  Witnessing subsystems have proven to be a useful concept in the analysis of
probabilistic systems, for example as diagnostic information on why a given
property holds or as input to refinement algorithms. This paper introduces
witnessing subsystems for reachability problems in probabilistic timed automata
(PTA). Using a new operation on difference bounds matrices, it is shown how
Farkas certificates of finite-state bisimulation quotients of a PTA can be
translated into witnessing subsystems. We present algorithms for the
computation of minimal witnessing subsystems under three notions of minimality,
which capture the timed behavior from different perspectives, and discuss their
complexity.
