# Backward through Time, Algebraically

arXiv: https://arxiv.org/abs/2608.17087 · HTML: https://mlsanity.com/paper/2608.17087/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Online normalizer calculation for softmax](https://mlsanity.com/paper/1805.02867/) (2018)
- [Backpropagation through Signal Temporal Logic Specifications: Infusing Logical Structure into Gradient-Based Methods](https://mlsanity.com/paper/2008.00097/) (2020)
- [On Robustness Metrics for Learning STL Tasks](https://mlsanity.com/paper/2003.06041/) (2020)
- [A Smooth Robustness Measure of Signal Temporal Logic for Symbolic Control](https://mlsanity.com/paper/2006.05239/) (2020)

## Abstract (arXiv)

Linear temporal logic is a modal extension of propositional logic that allows one to state how a system should behave over time. Its canonical domain is the booleans, but discretely-valued judgements are of little use in steering softly-valued systems (neural policies, adaptive controllers, sequence models, etc). In such cases, the goal formula's (dis)satisfaction becomes a training signal, and differentiability becomes a prime concern. Candidate differentiable semantics abound, but navigating them is tricky. Implementations, where available, are shallow embeddings, demanding an upfront commitment to a single semantic algebra and its (usually implicit) conduct. The paper casts the reader as a functional programmer asked to come to terms with this predicament, and refusing. Out of that refusal comes an evaluation engine that is algebra-generic and amenable to differentiation, together with an executable specification of the algebras it can accept. Various algebras are implemented and audited for their behavior, both forward and backward. Each algebra turns out to be a choice of which direction to disappoint, and how. Everything described (and more) is part of the PyTorch library telos, to be found at https://github.com/konstantinosKokos/telos.