# Explaining Multi-stage Tasks by Learning Temporal Logic Formulas from Suboptimal Demonstrations

arXiv: https://arxiv.org/abs/2006.02411 · HTML: https://mlsanity.com/paper/2006.02411/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2020.

## Built on by (most cited first, 13 of 13)

- [Data-Efficient Learning of Natural Language to Linear Temporal Logic Translators for Robot Task Specification](https://mlsanity.com/paper/2303.08006/) (2023)
- [Learning Interpretable Temporal Properties from Positive Examples Only](https://mlsanity.com/paper/2209.02650/) (2022)
- [Scalable Anytime Algorithms for Learning Fragments of Linear Temporal Logic](https://mlsanity.com/paper/2110.06726/) (2021)
- [Gaussian Process Constraint Learning for Scalable Chance-Constrained Motion Planning from Demonstrations](https://mlsanity.com/paper/2112.04612/) (2021)
- [Learning Formal Specifications from Membership and Preference Queries](https://mlsanity.com/paper/2307.10434/) (2023)
- [Learning Reward Machines from Partially Observed Policies](https://mlsanity.com/paper/2502.03762/) (2025)
- [Temporal and Object Quantification Networks](https://mlsanity.com/paper/2106.05891/) (2021)
- [L^*LM: Learning Automata from Examples using Natural Language Oracles](https://mlsanity.com/paper/2402.07051/) (2024)
- [Demonstration Informed Specification Search](https://mlsanity.com/paper/2112.10807/) (2021)
- [Pointwise-in-Time Explanation for Linear Temporal Logic Rules](https://mlsanity.com/paper/2306.13956/) (2023)
- [A Safe Preference Learning Approach for Personalization with Applications to Autonomous Vehicles](https://mlsanity.com/paper/2311.02099/) (2023)
- [LTLf Learning Meets Boolean Set Cover](https://mlsanity.com/paper/2509.24616/) (2025)
- [WFA-IRL: Inverse Reinforcement Learning of Autonomous Behaviors Encoded as Weighted Finite Automata](https://mlsanity.com/paper/2103.05895/) (2021)

## Abstract (arXiv)

We present a method for learning multi-stage tasks from demonstrations by learning the logical structure and atomic propositions of a consistent linear temporal logic (LTL) formula. The learner is given successful but potentially suboptimal demonstrations, where the demonstrator is optimizing a cost function while satisfying the LTL formula, and the cost function is uncertain to the learner. Our algorithm uses the Karush-Kuhn-Tucker (KKT) optimality conditions of the demonstrations together with a counterexample-guided falsification strategy to learn the atomic proposition parameters and logical structure of the LTL formula, respectively. We provide theoretical guarantees on the conservativeness of the recovered atomic proposition sets, as well as completeness in the search for finding an LTL formula consistent with the demonstrations. We evaluate our method on high-dimensional nonlinear systems by learning LTL formulas explaining multi-stage tasks on 7-DOF arm and quadrotor systems and show that it outperforms competing methods for learning LTL formulas from positive examples.