[Paper Review] Conformal Prediction for STL Runtime Verification

Lars Lindemann, Qin Xin+2 more|arXiv (Cornell University)|Nov 3, 2022

Software Reliability and Analysis Research4 citations

TL;DR

This paper proposes two conformal prediction-based algorithms for predictive runtime verification of stochastic cyber-physical systems using signal temporal logic (STL). By combining trajectory predictors (e.g., RNNs, LSTMs) with conformal prediction, the methods generate valid prediction regions for STL satisfaction measures with user-defined confidence, offering formal guarantees without distributional assumptions or retraining for specification changes.

ABSTRACT

We are interested in predicting failures of cyber-physical systems during their operation. Particularly, we consider stochastic systems and signal temporal logic specifications, and we want to calculate the probability that the current system trajectory violates the specification. The paper presents two predictive runtime verification algorithms that predict future system states from the current observed system trajectory. As these predictions may not be accurate, we construct prediction regions that quantify prediction uncertainty by using conformal prediction, a statistical tool for uncertainty quantification. Our first algorithm directly constructs a prediction region for the satisfaction measure of the specification so that we can predict specification violations with a desired confidence. The second algorithm constructs prediction regions for future system states first, and uses these to obtain a prediction region for the satisfaction measure. To the best of our knowledge, these are the first formal guarantees for a predictive runtime verification algorithm that applies to widely used trajectory predictors such as RNNs and LSTMs, while being computationally simple and making no assumptions on the underlying distribution. We present numerical experiments of an F-16 aircraft and a self-driving car.

Motivation & Objective

• Address the lack of formal verification guarantees in predictive runtime verification for stochastic systems under real-time constraints.
• Enable early detection of specification violations in cyber-physical systems with high confidence during runtime.
• Provide formal probabilistic guarantees for prediction-based verification without requiring assumptions on the underlying system distribution or predictor model.
• Support the use of complex predictors like RNNs and LSTMs in safety-critical applications without retraining when specifications change.
• Ensure computational simplicity and robustness to noise in system dynamics, such as added control signal perturbations.

Proposed method

• Use trajectory predictors (e.g., RNNs, LSTMs) to forecast future system states from observed system prefixes.
• Apply conformal prediction to construct prediction regions for future system states, ensuring finite-sample validity under i.i.d. assumptions.
• For the direct method, compute the satisfaction measure of the STL specification on predicted trajectories and construct a prediction region for the true satisfaction measure using nonconformity scores.
• For the indirect method, first generate prediction regions for future states, then derive a lower bound on the true satisfaction measure using these state regions.
• Use calibration data to compute nonconformity scores and determine the prediction region threshold $ C $, ensuring that the true satisfaction measure lies within $ C $ with a user-defined confidence level.
• Leverage the robust semantics of STL to quantify how reliably a system satisfies or violates a specification, enabling precise failure prediction.

Experimental results

Research questions

• RQ1 Can predictive runtime verification with trajectory predictors provide formal probabilistic guarantees on specification violation detection?
• RQ2 How can conformal prediction be integrated with STL to ensure valid uncertainty quantification for future system behavior?
• RQ3 To what extent do the proposed methods maintain validity when using complex predictors like LSTMs without distributional assumptions?
• RQ4 How does the performance of the direct versus indirect conformal prediction approach compare in terms of prediction region size and verification accuracy?
• RQ5 Can the framework handle noisy or variable system dynamics, such as those introduced by stochastic control inputs?

Key findings

• For the IL controller and specification $\phi_1$, 99 out of 100 test trajectories satisfied $\rho^{\phi_1}(\hat{x},\tau_0) > C \Rightarrow (x,\tau_0) \models \phi_1$, confirming Theorem 1.
• For the CBF controller and $\phi_1$, all 100 test trajectories satisfied $\rho^{\phi_1}(\hat{x},\tau_0) > C \Rightarrow (x,\tau_0) \models \phi_1$, demonstrating strong verification reliability.
• The prediction region $C$ was smaller for the CBF controller, likely due to lower trajectory variability, enabling more accurate LSTM predictions.
• For $\phi_2$, 99/100 trajectories under the IL controller and 98/100 under the CBF controller satisfied the implication $\rho^{\phi_2}(\hat{x},\tau_0) > C \Rightarrow (x,\tau_0) \models \phi_2$, validating the method across different specifications.
• Equation (3) was satisfied by 95/100 trajectories under the IL controller and 95/100 under the CBF controller for $\phi_1$, and by 98/100 and 92/100 respectively for $\phi_2$, confirming the calibration of prediction regions.
• The framework remained robust to Gaussian noise in control signals, with poorly trained LSTMs producing larger prediction regions, indicating sensitivity to predictor quality.