IID Relaxation by Logical Expressivity: A Research Agenda for Fitting Logics to Neurosymbolic Requirements

TL;DR

The paper tackles the mismatch between Neurosymbolic knowledge and ML IID assumptions by proposing an IID-relaxation hierarchy of logics. It defines a progression from $L_G$ to $L_{G.Sol}$, $L_{Sol}$, $L_{FP}$, and the Guarded Fragment ($GF$), illustrating how each adds operators to relax IID constraints; in this view, given a $d$-dimensional input space and $k$ labels, ML maps $x \in \mathbb{R}^d$ to $y \in \{0,1\}^k$. The authors discuss implications for loss calculations, batch construction, and modal-truth-inspired losses under a model-theoretic invariance lens. They outline a research agenda to develop dependency-aware ML routines and semantic categorization of NeSy formalisms that align background knowledge with practical distribution constraints.

Abstract

Neurosymbolic background knowledge and the expressivity required of its logic can break Machine Learning assumptions about data Independence and Identical Distribution. In this position paper we propose to analyze IID relaxation in a hierarchy of logics that fit different use case requirements. We discuss the benefits of exploiting known data dependencies and distribution constraints for Neurosymbolic use cases and argue that the expressivity required for this knowledge has implications for the design of underlying ML routines. This opens a new research agenda with general questions about Neurosymbolic background knowledge and the expressivity required of its logic.

Figures (2)

• Figure 1: Dataset for a streetlight detection use case, a Knowledge Graph of inter-, and intra-sample relations. Left: samples are consecutive images with bounding box object annotations in red. Middle: top view diagram with overlapping samples taken at regular intervals. Right: the resulting knowledge graph, black arrows represent intra-sample spatial relationships, dotted arrows are inter-sample relationships of sample contiguity, dotted blue lines indicate shared membership of object equivalence classes.
• Figure 2: A hierarchy of FOL fragments and their corresponding IID relaxations. The expressivity of ground atom conjunctions is sufficient for the symbolic content of standard (multi-label) ML datasets but too weak to describe IID violations. Arrows indicate addition of logical operators that result in incremental IID relaxations. The relevance of the Solitary Fragment, itself a Fixed Parameter fragment, is its ability to relax ID while maintaining full Independence. Intersections of Guarded- and Fixed Parameter languages (indicated by overlapping regions) each capture a notion of IID relaxation.