prediction 1455.4.2 Robust Safety Assurances Through Forward Reachability 1485.5 Frontiers, Open Questions, and Promising Directions 150References 1516 Controllable Generation via Locally Constrained Resampling 159Kareem Ahmed, Kai-Wei Chang, and Guy Van den Broeck6.1 Introduction 1596.2 Background 1606.2.1 Notation and Preliminaries 1606.2.2 A Probability Distribution over Sentences 1616.2.3 The State of Conditional Sampling 1626.3 Locally Constrained Resampling: A Tale of Two Distributions 1636.3.1 Inducing a Local Tractable Distribution 1646.3.2 Tractable Operations via Compilation 1656.3.3 Intermezzo: Constraint Circuits and DFAs 1686.3.4 Correcting Sample Bias: Importance Sampling… and Resampling 1686.4 Related Work 1706.5 Experimental Evaluation 1716.6 Conclusion and Future Work 175Appendix A Controllable Generation via Locally Constrained Resampling 175A. 1 Language Detoxification 175A. 2 Sudoku 176A. 3 Warcraft Shortest Path 176A. 4 Broader Impact 177References 1777 Tractable and Expressive Generative Modeling with Probabilistic Flow Circuits 183Sahil Sidheekh and Sriraam Natarajan7.1 Introduction 1837.2 Tractable Probabilistic Modeling 1887.2.1 Inference Queries 1897.2.2 The Expressivity-tractability Trade-off 1907.3 Probabilistic Circuits 1917.3.1 Defining a Probabilistic Circuit 1927.3.2 Structural Properties 1937.3.3 Tractable Inference with PCs 1947.3.4 Parameter Learning for PCs 1957.3.5 Structure Learning for PCs 1957.4 Normalizing Flows: A Primer 1977.4.1 Sampling and Inference in Flows 1997.5 Integrating Normalizing Flows and PC 2007.5.1 The Challenge 2007.5.2 τ-Decomposability 2017.6 Probabilistic Flow Circuits 2057.7 Experiments and Results 2107.7.1 Modeling Complex 3D Manifolds 2117.7.2 Scaling to High-dimensional Data 2127.7.3 Sample Generation and Inference 2157.7.4 Ablation: Influence of PC Complexity 2157.8 Conclusion and Discussion 2167.8.1 Key Takeaways 2177.8.2 Limitations and Future Directions 217Acknowledgements 218References 2198 Toward Verifiable and Scalable In-context Fine-tuning in Neurosymbolic AI 223Neel P. Bhatt, Alvaro Velasquez, Zhangyang Wang, and Ufuk Topcu8.1 Introduction 2238.2 Neurosymbolic Fine-tuning Using Automated Feedback from Formal Verification 2258.2.1 Introduction 2258.2.2 Preliminaries 2268.2.3 Methodology 2278.2.4 Experimental Results 2328.2.5 Conclusion 2408.3 Uncertainty-aware Fine-tuning and Inference for Multimodal Foundation Models 2428.3.1 Introduction 2428.3.2 Conformal Prediction 2428.3.3 Perception Uncertainty 2448.3.4 Decision Uncertainty 2458.3.5 Estimating Decision Uncertainty Score 2488.3.6 Targeted Interventions 2488.3.7 Experiments 2518.3.8 Automated Refinement 2538.3.9 Conclusion 2578.4 Toward a Hybrid Architecture: Dynamic Interleaving of Neural and Symbolic Reasoning 2578.5 Conclusion and Future Directions 2608.5.1 Extending the Scope: Symbolic Tool Use for Mathematical Reasoning 261References 262Part II Advanced Topics 2679 Physics-informed Deep Learning 269Nithin Chalapathi, Yiheng Du, Sanjeev Raja, and Aditi S. Krishnapriyan9.1 Introduction 2699.1.1 Data Generation in Physics-informed Machine Learning 2719.1.2 Architectures 2749.1.3 Training Objectives 2829.1.4 Open Challenges 2889.1.5 Connections to