Temporal Logic as Filtering

TL;DR

This paper establishes a novel connection between metric temporal logic and linear time-invariant filtering, enabling logical interpretation of signal processing techniques and providing both qualitative and quantitative semantics for MTL.

Contribution

It introduces a new perspective by interpreting metric temporal logic as filtering, unifying logical inference with signal feature detection, and provides a validated quantitative semantics.

Findings

Filtering semantics match classical MTL semantics

Quantitative semantics measure formula satisfaction frequency

Implementation in Matlab demonstrates practical applicability

Abstract

We show that metric temporal logic can be viewed as linear time-invariant filtering, by interpreting addition, multiplication, and their neutral elements, over the (max,min,0,1) idempotent dioid. Moreover, by interpreting these operators over the field of reals (+,*,0,1), one can associate various quantitative semantics to a metric-temporal-logic formula, depending on the filter's kernel used: square, rounded-square, Gaussian, low-pass, band-pass, or high-pass. This remarkable connection between filtering and metric temporal logic allows us to freely navigate between the two, and to regard signal-feature detection as logical inference. To the best of our knowledge, this connection has not been established before. We prove that our qualitative, filtering semantics is identical to the classical MTL semantics. We also provide a quantitative semantics for MTL, which measures the normalized, maximum number of times a formula is satisfied within its associated kernel, by a given signal. We show that this semantics is sound, in the sense that, if its measure is 0, then the formula is not satisfied, and it is satisfied otherwise. We have implemented both of our semantics in Matlab, and illustrate their properties on various formulas and signals, by plotting their computed measures.