### September 30, 2024

**Aldric Degorre**(IRIF, Université Paris-Cité)

Joint work with Eugene Asarin, Cătălin Dima and Bernardo Jacobo Inclán.

Timed languages contain sequences of discrete events (“letters”) separated by real-valued delays, they can be recognized by timed automata, and represent behaviors of various real-time systems.

The notion of bandwidth of a timed language characterizes the amount of information per time unit, encoded in words of the language observed with some precision $\varepsilon$.

In this talk, first we identify three classes of deterministic timed automata according to the asymptotics of the bandwidth of their languages with respect to this precision $\varepsilon$: automata are either

– meager, with an $O(1)$ bandwidth,

– normal, with a $\Theta\left(\log\frac{1}{\varepsilon}\right)$ bandwidth,

– or obese, with $\Theta\left(\frac{1}{\varepsilon}\right)$ bandwidth.

We characterize the 3 classes thanks to 2 structural criteria, showing that the above asympotical classification is an actual partiion (there is no intermediate asymptotic class). The classification problem of a deterministic timed automaton is \PSPACE-complete.

Both criteria are formulated using morphisms from paths of the timed automaton to some finite monoids extending Puri’s orbit graphs; the proofs are based on Simon’s factorization forest theorem.

Next, we propose a method, based on a finite-state simply-timed abstraction, to compute the actual value of the bandwidth of meager automata.

The states of this abstraction correspond to barycenters of the faces of the simplices in the region automaton.

Then the bandwidth is computed as $\log 1/|z_0|$ where $z_0$ is the smallest root (in modulus) of the characteristic polynomial of this finite-state abstraction.