Atomistic Modeling 289References 29110 Causal Representation Learning 307Burak Varıcı, Chandler Squires, and Pradeep Ravikumar10.1 Introduction 30710.2 Background 31010.2.1 Model Classes and Identifiability 31110.2.2 Causal Graphical Models and Interventions 31210.2.3 Causal Representation Models 31410.2.4 CRL Identifiability and Equivalence Classes 31510.3 Interventional CRL 31710.4 CRL with Linear SCMs 32010.4.1 Linear Mixing on Linear Latent SCMs 32110.4.2 General Mixing on Linear Latent SCMs 32310.5 CRL with General SCMs 32410.5.1 Linear Mixing on General Latent SCMs 32610.5.2 Multi-node Interventions 33010.5.3 General Mixing on General Latent SCMs 33210.6 Experiments 33510.6.1 Linear Mixing with Synthetic Data 33610.6.2 Experiments on Image Data 33710.7 Other Approaches 33910.8 Summary 340References 34111 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 Manohar11.1 Introduction 34711.2 Background 34811.2.1 Neurosymbolic Artificial Intelligence 34811.2.2 Emerging Hardware Computing Platforms 35011.3 Trends and Challenges 35111.3.1 Enhance Reasoning and Generalization 35111.3.2 Enable Compositionality 35211.3.3 Handle Uncertainty 35311.3.4 Improve System Efficiency 35411.3.5 Demonstrate Full-stack NeSy Systems 35411.4 Applications and Future Topics 35511.5 Conclusions 356References 35612 Programmatic Reinforcement Learning 365Swarat Chaudhuri12.1 Introduction 36512.2 Programmatic RL 36712.3 Imitation-projected Policy Gradients 36912.4 Related Work 37312.5 Conclusion 374References 376Part III Applications 38113 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. Morrison13.1 Motivation and Overview 38313.2 An Example of Symbolic IE 38613.2.1 Introduction 38613.2.2 Approach 38713.2.3 Intrinsic Evaluation: Machine Reading Performance 39413.2.4 Extrinsic Evaluation: Discovery of Biological Hypotheses 39613.2.5 Conclusion 40113.3 Problems of Symbolic IE Systems 40113.4 Generating Rules 40213.4.1 Introduction 40213.4.2 Approach 40313.4.3 Evaluation 40513.4.4 Conclusion 40913.5 Matching Rules 40913.5.1 Introduction 40913.5.2 Approach 41113.5.3 Evaluation 41513.5.4 Conclusion 42113.6 Take Away 421References 42214 Neurosymbolic AI for Legal AI-TRISM: Trustworthy, Reliable, Interpretable, Safe Models 429Deepa Tilwani, Yash Saxena, Ankur Padia, Srinivasan Parthasarathy, and Manas Gaur14.1 Introduction 42914.1.1 Neurosymbolic RAG 43114.1.2 Advantages of Using Neurosymbolic RAG 43214.2 Limitation of Using LLM as Legal Assistant 43314.3 Neurosymbolic AI for Legal Domain 43414.4 AI-TRISM with Neurosymbolic AI 43614.4.1 KG Construction 43614.4.2 Graph Construction Methodology 43714.5 Symbiosis of LLM and KG for Neurosymbolic RAG in Legal Domain 43914.6 Related Work 44214.6.1 KG Construction 44214.6.2 Legal Classification 44414.6.3 Legal Question Answering 44414.6.4 Legal Article and Case Retrieval 44514.6.5 Citation Recommendation and Interoperability 44514.6.6 Other Related Work 446Acknowledgement 447References 447Index 455

