Temporal logics with incommensurable distances are undecidable

Temporal Logic based on the two modalities “Since” and “Until” (TL) is the most popular logic for the specification of reactive systems. It is often called the linear time temporal logic. However, metric properties of real time cannot be expressed in this logic. The simplest modalities with metric properties are “X will happen within δ units of time”. The extension of TL by all these modalities with rational δ is decidable. We show that the extension of the linear time temporal logic by two modalities “X will happen within one unit of time”, “X will happen within τ unit of time” is undecidable, whenever τ is irrational.