### Abstract

Timed automata are a widely used formalism for modeling real-time
systems, which is employed in a class of successful model checkers
such as UPPAAL [LPY97], HyTech [HHWt97] or Kronos [Yov97]. This work
formalizes the theory for the subclass of diagonal-free timed
automata, which is sufficient to model many interesting problems. We
first define the basic concepts and semantics of diagonal-free timed
automata. Based on this, we prove two types of decidability results
for the language emptiness problem. The first is the classic result
of Alur and Dill [AD90, AD94], which uses a finite partitioning of
the state space into so-called `regions`. Our second result focuses
on an approach based on `Difference Bound Matrices (DBMs)`, which is
practically used by model checkers. We prove the correctness of the
basic forward analysis operations on DBMs. One of these operations is
the Floyd-Warshall algorithm for the all-pairs shortest paths problem.
To obtain a finite search space, a widening operation has to be used
for this kind of analysis. We use Patricia Bouyer's [Bou04] approach
to prove that this widening operation is correct in the sense that
DBM-based forward analysis in combination with the widening operation
also decides language emptiness. The interesting property of this
proof is that the first decidability result is reused to obtain the
second one.

BSD License