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 xvAbout the Authors xxiPart I Fundamentals 11 What Is Neurosymbolic AI? An Overview and Frontier Problems 3Alvaro Velasquez, Lucas White, Patrick Cooper, Antony Zhao, and Lekai Chen1.1 Introduction 31.2 Neurosymbolic Artificial Intelligence 41.2.1 Explicit to Implicit: From Symbolic Representations to Neural Networks 51.2.2 Implicit to Explicit: From Neural Networks to Symbolic Representations 61.3 Frontiers Problems 71.3.1 Neurosymbolic AI for Synthetic Biology 71.3.2 Neurosymbolic AI for Robust Autonomy 91.3.3 Neurosymbolic AI for Creative Scientific Discovery 111.4 Conclusion 11References 122 Reasoning in Neurosymbolic AI 15Son Tran, Edjard Mota, and Artur d’Avila Garcez2.1 What Is Reasoning in Neural Networks? 152.1.1 Reasoning in LLMs 162.1.2 AI from a Neurosymbolic Perspective 192.2 Background: Logic and RBMs 212.2.1 Illustrating Logical Reasoning with the Sudoku Puzzle 232.2.2 Sudoku with Strategies of Sampling 262.2.3 Restricted Boltzmann Machines 272.3 Symbolic Reasoning with Energy-based Neural Networks 282.3.1 Related Work 282.3.2 Knowledge Representation in RBMs 302.3.3 Reasoning in RBMs 332.3.4 Logical Boltzmann Machines 362.3.5 Experimental Results 392.3.6 Extensions of LBMs 432.4 LBMs for MaxSAT 492.4.1 LBM with Dual Annealing 522.4.2 Experimental Results of LBM for MaxSAT 522.5 Integrating Learning and Reasoning in LBMs 542.6 Challenges for Neurosymbolic AI 572.6.1 Nonmonotonic Logic 582.6.2 Planning 582.6.3 Learning from Its Mistakes 592.7 Conclusion 60References 623 Neurosymbolic Assurance Using Concept Probes in Foundation Models 69Ramneet Kaur, Anirban Roy, and Susmit Jha3.1 Introduction 693.2 Neural Features and Concept Probes 713.3 Foundation Models as Specification Lens 723.4 Symbolic Specification of ML Models Using Concept Probes 753.5 Implementation and Evaluation 783.6 Conclusion and Open Challenges 86References 874 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 Karsai4.1 Introduction 894.2 Problem Formulation and Challenges: Maneuver Control for Autonomous Vehicles 904.3 Software Architecture: Components and Interactions 914.4 Probabilistic World Model 934.4.1 Obstacle Map Calculation 944.4.2 Reward Map Calculation 964.5 Planner 974.5.1 Formalization 984.5.2 Online Planning Through Monte Carlo Search 984.5.3 Scalability Through Hierarchical Planning 1004.5.4 Evaluation and Analysis 1014.5.5 Neurosymbolic Extensions for Planning Under Partial Observability 1014.6 Trajectory Control with Evolving Behavior Trees (EBTs) 1034.6.1 Safe Autonomous UAV Navigation 1034.6.2 Safe EBTs for Navigation 1044.6.3 Evaluation 1064.7 Assurance for Neurosymbolic Systems 1084.7.1 Neurosymbolic Verification with BehaVerify 1094.7.2 Assurance on Grid Abstractions 1114.7.3 Timing Results and Conclusions 1124.7.4 Future Work 1134.8 Conclusions 114References 1155 Safe Neurosymbolic Learning and Control 119Somil Bansal and Jaime F. Fisac5.1 Problem Setup 1195.1.1 Dynamical Safety Problem 1205.1.2 Running Example: Air Collision Avoidance 1225.2 Hamilton–Jacobi (HJ) Reachability 1235.2.1 Methods to Solve HJI-VI and Compute Unsafe Set 1265.2.2 Running Example: Air Collision Avoidance 1275.3 A Neurosymbolic Perspective on Learning Safe Controllers 1295.3.1 Self-supervised Neurosymbolic Learning for Synthesizing Safe Controllers 1295.3.2 Neurosymbolic Reinforcement Learning for Synthesizing Safe Controllers 1355.3.3 Connections Between Reinforcement and Self-supervised