Metrics for Signal Temporal Logic Formulae

TL;DR

This paper introduces a formal metric space for Signal Temporal Logic (STL) formulae, enabling comparison and analysis of temporal properties in cyber-physical systems, with applications in design evaluation and logic inference.

Contribution

It proposes the first formal metrics for STL formulae based on Hausdorff and symmetric difference measures, along with algorithms for their computation.

Findings

Metrics enable comparison of STL formulae.

Applications include system design quality assessment.

Metrics facilitate formal evaluation of temporal logic inference.

Abstract

Signal Temporal Logic (STL) is a formal language for describing a broad range of real-valued, temporal properties in cyber-physical systems. While there has been extensive research on verification and control synthesis from STL requirements, there is no formal framework for comparing two STL formulae. In this paper, we show that under mild assumptions, STL formulae admit a metric space. We propose two metrics over this space based on i) the Pompeiu-Hausdorff distance and ii) the symmetric difference measure, and present algorithms to compute them. Alongside illustrative examples, we present applications of these metrics for two fundamental problems: a) design quality measures: to compare all the temporal behaviors of a designed system, such as a synthetic genetic circuit, with the "desired" specification, and b) loss functions: to quantify errors in Temporal Logic Inference (TLI) as a first step to establish formal performance guarantees of TLI algorithms.