Distilling Formal Logic into Neural Spaces: A Kernel Alignment Approach for Signal Temporal Logic
Accepted at NeSy 2026 as an oral presentation 🎉
The Idea
Typical knowledge distillation compresses a big neural model into a smaller one. We do something different: our “expert” isn’t a neural network at all, it’s a mathematical kernel built on top of formal logic.
This kernel is provably correct: it captures the true meaning of logical formulas with mathematical guarantees. The problem? It’s expensive to compute and doesn’t scale.
So instead of distilling a big neural model into a smaller neural model, we distill the geometric structure (i.e. relative positions) of a symbolic kernel into a Transformer encoder. We’re not compressing parameters, we’re transferring mathematical meaning into neural space.
Using a teacher-student setup with a kernel-weighted geometric alignment objective, we train the encoder to mirror the semantic distances defined by the symbolic kernel. Errors are penalized proportionally to their semantic discrepancy, not just their magnitude.
Why It Matters
The result is a model that:
- runs in a single forward pass
- produces semantically faithful embeddings of logical formulas
- can reconstruct the original formula from its embedding
- all without sacrificing the logical guarantees of the original kernel
This opens up scalable, trustworthy neuro-symbolic reasoning for domains where correctness and efficiency aren’t optional:
- 🚗 Autonomous driving & robotics (real-time monitoring of safety specs)
- 🏥 Healthcare (checking patient signals against clinical guidelines)
- ⚙️ Cyber-physical systems (fast verification & control synthesis)
And beyond STL, this approach generalizes to any domain with a meaningful but expensive similarity function, think genomic sequences, molecular graphs, or structured data where semantics matter more than surface form.
Co-authored with Gabriele Sarti, Gaia Saveri, and Luca Bortolussi.
If you want to explore how to transfer a slow, expensive similarity function into Transformers, let’s connect!
