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

Daniel Romero Schellhorn, Till Mossakowski

Keywords: neurosymbolic AI, monads, category theory, categorical logic, fuzzy logic, model theory

TL;DR: We unify the ULLER framework and semantics using Moggi's computational monads, thus obtaining a modular semantics.

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.

Track: Main Track

Paper Type: Long Paper

Resubmission: No

Software: https://github.com/cherryfunk/mULLER

Submission Number: 17