mULLER: A Modular Monad-Based Semantics of the Neurosymbolic ULLER Framework

Daniel Romero Schellhorn, Till Mossakowski
Proceedings of The 19th International Conference on Neurosymbolic Learning and Reasoning, PMLR 284:494-518, 2025.

Abstract

ULLER (Unified Language for LEarning and Reasoning) provides a single first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics—classical, fuzzy, and probabilistic—each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in func- tional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalized quantifi- cation in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular imple- mentation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.

Cite this Paper

BibTeX
@InProceedings{pmlr-v284-schellhorn25a, title = {mULLER: A Modular Monad-Based Semantics of the Neurosymbolic ULLER Framework}, author = {Schellhorn, Daniel Romero and Mossakowski, Till}, booktitle = {Proceedings of The 19th International Conference on Neurosymbolic Learning and Reasoning}, pages = {494--518}, year = {2025}, editor = {H. Gilpin, Leilani and Giunchiglia, Eleonora and Hitzler, Pascal and van Krieken, Emile}, volume = {284}, series = {Proceedings of Machine Learning Research}, month = {08--10 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v284/main/assets/schellhorn25a/schellhorn25a.pdf}, url = {https://proceedings.mlr.press/v284/schellhorn25a.html}, abstract = {ULLER (Unified Language for LEarning and Reasoning) provides a single first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics—classical, fuzzy, and probabilistic—each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in func- tional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalized quantifi- cation in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular imple- mentation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.} }
Endnote
%0 Conference Paper %T mULLER: A Modular Monad-Based Semantics of the Neurosymbolic ULLER Framework %A Daniel Romero Schellhorn %A Till Mossakowski %B Proceedings of The 19th International Conference on Neurosymbolic Learning and Reasoning %C Proceedings of Machine Learning Research %D 2025 %E Leilani H. Gilpin %E Eleonora Giunchiglia %E Pascal Hitzler %E Emile van Krieken %F pmlr-v284-schellhorn25a %I PMLR %P 494--518 %U https://proceedings.mlr.press/v284/schellhorn25a.html %V 284 %X ULLER (Unified Language for LEarning and Reasoning) provides a single first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics—classical, fuzzy, and probabilistic—each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in func- tional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalized quantifi- cation in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular imple- mentation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.
APA
Schellhorn, D.R. & Mossakowski, T.. (2025). mULLER: A Modular Monad-Based Semantics of the Neurosymbolic ULLER Framework. Proceedings of The 19th International Conference on Neurosymbolic Learning and Reasoning, in Proceedings of Machine Learning Research 284:494-518 Available from https://proceedings.mlr.press/v284/schellhorn25a.html.

Related Material