Neurosymbolic Learning 1435.4 Safety Assurances for Learned Controllers 1445.4.1 Probabilistic Safety Assurances Through Conformal Prediction 1455.4.2 Robust Safety Assurances Through Forward Reachability 1485.5 Frontiers, Open Questions, and Promising Directions 150References 1516 Controllable Generation via Locally Constrained Resampling 159Kareem Ahmed, Kai-Wei Chang, and Guy Van den Broeck6.1 Introduction 1596.2 Background 1606.2.1 Notation and Preliminaries 1606.2.2 A Probability Distribution over Sentences 1616.2.3 The State of Conditional Sampling 1626.3 Locally Constrained Resampling: A Tale of Two Distributions 1636.3.1 Inducing a Local Tractable Distribution 1646.3.2 Tractable Operations via Compilation 1656.3.3 Intermezzo: Constraint Circuits and DFAs 1686.3.4 Correcting Sample Bias: Importance Sampling… and Resampling 1686.4 Related Work 1706.5 Experimental Evaluation 1716.6 Conclusion and Future Work 175Appendix A Controllable Generation via Locally Constrained Resampling 175A. 1 Language Detoxification 175A. 2 Sudoku 176A. 3 Warcraft Shortest Path 176A. 4 Broader Impact 177References 1777 Tractable and Expressive Generative Modeling with Probabilistic Flow Circuits 183Sahil Sidheekh and Sriraam Natarajan7.1 Introduction 1837.2 Tractable Probabilistic Modeling 1887.2.1 Inference Queries 1897.2.2 The Expressivity-tractability Trade-off 1907.3 Probabilistic Circuits 1917.3.1 Defining a Probabilistic Circuit 1927.3.2 Structural Properties 1937.3.3 Tractable Inference with PCs 1947.3.4 Parameter Learning for PCs 1957.3.5 Structure Learning for PCs 1957.4 Normalizing Flows: A Primer 1977.4.1 Sampling and Inference in Flows 1997.5 Integrating Normalizing Flows and PC 2007.5.1 The Challenge 2007.5.2 τ-Decomposability 2017.6 Probabilistic Flow Circuits 2057.7 Experiments and Results 2107.7.1 Modeling Complex 3D Manifolds 2117.7.2 Scaling to High-dimensional Data 2127.7.3 Sample Generation and Inference 2157.7.4 Ablation: Influence of PC Complexity 2157.8 Conclusion and Discussion 2167.8.1 Key Takeaways 2177.8.2 Limitations and Future Directions 217Acknowledgements 218References 2198 Toward Verifiable and Scalable In-context Fine-tuning in Neurosymbolic AI 223Neel P. Bhatt, Alvaro Velasquez, Zhangyang Wang, and Ufuk Topcu8.1 Introduction 2238.2 Neurosymbolic Fine-tuning Using Automated Feedback from Formal Verification 2258.2.1 Introduction 2258.2.2 Preliminaries 2268.2.3 Methodology 2278.2.4 Experimental Results 2328.2.5 Conclusion 2408.3 Uncertainty-aware Fine-tuning and Inference for Multimodal Foundation Models 2428.3.1 Introduction 2428.3.2 Conformal Prediction 2428.3.3 Perception Uncertainty 2448.3.4 Decision Uncertainty 2458.3.5 Estimating Decision Uncertainty Score 2488.3.6 Targeted Interventions 2488.3.7 Experiments 2518.3.8 Automated Refinement 2538.3.9 Conclusion 2578.4 Toward a Hybrid Architecture: Dynamic Interleaving of Neural and Symbolic Reasoning 2578.5 Conclusion and Future Directions 2608.5.1 Extending the Scope: Symbolic Tool Use for Mathematical Reasoning 261References 262Part II Advanced Topics 2679 Physics-informed Deep Learning 269Nithin Chalapathi, Yiheng Du, Sanjeev Raja, and Aditi S. Krishnapriyan9.1 Introduction 2699.1.1 Data Generation in