This thesis explores novel approaches to compositional reasoning in AI leveraging the mathematics of quantum theory as a general probabilistic theory. Starting from the quantum picturialism paradigm, offering a diagrammatic category-theoretic language, I show that quantum theory provides practical modelling and computational benefits for AI. A literature survey connect various applications, from quantum game theory and satisfiability to ML, NLP and cognition. How to formally represent and reason with concepts is a longstanding challenge in cognitive science and AI. My thesis studies the Quantum Model of Concepts (QMC), which provides conceptual space theory with quantum theoretical semantics. The diagrammatic language serves as a compositional framework for both, exposing common structures and facilitating insights between domains. I implement the model as a hybrid quantum-classical architecture on real quantum hardware to explore how QMCs can form practical intermediate, compositional representations for artificial agents combining symbolic and subsymbolic reasoning. Addressing the symbol grounding problem, I show that QMC representations can be learned from raw data in a (self-)supervised subsymbolic way, but that composite concepts can also be grounded in simpler ones to be interpretable and data-efficient. By transforming quantum concepts into probabilistic generative processes, the QMC can solve visual relational Blackbird puzzles involving abstraction and perceptual uncertainty, similar to Raven’s Progressive Matrices.
@phdthesis{gauderisQuantumTheoryKnowledge2023,title={Quantum {{Theory}} in {{Knowledge Representation}}: {{A Novel Approach}} to {{Reasoning}} with a {{Quantum Model}} of {{Concepts}}},shorttitle={Quantum {{Theory}} in {{Knowledge Representation}}},author={Gauderis, Ward and Wiggins, Geraint},year={2023},month=aug,address={Brussels, Belgium},school={Vrije Universiteit Brussel},}
We propose χ-net, an intrinsically interpretable architecture combining the compositional multilinear structure of tensor networks with the expressivity and efficiency of deep neural networks. χ-nets retain equal accuracy compared to their baseline counterparts. Our novel, efficient diagonalisation algorithm, ODT, reveals linear low-rank structure in a multilayer SVHN model. We leverage this toward formal weightbased interpretability and model compression.
@inproceedings{dooms_compositionality_2024,title={Compositionality {Unlocks} {Deep} {Interpretable} {Models}},url={https://openreview.net/forum?id=bXAt5iZ69l},urldate={2025-02-17},booktitle={Connecting {Low}-{Rank} {Representations} in {AI}: {At} the 39th {Annual} {AAAI} {Conference} on {Artificial} {Intelligence}},author={Dooms, Thomas and Gauderis, Ward and Wiggins, Geraint and Mogrovejo, Jose Antonio Oramas},month=nov,year={2024},}
This paper presents Bayesian Ultra-Q Learning, a variant of Q-Learning adapted for solving multi-agent games with independent learning agents. Bayesian Ultra-Q Learning is an extension of the Bayesian Hyper-Q Learning algorithm proposed by Tesauro that is more efficient for solving adaptive multi-agent games. While Hyper-Q agents merely update the Q-table corresponding to a single state, Ultra-Q leverages the information that similar states most likely result in similar rewards and therefore updates the Q-values of nearby states as well. We assess the performance of our Bayesian Ultra-Q Learning algorithm against three variants of Hyper-Q as defined by Tesauro, and against Infinitesimal Gradient Ascent (IGA) and Policy Hill Climbing (PHC) agents. We do so by evaluating the agents in two normal-form games, namely, the zero-sum game of rock-paper-scissors and a cooperative stochastic hill-climbing game. In rock-paper-scissors, games of Bayesian Ultra-Q agents against IGA agents end in draws where, averaged over time, all players play the Nash equilibrium, meaning no player can exploit another. Against PHC, neither Bayesian Ultra-Q nor Hyper-Q agents are able to win on average, which goes against the findings of Tesauro. In the cooperation game, Bayesian Ultra-Q converges in the direction of an optimal joint strategy and vastly outperforms all other algorithms including Hyper-Q, which are unsuccessful in finding a strong equilibrium due to relative overgeneralisation.
@inproceedings{gauderisEfficientBayesianUltraQ2023,title={Efficient {{Bayesian Ultra-Q Learning}} for {{Multi-Agent Games}}},url={https://alaworkshop2023.github.io/papers/ALA2023_paper_57.pdf},urldate={2024-11-23},booktitle={Proc. of the {{Adaptive}} and {{Learning Agents Workshop}} ({{ALA}} 2023 at {{AAMAS}})},author={Gauderis, Ward and Denoodt, Fabian and Silue, Bram and Vanvolsem, Pierre and Rosseau, Andries},year={2023},month=may,}
@inproceedings{bnaic,title={Quantum {{Theory}} in {{Knowledge Representation}}: {{A Novel Approach}} to {{Reasoning}} with a {{Quantum Model}} of {{Concepts}}},shorttitle={Quantum {{Theory}} in {{Knowledge Representation}}},booktitle={Pre-Proceedings of {{BNAIC}}/{{BeNeLearn}} 2024},year={2024},month=nov,address={Utrecht},author={Gauderis, Ward and Wiggins, Geraint},url={https://bnaic2024.sites.uu.nl/wp-content/uploads/sites/986/2024/11/Quantum-Theory-in-Knowledge-Representation-A-Novel-Approach-to-Reasoning-with-a-Quantum-Model-of-Concepts.pdf},}
