Welcome to the course Topics in the Foundations of AI! It is given during the summer semester 2026 at LMU Munich as part of the Master in Logic and Philosophy of Science. (Past editions: summer 2023, 2024, 2026.)
Motivation
In recent years, artificial intelligence and, in particular, machine learning made great—but also disconcerting—progress. However, their foundations are, unlike other areas of computer science, less well understood. This situation is sometimes compared to being able to build steam engines without having a theory of thermodynamics. This course provides an introduction to the foundations of AI.
After an introduction to AI, we first review computability theory as the established theory of symbolic AI. This serves as a benchmark for what a theory should deliver when we turn to modern deep-learning-based AI. Here, a general theory is not yet discovered, but the topic of much research. We overview the established statistical learning theory and classical results (No-Free-Lunch Theorems, Universal Approximation, etc.), and then we discuss recent developments. While this theory describes the abilities and limits of AI systems as a whole, we may also ask what one specific AI system is doing. So, at the end, we turn to interpretable AI: explaining what an AI system is doing in human-understandable terms.
The course aims to convey not only the knowledge of these extant approaches, but also the skill to mathematically develop and philosophically assess them.
General information
The instructor of the seminar is me, Levin Hornischer. During the semester, we meet on Wednesdays from 12:15 to 14:00 in room 021 (Ludwigstr. 31). Below you find a schedule of when we cover which topic.
Reader
You find the latest edition of the reader in this file: found-ai.pdf. It will be updated as the course progresses.
Formalities
All the organizational details for the course are described in this file: formalities.pdf.
Schedule
The schedule below describes in which week we will cover which material. Here ‘chapter’ refers to the chapter of the reader.
| Week | Date | Chapter | Topic | Main reading | Additional material |
|---|---|---|---|---|---|
| 1 | 15 Apr | 1 | Course intro | - | - |
| 2 | 22 Apr | 2 | Introduction to AI | Russell and Norvig 2021, ch. 1 | Do coding exercise and browse ‘further material’ in ch. 2 to solidify the concepts discussed in class |
| 3 | 29 Apr | 3 | Computability theory | Flasiński (2016, ch. 2) , Immerman (2021) | |
| 4 | 6 May | 3 | Computability theory | Lecture notes, ch. 3 | |
| 5 | 13 May | 3 | Complexity theory | Aaronson 2011, especially ch. 1-4 and 7 | |
| 6 | 20 May | 4, 5 | Desiderata for theory of ML, statistical learning theory | Shalev-Shwartz and Ben-David (2014), ch. 2 and 3 | |
| 7 | 27 May | 5 | The No-Free-Lunch theorem | Shalev-Shwartz and Ben-David (2014, ch. 5 and sec. 6.1-4) | |
| 8 | 3 Jun | 6 | Statistical learning for deep learning | Berner et al. (2022) pages 1–31 | |
| 9 | 10 Jun | 6 | Statistical learning for deep learning | Belkin (2021) | |
| 10 | 17 Jun | 8 | Statistical mechanics and deep learning | Simon et al. 2026 | |
| 11 | 24 Jun | 11 | Mechanistic interpretability | Geiger et al. (2025), sec. 1-2 | Chalmers (2025) |
| 12 | 1 Jul | 11 | Mechanistic interpretability | Geiger et al. (2025), sec. 3-4 | |
| 13 | 8 Jul | 10 | Topological data analysis | Hensel et al. (2021) | |
| 14 | 15 Jul | - | Term paper discussion | - | - |
Essay topics
Below are some possible essay topics.
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On the idea of a theory of machine learning. It is common to hear that we need a theory of machine learning. But it is less clear how such a theory should look like. What should it be able to do? Maybe learning guarantees: that this neural network architecture with that dataset will learn the true function after so and so many steps? Or interpretability: that a human-understandable description of what the neural network does can be given so and so? Or verification: that this machine learning system will only produce safe (i.e., fair, unbiased, non-harmful, etc.) behavior? Looking at the philosophy of science literature, can you find—and argue for and against—desiderata for a theory of ML? Does the theory that we have for symbolic AI meet those desiderata? Also compare this to our discussion in class.
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Assessing the statistical mechanics approach to neural networks. Assume you have a method to analytically compute the probability distribution of which neural network you end up with conditioned on the learning algorithm and training data that you use—discussed by Roberts and Yaida (2022, p. 7). Which aspects of a theory of machine learning have you then achieved? Which are still missing? Besides the many analogies between statistical mechanics in physics and neural networks in machine learning, can you also think of disanalogies?
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The No-Free-Lunch theorem and the problem of induction. There is an extended discussion on the interpretation of the No-Free-Lunch theorem and how it connects to the philosophical problem of induction (see, e.g., Sterkenburg and Grünwald 2021). What can you say for the specific case where the learner is a neural network: how does it cope with the problem of induction? Also see the next question.
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In what sense do neural networks pick ‘simple’ functions? A prominent idea to explain why modern, over-parametrized neural networks generalize well is that among the many choices of parameters with which they would fit to the training data, they pick the ‘simplest’ ones, which hence are the ones that generalize, i.e., ‘fit the reality’ (see, e.g., Belkin 2021). But what sense of ‘simple’ is at play here, and can it be formalized in a noncircular way? (Also see open direction 3 of Simon et al. 2026.)
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What are the ‘features’ that neural networks learn? Mechanistic interpretability aims for a faithful macro-level description of the micro-level behavior of neural networks. To do so, it aims to identify ‘features’ and ‘circuits’ found in the micro-level behavior. Geiger et al. 2025 formalize this in terms of causal abstraction. Assess to what extend this answers the open direction 4 of Simon et al. 2026.
Just to be sure, these suggested topics are meant as first ideas. It is part of the task of writing an essay to turn an interesting aspect of the suggested topic into a precise research question and collect the relevant literature on it. Please take a look at the grading criteria mentioned in the file formalities.pdf to get a clear idea of what a good essay is expected to look like.