Physics-informed Machine Learning 2719.1.2 Architectures 2749.1.3 Training Objectives 2829.1.4 Open Challenges 2889.1.5 Connections to Atomistic Modeling 289References 29110 Causal Representation Learning 307Burak Varıcı, Chandler Squires, and Pradeep Ravikumar10.1 Introduction 30710.2 Background 31010.2.1 Model Classes and Identifiability 31110.2.2 Causal Graphical Models and Interventions 31210.2.3 Causal Representation Models 31410.2.4 CRL Identifiability and Equivalence Classes 31510.3 Interventional CRL 31710.4 CRL with Linear SCMs 32010.4.1 Linear Mixing on Linear Latent SCMs 32110.4.2 General Mixing on Linear Latent SCMs 32310.5 CRL with General SCMs 32410.5.1 Linear Mixing on General Latent SCMs 32610.5.2 Multi-node Interventions 33010.5.3 General Mixing on General Latent SCMs 33210.6 Experiments 33510.6.1 Linear Mixing with Synthetic Data 33610.6.2 Experiments on Image Data 33710.7 Other Approaches 33910.8 Summary 340References 34111 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 Manohar11.1 Introduction 34711.2 Background 34811.2.1 Neurosymbolic Artificial Intelligence 34811.2.2 Emerging Hardware Computing Platforms 35011.3 Trends and Challenges 35111.3.1 Enhance Reasoning and Generalization 35111.3.2 Enable Compositionality 35211.3.3 Handle Uncertainty 35311.3.4 Improve System Efficiency 35411.3.5 Demonstrate Full-stack NeSy Systems 35411.4 Applications and Future Topics 35511.5 Conclusions 356References 35612 Programmatic Reinforcement Learning 365Swarat Chaudhuri12.1 Introduction 36512.2 Programmatic RL 36712.3 Imitation-projected Policy Gradients 36912.4 Related Work 37312.5 Conclusion 374References 376Part III Applications 38113 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. Morrison13.1 Motivation and Overview 38313.2 An Example of Symbolic IE 38613.2.1 Introduction 38613.2.2 Approach 38713.2.3 Intrinsic Evaluation: Machine Reading Performance 39413.2.4 Extrinsic Evaluation: Discovery of Biological Hypotheses 39613.2.5 Conclusion 40113.3 Problems of Symbolic IE Systems 40113.4 Generating Rules 40213.4.1 Introduction 40213.4.2 Approach 40313.4.3 Evaluation 40513.4.4 Conclusion 40913.5 Matching Rules 40913.5.1 Introduction 40913.5.2 Approach 41113.5.3 Evaluation 41513.5.4 Conclusion 42113.6 Take Away 421References 42214 Neurosymbolic AI for Legal AI-TRISM: Trustworthy, Reliable, Interpretable, Safe Models 429Deepa Tilwani, Yash Saxena, Ankur Padia, Srinivasan Parthasarathy, and Manas Gaur14.1 Introduction 42914.1.1 Neurosymbolic RAG 43114.1.2 Advantages of Using Neurosymbolic RAG 43214.2 Limitation of Using LLM as Legal Assistant 43314.3 Neurosymbolic AI for Legal Domain 43414.4 AI-TRISM with Neurosymbolic AI 43614.4.1 KG Construction 43614.4.2 Graph Construction Methodology 43714.5 Symbiosis of LLM and KG for Neurosymbolic RAG in Legal Domain 43914.6 Related Work 44214.6.1 KG Construction 44214.6.2 Legal Classification 44414.6.3 Legal Question Answering 44414.6.4 Legal Article and Case Retrieval 44514.6.5 Citation Recommendation and Interoperability 44514.6.6 Other Related Work 446Acknowledgement 447References 447Index 455

