Thomas Breuel

Summary

This course bridges large language models and symbolic reasoning by examining transformer architectures, reasoning techniques, interpretability studies, benchmarks, and results in neursymbolic AI; grounding foundations in logic, mathematics, and symbolic AI methods; and demonstrating hybrid approaches that integrate knowledge graphs, rule-based inference, and neural architectures to enhance explainability, factual accuracy, and reasoning performance.

Syllabus

* Course Overview and Motivation

• Historical context: rise of LLMs via scaling and heuristics
• Limitations: brittleness and lack of explicit reasoning mechanisms
• Course goal: integrating logic and symbolic methods to enhance LLM performance, neurosymbolic AI

* LLM Architectures

• Transformer fundamentals: self-attention, positional encoding, feedforward networks
• Model variants: decoder-only, encoder-decoder, and encoder-only architectures
• Scaling laws: tradeoffs between model size, data volume, and computational resources
• Training paradigms: autoregressive vs. masked language modeling

* Current Approaches to Reasoning in LLMs

• Chain-of-thought prompting for informal stepwise reasoning
• Programmatic prompting: embedding code interpreters for proofs and calculations
• Mathematical prompting techniques: solving algebraic and calculus problems
• Logical prompting: designing prompts for deductive and inductive inference

* Empirical Studies of LLM Reasoning

• Probing hidden states and attention patterns for reasoning traces
• Case studies: arithmetic problem-solving and commonsense reasoning examples
• Interpretability methods: attention visualization and concept attribution
• Analysis of failure modes: hallucinations, inconsistency, and reasoning breakdowns

* Benchmarks for LLM Reasoning

• GSM8K: grade-school math problem benchmark
• Big-Bench Hard (BBH): diverse reasoning and logic tasks
• Theorem-proving benchmarks: ProofWiki and HOL Light

* Foundations of Logic and Reasoning in Philosophy and Mathematics

• Propositional logic and predicate logic essentials
• Formal proof systems: natural deduction and sequent calculus
• Philosophical logics: modal, deontic, and epistemic frameworks

* Symbolic AI Methods and Principles

• Rule-based expert systems: knowledge representation and inference rules
• Goals-Scripts-Plans understanding: modeling agent goals, actions, and outcomes
• Conceptual Dependency Theory: representing meaning via primitive actions and entities
• Semantic representations: frames, scripts, plans, semantic networks, and case frames
• Forward chaining vs. backward chaining paradigms
• Logic programming with Prolog: unification, backtracking, and resolution
• Ontology languages: OWL, RDF Schema, and description logics

* Symbolic AI Knowledge Sources and Limitations

• Linked open data sources: Wikidata, DBpedia, Freebase as symbolic knowledge bases
• Commonsense knowledge resources: ConceptNet, ATOMIC, WordNet as commonsense datasets
• Core limitations: symbol grounding, ontology mismatches, and brittleness- Syntax, Semantics, Knowledge Bases, and Ontologies
• Formal grammars: context-free grammars and dependency parsing
• Semantic role labeling and frame semantics
• Knowledge graphs: RDF triples, SPARQL queries, and graph databases
• Ontology design: taxonomy creation and schema alignment
• Integration strategies with symbolic AI engines

* Representation of Knowledge and Reasoning in LLMs

• Embedding spaces as implicit knowledge repositories
• Implicit vs. explicit retrieval: memory networks and external stores
• Retrieval-Augmented Generation (RAG) and graph-based RAG techniques
• Probing methods for relational and factual knowledge extraction

* Integrating Symbolic Reasoning and LLMs

• Symbolic inference as external tools: API-based reasoning pipelines
• Synthetic data generation: using LLMs to create logic training sets
• Hybrid neural-symbolic architectures: neural theorem provers
• Graph-based RAG with symbolic constraint enforcement
• Symbolic verification to improve inference accuracy
• Future directions: neurosymbolic synergy and automated reasoning enhancement

References

Stuart Russell & Peter Norvig, Artificial Intelligence: A Modern Approach.
[Comprehensive coverage of search, logic, planning, knowledge representation, ontologies, expert systems, and probabilistic reasoning—widely used as the standard AI reference.]

Ronald Brachman & Hector Levesque, Knowledge Representation and Reasoning.
[In-depth treatment of semantic networks, frames, description logics, rule systems, and the foundations of symbolic inference and ontology design.]

Lewis Tunstall, Leandro von Werra & Thomas Wolf, Natural Language Processing with Transformers: Building Language Applications with Hugging Face.
[Detailed exploration of transformer architectures, pretraining/fine-tuning methods, prompting strategies, and production-grade examples for large-scale language models.]

Pre-requisites

Participants should have a working knowledge of transformer-based language models (e.g., attention mechanisms, pretraining/fine-tuning workflows), proficiency in Python programming, and familiarity with linear algebra and probability. No prior background in symbolic AI is required, as logic and symbolic methods will be introduced during the course.

Short bio

Thomas Breuel works on deep learning and computer vision at NVIDIA Research. Prior to NVIDIA, he was a full professor of computer science at the University of Kaiserslautern (Germany), where he also led a research group on document analysis, computer vision, and deep learning. Earlier, he worked as a researcher at Google, Xerox PARC, the IBM Almaden Research Center, IDIAP Switzerland. He is an alumnus of Massachusetts Institute of Technology and Harvard University. Contact Info: www.9×9.com.