Neuro Symbolic Reasoning and Learning

Foreword
Acknowledgements
Contents
1 New Ideas in Neuro Symbolic Reasoning and Learning
	References
2 Brief Introduction to Propositional Logic and Predicate Calculus
	2.1 Introduction
	2.2 Propositional Logic
		2.2.1 Syntax and Semantics
		2.2.2 Consistency and Entailment
		2.2.3 Entailment and the Fixpoint Operator
	2.3 Predicate Calculus
	2.4 Conclusion
	References
3 Fuzzy and Annotated Logic for Neuro Symbolic Artificial Intelligence
	3.1 Introduction
	3.2 Generalized Annotated Programs
		3.2.1 Syntax of GAPs
		3.2.2 Semantics of GAPs
		3.2.3 Satisfaction
		3.2.4 Theoretical Results for Annotated Logic on a Lower Semi-Lattice
	3.3 Fuzzy Logic
		3.3.1 Properties of Fuzzy Operators Relevant to NSR
		3.3.2 Fuzzy Negation and Considerations for Unit Interval Annotations
		3.3.3 T-Norms
		3.3.4 T-Conorms
		3.3.5 Fuzzy Implication
		3.3.6 Aggregators
		3.3.7 Product Real Logic
		3.3.8 Conorms to Approximate the GAPs Fixpoint Operator
	3.4 Chapter Conclusion
	References
4 LTN: Logic Tensor Networks
	4.1 Introduction and Underlying Language
	4.2 From Real Logic to Logic Tensor Networks
		4.2.1 Representing Knowledge
	4.3 LTN Tasks
		4.3.1 Satisfiability and Learning
		4.3.2 Querying
		4.3.3 Reasoning
	4.4 Use Cases
		4.4.1 Basic Use Cases: Classification and Clustering
		4.4.2 Other Use Cases
	4.5 Discussion
	4.6 Chapter Conclusions
	References
5 Neuro Symbolic Reasoning with Ontological Networks
	5.1 Introduction and Underlying Language
	5.2 Recursive Reasoning Networks
		5.2.1 RRNs: Intuitive Overview
		5.2.2 RRNs: A Closer Look
	5.3 Use Cases
	5.4 Related Approaches
	5.5 Chapter Conclusions
	References
6 LNN: Logical Neural Networks
	6.1 Introduction
	6.2 Logic and Inference in LNNs
	6.3 Training LNNs
	6.4 Discussion
	6.5 Chapter Conclusion
	References
7 NeurASP
	7.1 Introduction
	7.2 ASP: Answer Set Programming
	7.3 NeurASP
		7.3.1 Semantics
		7.3.2 Inference in NeurASP
		7.3.3 Learning in NeurASP
	7.4 Discussion
	References
8 Neuro Symbolic Learning with Differentiable Inductive Logic Programming
	8.1 Introduction
	8.2 ILP Framework
		8.2.1 Problem Formulation
		8.2.2 Solving ILP Problems
	8.3 A Neural Framework for ILP
		8.3.1 Loss-Based ILP
		8.3.2 Architecture
		8.3.3 Complexity
		8.3.4 Training Considerations and Empirical Results
	8.4 Extensions to δILP
	8.5 Conclusion
	References
9 Understanding SATNet: Constraint Learning and Symbol Grounding
	9.1 Introduction
	9.2 SATNet to Learn Instances of MAXSAT
		9.2.1 Problem Relaxation
		9.2.2 SATNet Forward Pass
		9.2.3 Learning the Relaxed Constraint Matrix
		9.2.4 Experimental Findings
	9.3 Symbol Grounding
		9.3.1 Rebuttal on SATNet\'s Performance on Visual Sudoku
	9.4 Discussion
	9.5 Chapter Conclusion
	References
10 Neuro Symbolic AI for Sequential Decision Making
	10.1 Introduction
	10.2 Deep Symbolic Policy Learning
		10.2.1 Deep Symbolic Regression
		10.2.2 Deep Symbolic Policy Learning
	10.3 Verifying Neural-Based Models
		10.3.1 STLNet
	10.4 Discussion
	10.5 Chapter Conclusion
	References
11 Neuro Symbolic Applications
	11.1 Introduction
	11.2 Neuro Symbolic Reasoning in Visual Question Answering
	11.3 Neuro Symbolic Reasoning involving Natural Language Processing
		11.3.1 Concept Learning
		11.3.2 Reasoning Over Text
		11.3.3 Using ASP in Reasoning Over Text
		11.3.4 Solving Logic Grid Puzzles Described in Text
	11.4 Neuro-Symbolic Reinforcement Learning
	11.5 Chapter Conclusion
	References