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 xvAbout the Authors xxiPart I Fundamentals 11 What Is Neurosymbolic AI? An Overview and Frontier Problems 3Alvaro Velasquez, Lucas White, Patrick Cooper, Antony Zhao, and Lekai Chen1.1 Introduction 31.2 Neurosymbolic Artificial Intelligence 41.2.1 Explicit to Implicit: From Symbolic Representations to Neural Networks 51.2.2 Implicit to Explicit: From Neural Networks to Symbolic Representations 61.3 Frontiers Problems 71.3.1 Neurosymbolic AI for Synthetic Biology 71.3.2 Neurosymbolic AI for Robust Autonomy 91.3.3 Neurosymbolic AI for Creative Scientific Discovery 111.4 Conclusion 11References 122 Reasoning in Neurosymbolic AI 15Son Tran, Edjard Mota, and Artur d’Avila Garcez2.1 What Is Reasoning in Neural Networks? 152.1.1 Reasoning in LLMs 162.1.2 AI from a Neurosymbolic Perspective 192.2 Background: Logic and RBMs 212.2.1 Illustrating Logical Reasoning with the Sudoku Puzzle 232.2.2 Sudoku with Strategies of Sampling 262.2.3 Restricted Boltzmann Machines 272.3 Symbolic Reasoning with Energy-based Neural Networks 282.3.1 Related Work 282.3.2 Knowledge Representation in RBMs 302.3.3 Reasoning in RBMs 332.3.4 Logical Boltzmann Machines 362.3.5 Experimental Results 392.3.6 Extensions of LBMs 432.4 LBMs for MaxSAT 492.4.1 LBM with Dual Annealing 522.4.2 Experimental Results of LBM for MaxSAT 522.5 Integrating Learning and Reasoning in LBMs 542.6 Challenges for Neurosymbolic AI 572.6.1 Nonmonotonic Logic 582.6.2 Planning 582.6.3 Learning from Its Mistakes 592.7 Conclusion 60References 623 Neurosymbolic Assurance Using Concept Probes in Foundation Models 69Ramneet Kaur, Anirban Roy, and Susmit Jha3.1 Introduction 693.2 Neural Features and Concept Probes 713.3 Foundation Models as Specification Lens 723.4 Symbolic Specification of ML Models Using Concept Probes 753.5 Implementation and Evaluation 783.6 Conclusion and Open Challenges 86References 874 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 Karsai4.1 Introduction 894.2 Problem Formulation and Challenges: Maneuver Control for Autonomous Vehicles 904.3 Software Architecture: Components and Interactions 914.4 Probabilistic World Model 934.4.1 Obstacle Map Calculation 944.4.2 Reward Map Calculation 964.5 Planner 974.5.1 Formalization 984.5.2 Online Planning Through Monte Carlo Search 984.5.3 Scalability Through Hierarchical Planning 1004.5.4 Evaluation and Analysis 1014.5.5 Neurosymbolic Extensions for Planning Under Partial Observability 1014.6 Trajectory Control with Evolving Behavior Trees (EBTs) 1034.6.1 Safe Autonomous UAV Navigation 1034.6.2 Safe EBTs for Navigation 1044.6.3 Evaluation 1064.7 Assurance for Neurosymbolic Systems 1084.7.1 Neurosymbolic Verification with BehaVerify 1094.7.2 Assurance on Grid Abstractions 1114.7.3 Timing Results and Conclusions 1124.7.4 Future Work 1134.8 Conclusions 114References 1155 Safe Neurosymbolic Learning and Control 119Somil Bansal and Jaime F. Fisac5.1 Problem Setup 1195.1.1 Dynamical Safety Problem 1205.1.2 Running Example: Air Collision Avoidance 1225.2 Hamilton–Jacobi (HJ) Reachability 1235.2.1 Methods to Solve HJI-VI and Compute Unsafe Set 1265.2.2 Running Example: Air Collision Avoidance 1275.3 A Neurosymbolic Perspective on Learning Safe Controllers 1295.3.1 Self-supervised Neurosymbolic Learning for Synthesizing Safe Controllers 1295.3.2 Neurosymbolic Reinforcement Learning for Synthesizing Safe Controllers 1355.3.3 Connections Between Reinforcement and Self-supervised Neurosymbolic Learning 1435.4 Safety Assurances for Learned Controllers 1445.4.1 Probabilistic Safety Assurances Through Conformal Prediction 1455.4.2 Robust Safety Assurances Through Forward Reachability 1485.5 Frontiers, Open Questions, and Promising Directions 150References 1516 Controllable Generation via Locally Constrained Resampling 159Kareem Ahmed, Kai-Wei Chang, and Guy Van den Broeck6.1 Introduction 1596.2 Background 1606.2.1 Notation and Preliminaries 1606.2.2 A Probability Distribution over Sentences 1616.2.3 The State of Conditional Sampling 1626.3 Locally Constrained Resampling: A Tale of Two Distributions 1636.3.1 Inducing a Local Tractable Distribution 1646.3.2 Tractable Operations via Compilation 1656.3.3 Intermezzo: Constraint Circuits and DFAs 1686.3.4 Correcting Sample Bias: Importance Sampling… and Resampling 1686.4 Related Work 1706.5 Experimental Evaluation 1716.6 Conclusion and Future Work 175Appendix A Controllable Generation via Locally Constrained Resampling 175A. 1 Language Detoxification 175A. 2 Sudoku 176A. 3 Warcraft Shortest Path 176A. 4 Broader Impact 177References 1777 Tractable and Expressive Generative Modeling with Probabilistic Flow Circuits 183Sahil Sidheekh and Sriraam Natarajan7.1 Introduction 1837.2 Tractable Probabilistic Modeling 1887.2.1 Inference Queries 1897.2.2 The Expressivity-tractability Trade-off 1907.3 Probabilistic Circuits 1917.3.1 Defining a Probabilistic Circuit 1927.3.2 Structural Properties 1937.3.3 Tractable Inference with PCs 1947.3.4 Parameter Learning for PCs 1957.3.5 Structure Learning for PCs 1957.4 Normalizing Flows: A Primer 1977.4.1 Sampling and Inference in Flows 1997.5 Integrating Normalizing Flows and PC 2007.5.1 The Challenge 2007.5.2 τ-Decomposability 2017.6 Probabilistic Flow Circuits 2057.7 Experiments and Results 2107.7.1 Modeling Complex 3D Manifolds 2117.7.2 Scaling to High-dimensional Data 2127.7.3 Sample Generation and Inference 2157.7.4 Ablation: Influence of PC Complexity 2157.8 Conclusion and Discussion 2167.8.1 Key Takeaways 2177.8.2 Limitations and Future Directions 217Acknowledgements 218References 2198 Toward Verifiable and Scalable In-context Fine-tuning in Neurosymbolic AI 223Neel P. Bhatt, Alvaro Velasquez, Zhangyang Wang, and Ufuk Topcu8.1 Introduction 2238.2 Neurosymbolic Fine-tuning Using Automated Feedback from Formal Verification 2258.2.1 Introduction 2258.2.2 Preliminaries 2268.2.3 Methodology 2278.2.4 Experimental Results 2328.2.5 Conclusion 2408.3 Uncertainty-aware Fine-tuning and Inference for Multimodal Foundation Models 2428.3.1 Introduction 2428.3.2 Conformal Prediction 2428.3.3 Perception Uncertainty 2448.3.4 Decision Uncertainty 2458.3.5 Estimating Decision Uncertainty Score 2488.3.6 Targeted Interventions 2488.3.7 Experiments 2518.3.8 Automated Refinement 2538.3.9 Conclusion 2578.4 Toward a Hybrid Architecture: Dynamic Interleaving of Neural and Symbolic Reasoning 2578.5 Conclusion and Future Directions 2608.5.1 Extending the Scope: Symbolic Tool Use for Mathematical Reasoning 261References 262Part II Advanced Topics 2679 Physics-informed Deep Learning 269Nithin Chalapathi, Yiheng Du, Sanjeev Raja, and Aditi S. Krishnapriyan9.1 Introduction 2699.1.1 Data Generation in Physics-informed Machine Learning 2719.1.2 Architectures 2749.1.3 Training Objectives 2829.1.4 Open Challenges 2889.1.5 Connections to Atomistic Modeling 289References 29110 Causal Representation Learning 307Burak Varıcı, Chandler Squires, and Pradeep Ravikumar10.1 Introduction 30710.2 Background 31010.2.1 Model Classes and Identifiability 31110.2.2 Causal Graphical Models and Interventions 31210.2.3 Causal Representation Models 31410.2.4 CRL Identifiability and Equivalence Classes 31510.3 Interventional CRL 31710.4 CRL with Linear SCMs 32010.4.1 Linear Mixing on Linear Latent SCMs 32110.4.2 General Mixing on Linear Latent SCMs 32310.5 CRL with General SCMs 32410.5.1 Linear Mixing on General Latent SCMs 32610.5.2 Multi-node Interventions 33010.5.3 General Mixing on General Latent SCMs 33210.6 Experiments 33510.6.1 Linear Mixing with Synthetic Data 33610.6.2 Experiments on Image Data 33710.7 Other Approaches 33910.8 Summary 340References 34111 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 Manohar11.1 Introduction 34711.2 Background 34811.2.1 Neurosymbolic Artificial Intelligence 34811.2.2 Emerging Hardware Computing Platforms 35011.3 Trends and Challenges 35111.3.1 Enhance Reasoning and Generalization 35111.3.2 Enable Compositionality 35211.3.3 Handle Uncertainty 35311.3.4 Improve System Efficiency 35411.3.5 Demonstrate Full-stack NeSy Systems 35411.4 Applications and Future Topics 35511.5 Conclusions 356References 35612 Programmatic Reinforcement Learning 365Swarat Chaudhuri12.1 Introduction 36512.2 Programmatic RL 36712.3 Imitation-projected Policy Gradients 36912.4 Related Work 37312.5 Conclusion 374References 376Part III Applications 38113 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. Morrison13.1 Motivation and Overview 38313.2 An Example of Symbolic IE 38613.2.1 Introduction 38613.2.2 Approach 38713.2.3 Intrinsic Evaluation: Machine Reading Performance 39413.2.4 Extrinsic Evaluation: Discovery of Biological Hypotheses 39613.2.5 Conclusion 40113.3 Problems of Symbolic IE Systems 40113.4 Generating Rules 40213.4.1 Introduction 40213.4.2 Approach 40313.4.3 Evaluation 40513.4.4 Conclusion 40913.5 Matching Rules 40913.5.1 Introduction 40913.5.2 Approach 41113.5.3 Evaluation 41513.5.4 Conclusion 42113.6 Take Away 421References 42214 Neurosymbolic AI for Legal AI-TRISM: Trustworthy, Reliable, Interpretable, Safe Models 429Deepa Tilwani, Yash Saxena, Ankur Padia, Srinivasan Parthasarathy, and Manas Gaur14.1 Introduction 42914.1.1 Neurosymbolic RAG 43114.1.2 Advantages of Using Neurosymbolic RAG 43214.2 Limitation of Using LLM as Legal Assistant 43314.3 Neurosymbolic AI for Legal Domain 43414.4 AI-TRISM with Neurosymbolic AI 43614.4.1 KG Construction 43614.4.2 Graph Construction Methodology 43714.5 Symbiosis of LLM and KG for Neurosymbolic RAG in Legal Domain 43914.6 Related Work 44214.6.1 KG Construction 44214.6.2 Legal Classification 44414.6.3 Legal Question Answering 44414.6.4 Legal Article and Case Retrieval 44514.6.5 Citation Recommendation and Interoperability 44514.6.6 Other Related Work 446Acknowledgement 447References 447Index 455